-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathindex.html
More file actions
683 lines (669 loc) · 121 KB
/
index.html
File metadata and controls
683 lines (669 loc) · 121 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=no">
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black">
<meta name="mobile-web-app-capable" content="yes">
<title>
Decision 2 Notes - HackMD
</title>
<link rel="icon" type="image/png" href="https://tpb1908-hackmd.herokuapp.com/favicon.png">
<link rel="apple-touch-icon" href="https://tpb1908-hackmd.herokuapp.com/apple-touch-icon.png">
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.7/css/bootstrap.min.css" integrity="sha256-916EbMg70RQy9LHiGkXzG8hSg9EdNy97GazNG/aiY1w=" crossorigin="anonymous" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/4.7.0/css/font-awesome.min.css" integrity="sha256-eZrrJcwDc/3uDhsdt61sL2oOBY362qM3lon1gyExkL0=" crossorigin="anonymous" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/ionicons/2.0.1/css/ionicons.min.css" integrity="sha256-3iu9jgsy9TpTwXKb7bNQzqWekRX7pPK+2OLj3R922fo=" crossorigin="anonymous" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/octicons/3.5.0/octicons.min.css" integrity="sha256-QiWfLIsCT02Sdwkogf6YMiQlj4NE84MKkzEMkZnMGdg=" crossorigin="anonymous" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/prism/1.5.1/themes/prism.min.css" integrity="sha256-vtR0hSWRc3Tb26iuN2oZHt3KRUomwTufNIf5/4oeCyg=" crossorigin="anonymous" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.7.0/styles/github-gist.min.css" integrity="sha256-tAflq+ymku3Khs+I/WcAneIlafYgDiOQ9stIHH985Wo=" crossorigin="anonymous" />
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/emojify.js/1.1.0/css/basic/emojify.min.css" integrity="sha256-UOrvMOsSDSrW6szVLe8ZDZezBxh5IoIfgTwdNDgTjiU=" crossorigin="anonymous" />
<style>
@import url(https://fonts.googleapis.com/css?family=Source+Sans+Pro:400,400italic,600,600italic,300italic,300|Source+Serif+Pro|Source+Code+Pro:400,300,500&subset=latin,latin-ext);.markdown-body{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Helvetica,Arial,sans-serif;font-size:16px;line-height:1.5;word-wrap:break-word}.markdown-body:after,.markdown-body:before{display:table;content:""}.markdown-body:after{clear:both}.markdown-body>:first-child{margin-top:0!important}.markdown-body>:last-child{margin-bottom:0!important}.markdown-body a:not([href]){color:inherit;text-decoration:none}.markdown-body .absent{color:#c00}.markdown-body .anchor{float:left;padding-right:4px;margin-left:-20px;line-height:1}.markdown-body .anchor:focus{outline:none}.markdown-body blockquote,.markdown-body dl,.markdown-body ol,.markdown-body p,.markdown-body pre,.markdown-body table,.markdown-body ul{margin-top:0;margin-bottom:16px}.markdown-body hr{height:.25em;padding:0;margin:24px 0;background-color:#e7e7e7;border:0}.markdown-body blockquote{padding:0 1em;color:#777;border-left:.25em solid #ddd}.markdown-body blockquote>:first-child{margin-top:0}.markdown-body blockquote>:last-child{margin-bottom:0}.markdown-body .loweralpha{list-style-type:lower-alpha}.markdown-body h1,.markdown-body h2,.markdown-body h3,.markdown-body h4,.markdown-body h5,.markdown-body h6{margin-top:24px;margin-bottom:16px;font-weight:600;line-height:1.25}.markdown-body h1 .octicon-link,.markdown-body h2 .octicon-link,.markdown-body h3 .octicon-link,.markdown-body h4 .octicon-link,.markdown-body h5 .octicon-link,.markdown-body h6 .octicon-link{color:#000;vertical-align:middle;visibility:hidden}.markdown-body h1:hover .anchor,.markdown-body h2:hover .anchor,.markdown-body h3:hover .anchor,.markdown-body h4:hover .anchor,.markdown-body h5:hover .anchor,.markdown-body h6:hover .anchor{text-decoration:none}.markdown-body h1:hover .anchor .octicon-link,.markdown-body h2:hover .anchor .octicon-link,.markdown-body h3:hover .anchor .octicon-link,.markdown-body h4:hover .anchor .octicon-link,.markdown-body h5:hover .anchor .octicon-link,.markdown-body h6:hover .anchor .octicon-link{visibility:visible}.markdown-body h1 code,.markdown-body h1 tt,.markdown-body h2 code,.markdown-body h2 tt,.markdown-body h3 code,.markdown-body h3 tt,.markdown-body h4 code,.markdown-body h4 tt,.markdown-body h5 code,.markdown-body h5 tt,.markdown-body h6 code,.markdown-body h6 tt{font-size:inherit}.markdown-body h1{font-size:2em}.markdown-body h1,.markdown-body h2{padding-bottom:.3em;border-bottom:1px solid #eee}.markdown-body h2{font-size:1.5em}.markdown-body h3{font-size:1.25em}.markdown-body h4{font-size:1em}.markdown-body h5{font-size:.875em}.markdown-body h6{font-size:.85em;color:#777}.markdown-body ol,.markdown-body ul{padding-left:2em}.markdown-body ol.no-list,.markdown-body ul.no-list{padding:0;list-style-type:none}.markdown-body ol ol,.markdown-body ol ul,.markdown-body ul ol,.markdown-body ul ul{margin-top:0;margin-bottom:0}.markdown-body li>p{margin-top:16px}.markdown-body li+li{margin-top:.25em}.markdown-body dl{padding:0}.markdown-body dl dt{padding:0;margin-top:16px;font-size:1em;font-style:italic;font-weight:700}.markdown-body dl dd{padding:0 16px;margin-bottom:16px}.markdown-body table{display:block;width:100%;overflow:auto;word-break:normal;word-break:keep-all}.markdown-body table th{font-weight:700}.markdown-body table td,.markdown-body table th{padding:6px 13px;border:1px solid #ddd}.markdown-body table tr{background-color:#fff;border-top:1px solid #ccc}.markdown-body table tr:nth-child(2n){background-color:#f8f8f8}.markdown-body img{max-width:100%;box-sizing:content-box;background-color:#fff}.markdown-body img[align=right]{padding-left:20px}.markdown-body img[align=left]{padding-right:20px}.markdown-body .emoji{max-width:none;vertical-align:text-top;background-color:transparent}.markdown-body span.frame{display:block;overflow:hidden}.markdown-body span.frame>span{display:block;float:left;width:auto;padding:7px;margin:13px 0 0;overflow:hidden;border:1px solid #ddd}.markdown-body span.frame span img{display:block;float:left}.markdown-body span.frame span span{display:block;padding:5px 0 0;clear:both;color:#333}.markdown-body span.align-center{display:block;overflow:hidden;clear:both}.markdown-body span.align-center>span{display:block;margin:13px auto 0;overflow:hidden;text-align:center}.markdown-body span.align-center span img{margin:0 auto;text-align:center}.markdown-body span.align-right{display:block;overflow:hidden;clear:both}.markdown-body span.align-right>span{display:block;margin:13px 0 0;overflow:hidden;text-align:right}.markdown-body span.align-right span img{margin:0;text-align:right}.markdown-body span.float-left{display:block;float:left;margin-right:13px;overflow:hidden}.markdown-body span.float-left span{margin:13px 0 0}.markdown-body span.float-right{display:block;float:right;margin-left:13px;overflow:hidden}.markdown-body span.float-right>span{display:block;margin:13px auto 0;overflow:hidden;text-align:right}.markdown-body code,.markdown-body tt{padding:0;padding-top:.2em;padding-bottom:.2em;margin:0;font-size:85%;background-color:rgba(0,0,0,.04);border-radius:3px}.markdown-body code:after,.markdown-body code:before,.markdown-body tt:after,.markdown-body tt:before{letter-spacing:-.2em;content:"\A0"}.markdown-body code br,.markdown-body tt br{display:none}.markdown-body del code{text-decoration:inherit}.markdown-body pre{word-wrap:normal}.markdown-body pre>code{padding:0;margin:0;font-size:100%;word-break:normal;white-space:pre;background:transparent;border:0}.markdown-body .highlight{margin-bottom:16px}.markdown-body .highlight pre{margin-bottom:0;word-break:normal}.markdown-body .highlight pre,.markdown-body pre{padding:16px;overflow:auto;font-size:85%;line-height:1.45;background-color:#f7f7f7;border-radius:3px}.markdown-body pre code,.markdown-body pre tt{display:inline;max-width:auto;padding:0;margin:0;overflow:visible;line-height:inherit;word-wrap:normal;background-color:transparent;border:0}.markdown-body pre code:after,.markdown-body pre code:before,.markdown-body pre tt:after,.markdown-body pre tt:before{content:normal}.markdown-body .csv-data td,.markdown-body .csv-data th{padding:5px;overflow:hidden;font-size:12px;line-height:1;text-align:left;white-space:nowrap}.markdown-body .csv-data .blob-line-num{padding:10px 8px 9px;text-align:right;background:#fff;border:0}.markdown-body .csv-data tr{border-top:0}.markdown-body .csv-data th{font-weight:700;background:#f8f8f8;border-top:0}.markdown-body kbd{display:inline-block;padding:3px 5px;font-size:11px;line-height:10px;color:#555;vertical-align:middle;background-color:#fcfcfc;border:1px solid #ccc;border-bottom-color:#bbb;border-radius:3px;box-shadow:inset 0 -1px 0 #bbb}.news .alert .markdown-body blockquote{padding:0 0 0 40px;border:0 none}.activity-tab .news .alert .commits,.activity-tab .news .markdown-body blockquote{padding-left:0}.task-list-item{list-style-type:none}.task-list-item label{font-weight:400}.task-list-item.enabled label{cursor:pointer}.task-list-item+.task-list-item{margin-top:3px}.task-list-item-checkbox{float:left;margin:.31em 0 .2em -1.3em!important;vertical-align:middle;cursor:default!important}.markdown-body{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Helvetica Neue,Helvetica,Arial,sans-serif;padding-top:40px;padding-bottom:40px;max-width:758px;overflow:visible!important}.markdown-body pre{border:inherit!important}.markdown-body code{color:inherit!important}.markdown-body pre code .wrapper{display:-moz-inline-flex;display:-ms-inline-flex;display:-o-inline-flex;display:inline-flex}.markdown-body pre code .gutter{float:left;overflow:hidden;-webkit-user-select:none;user-select:none}.markdown-body pre code .gutter.linenumber{text-align:right;position:relative;display:inline-block;cursor:default;z-index:4;padding:0 8px 0 0;min-width:20px;box-sizing:content-box;color:#afafaf!important;border-right:3px solid #6ce26c!important}.markdown-body pre code .gutter.linenumber>span:before{content:attr(data-linenumber)}.markdown-body pre code .code{float:left;margin:0 0 0 16px}.markdown-body .gist .line-numbers{border-left:none;border-top:none;border-bottom:none}.markdown-body .gist .line-data{border:none}.markdown-body .gist table{border-spacing:0;border-collapse:inherit!important}.markdown-body code[data-gist-id]{background:none;padding:0}.markdown-body code[data-gist-id]:after,.markdown-body code[data-gist-id]:before{content:""}.markdown-body code[data-gist-id] .blob-num{border:unset}.markdown-body code[data-gist-id] table{overflow:unset;margin-bottom:unset}.markdown-body code[data-gist-id] table tr{background:unset}.markdown-body[dir=rtl] pre{direction:ltr}.markdown-body[dir=rtl] code{direction:ltr;unicode-bidi:embed}.markdown-body .alert>p{margin-bottom:0}.markdown-body pre.flow-chart,.markdown-body pre.graphviz,.markdown-body pre.mermaid,.markdown-body pre.sequence-diagram{text-align:center;background-color:inherit;border-radius:0;white-space:inherit}.markdown-body pre.flow-chart>code,.markdown-body pre.graphviz>code,.markdown-body pre.mermaid>code,.markdown-body pre.sequence-diagram>code{text-align:left}.markdown-body pre.flow-chart>svg,.markdown-body pre.graphviz>svg,.markdown-body pre.mermaid>svg,.markdown-body pre.sequence-diagram>svg{max-width:100%;height:100%}.markdown-body pre>code.wrap{white-space:pre-wrap;white-space:-moz-pre-wrap;white-space:-pre-wrap;white-space:-o-pre-wrap;word-wrap:break-word}.markdown-body .alert>p,.markdown-body .alert>ul{margin-bottom:0}.vimeo,.youtube{cursor:pointer;display:table;text-align:center;background-position:50%;background-repeat:no-repeat;background-size:contain;background-color:#000;overflow:hidden}.vimeo,.youtube{position:relative;width:100%}.youtube{padding-bottom:56.25%}.vimeo img{width:100%;object-fit:contain;z-index:0}.youtube img{object-fit:cover;z-index:0}.vimeo iframe,.youtube iframe,.youtube img{width:100%;height:100%;position:absolute;top:0;left:0}.vimeo iframe,.youtube iframe{vertical-align:middle;z-index:1}.vimeo .icon,.youtube .icon{position:absolute;height:auto;width:auto;top:50%;left:50%;transform:translate(-50%,-50%);color:#fff;opacity:.3;-webkit-transition:opacity .2s;transition:opacity .2s;z-index:0}.vimeo:hover .icon,.youtube:hover .icon{opacity:.6;-webkit-transition:opacity .2s;transition:opacity .2s}.slideshare .inner,.speakerdeck .inner{position:relative;width:100%}.slideshare .inner iframe,.speakerdeck .inner iframe{position:absolute;top:0;bottom:0;left:0;right:0;width:100%;height:100%}.MJX_Assistive_MathML{display:none}.ui-infobar{position:relative;z-index:2;max-width:758px;margin-top:25px;margin-bottom:-25px;color:#777}.ui-toc{position:fixed;bottom:20px;z-index:10000}.ui-toc-label{opacity:.3;background-color:#ccc;border:none}.ui-toc-label,.ui-toc .open .ui-toc-label{-webkit-transition:opacity .2s;transition:opacity .2s}.ui-toc .open .ui-toc-label{opacity:1;color:#fff}.ui-toc-label:focus{opacity:.3;background-color:#ccc;color:#000}.ui-toc-label:hover{opacity:1;background-color:#ccc;-webkit-transition:opacity .2s;transition:opacity .2s}.ui-toc-dropdown{margin-top:23px;margin-bottom:20px;padding-left:10px;padding-right:10px;max-width:45vw;width:25vw;max-height:70vh;overflow:auto;text-align:inherit}.ui-toc-dropdown>.toc{max-height:calc(70vh - 100px);overflow:auto}.ui-toc-dropdown[dir=rtl] .nav{padding-right:0;letter-spacing:.0029em}.ui-toc-dropdown a{overflow:hidden;text-overflow:ellipsis;white-space:pre}.ui-toc-dropdown .nav>li>a{display:block;padding:4px 20px;font-size:13px;font-weight:500;color:#767676}.ui-toc-dropdown .nav>li:first-child:last-child > ul,.ui-toc-dropdown .toc.expand ul{display:block}.ui-toc-dropdown .nav>li>a:focus,.ui-toc-dropdown .nav>li>a:hover{padding-left:19px;color:#000;text-decoration:none;background-color:transparent;border-left:1px solid #000}.ui-toc-dropdown[dir=rtl] .nav>li>a:focus,.ui-toc-dropdown[dir=rtl] .nav>li>a:hover{padding-right:19px;border-left:none;border-right:1px solid #000}.ui-toc-dropdown .nav>.active:focus>a,.ui-toc-dropdown .nav>.active:hover>a,.ui-toc-dropdown .nav>.active>a{padding-left:18px;font-weight:700;color:#000;background-color:transparent;border-left:2px solid #000}.ui-toc-dropdown[dir=rtl] .nav>.active:focus>a,.ui-toc-dropdown[dir=rtl] .nav>.active:hover>a,.ui-toc-dropdown[dir=rtl] .nav>.active>a{padding-right:18px;border-left:none;border-right:2px solid #000}.ui-toc-dropdown .nav .nav{display:none;padding-bottom:10px}.ui-toc-dropdown .nav>.active>ul{display:block}.ui-toc-dropdown .nav .nav>li>a{padding-top:1px;padding-bottom:1px;padding-left:30px;font-size:12px;font-weight:400}.ui-toc-dropdown[dir=rtl] .nav .nav>li>a{padding-right:30px}.ui-toc-dropdown .nav .nav>li>ul>li>a{padding-top:1px;padding-bottom:1px;padding-left:40px;font-size:12px;font-weight:400}.ui-toc-dropdown[dir=rtl] .nav .nav>li>ul>li>a{padding-right:40px}.ui-toc-dropdown .nav .nav>li>a:focus,.ui-toc-dropdown .nav .nav>li>a:hover{padding-left:29px}.ui-toc-dropdown[dir=rtl] .nav .nav>li>a:focus,.ui-toc-dropdown[dir=rtl] .nav .nav>li>a:hover{padding-right:29px}.ui-toc-dropdown .nav .nav>li>ul>li>a:focus,.ui-toc-dropdown .nav .nav>li>ul>li>a:hover{padding-left:39px}.ui-toc-dropdown[dir=rtl] .nav .nav>li>ul>li>a:focus,.ui-toc-dropdown[dir=rtl] .nav .nav>li>ul>li>a:hover{padding-right:39px}.ui-toc-dropdown .nav .nav>.active:focus>a,.ui-toc-dropdown .nav .nav>.active:hover>a,.ui-toc-dropdown .nav .nav>.active>a{padding-left:28px;font-weight:500}.ui-toc-dropdown[dir=rtl] .nav .nav>.active:focus>a,.ui-toc-dropdown[dir=rtl] .nav .nav>.active:hover>a,.ui-toc-dropdown[dir=rtl] .nav .nav>.active>a{padding-right:28px}.ui-toc-dropdown .nav .nav>.active>.nav>.active:focus>a,.ui-toc-dropdown .nav .nav>.active>.nav>.active:hover>a,.ui-toc-dropdown .nav .nav>.active>.nav>.active>a{padding-left:38px;font-weight:500}.ui-toc-dropdown[dir=rtl] .nav .nav>.active>.nav>.active:focus>a,.ui-toc-dropdown[dir=rtl] .nav .nav>.active>.nav>.active:hover>a,.ui-toc-dropdown[dir=rtl] .nav .nav>.active>.nav>.active>a{padding-right:38px}.markdown-body[lang^=ja]{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Helvetica Neue,Helvetica,Arial,Hiragino Kaku Gothic Pro,\\30D2\30E9\30AE\30CE\89D2\30B4 Pro W3,Osaka,Meiryo,\\30E1\30A4\30EA\30AA,MS Gothic,"\FF2D\FF33 \30B4\30B7\30C3\30AF",sans-serif}.ui-toc-dropdown[lang^=ja]{font-family:Source Sans Pro,Helvetica,Arial,Meiryo UI,MS PGothic,"\FF2D\FF33 \FF30\30B4\30B7\30C3\30AF",sans-serif}.markdown-body[lang=zh-tw]{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Helvetica Neue,Helvetica,Arial,PingFang TC,Microsoft JhengHei,\\5FAE\8EDF\6B63\9ED1,sans-serif}.ui-toc-dropdown[lang=zh-tw]{font-family:Source Sans Pro,Helvetica,Arial,Microsoft JhengHei UI,\\5FAE\8EDF\6B63\9ED1UI,sans-serif}.markdown-body[lang=zh-cn]{font-family:-apple-system,BlinkMacSystemFont,Segoe UI,Roboto,Helvetica Neue,Helvetica,Arial,PingFang SC,Microsoft YaHei,\\5FAE\8F6F\96C5\9ED1,sans-serif}.ui-toc-dropdown[lang=zh-cn]{font-family:Source Sans Pro,Helvetica,Arial,Microsoft YaHei UI,\\5FAE\8F6F\96C5\9ED1UI,sans-serif}.ui-affix-toc{position:fixed;top:0;max-width:15vw;max-height:70vh;overflow:auto}.back-to-top,.expand-toggle,.go-to-bottom{display:block;padding:4px 10px;margin-top:10px;margin-left:10px;font-size:12px;font-weight:500;color:#999}.back-to-top:focus,.back-to-top:hover,.expand-toggle:focus,.expand-toggle:hover,.go-to-bottom:focus,.go-to-bottom:hover{color:#563d7c;text-decoration:none}.back-to-top,.go-to-bottom{margin-top:0}.ui-user-icon{width:20px;height:20px;display:block;border-radius:3px;margin-top:2px;margin-bottom:2px;margin-right:5px;background-position:50%;background-repeat:no-repeat;background-size:contain}.ui-user-icon.small{width:18px;height:18px;display:inline-block;vertical-align:middle;margin:0 0 .2em}small span{line-height:22px}small .dropdown{display:inline-block}small .dropdown a:focus,small .dropdown a:hover{text-decoration:none}.unselectable{-moz-user-select:none;-webkit-user-select:none;-o-user-select:none;user-select:none}@media print{blockquote,div,img,pre,table{page-break-inside:avoid!important}a[href]:after{font-size:12px!important}}.markdown-body.slides{position:relative;z-index:1;color:#222}.markdown-body.slides:before{content:"";display:block;position:absolute;top:0;left:0;right:0;bottom:0;z-index:-1;background-color:currentColor;box-shadow:0 0 0 50vw}.markdown-body.slides section[data-markdown]{position:relative;margin-bottom:1.5em;background-color:#fff;text-align:center}.markdown-body.slides section[data-markdown] code{text-align:left}.markdown-body.slides section[data-markdown]:before{content:"";display:block;padding-bottom:56.23%}.markdown-body.slides section[data-markdown]>div:first-child{position:absolute;top:50%;left:1em;right:1em;transform:translateY(-50%);max-height:100%;overflow:hidden}.markdown-body.slides section[data-markdown]>ul{display:inline-block}.markdown-body.slides>section>section+section:after{content:"";position:absolute;top:-1.5em;right:1em;height:1.5em;border:3px solid #777}body{font-smoothing:subpixel-antialiased!important;-webkit-font-smoothing:subpixel-antialiased!important;-moz-osx-font-smoothing:auto!important;text-shadow:0 0 1em transparent,1px 1px 1.2px rgba(0,0,0,.004);-webkit-overflow-scrolling:touch;font-family:Source Sans Pro,Helvetica,Arial,sans-serif;letter-spacing:.025em}.focus,:focus{outline:none!important}::-moz-focus-inner{border:0!important}body.modal-open{overflow-y:auto;padding-right:0!important}
</style>
<!-- HTML5 shim and Respond.js for IE8 support of HTML5 elements and media queries -->
<!-- WARNING: Respond.js doesn't work if you view the page via file:// -->
<!--[if lt IE 9]>
<script src="https://cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv.min.js" integrity="sha256-3Jy/GbSLrg0o9y5Z5n1uw0qxZECH7C6OQpVBgNFYa0g=" crossorigin="anonymous"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/respond.js/1.4.2/respond.min.js" integrity="sha256-g6iAfvZp+nDQ2TdTR/VVKJf3bGro4ub5fvWSWVRi2NE=" crossorigin="anonymous"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/es5-shim/4.5.9/es5-shim.min.js" integrity="sha256-8E4Is26QH0bD52WoQpcB+R/tcWQtpzlCojrybUd7Mxo=" crossorigin="anonymous"></script>
<![endif]-->
</head>
<body>
<div id="doc" class="markdown-body container-fluid"><h1 id="decision-2-notes"><a class="anchor hidden-xs" href="#decision-2-notes" title="decision-2-notes"><span class="octicon octicon-link"></span></a>Decision 2 Notes</h1><h2 id="critical-path-analysis"><a class="anchor hidden-xs" href="#critical-path-analysis" title="critical-path-analysis"><span class="octicon octicon-link"></span></a>Critical path analysis</h2><h3 id="activity-networks"><a class="anchor hidden-xs" href="#activity-networks" title="activity-networks"><span class="octicon octicon-link"></span></a>Activity networks</h3><p>The aim of an activity network is to determine the minimum value, in time or resources, with which a project can be completed.</p><p>The network should be drawn such that preceding activities are all shown to the left of the given activity.<br>
The activity name is written along with a number indicating the duration or cost of the activity.</p><ol>
<li>Draw a circle to represent the start of all those activities that have no immediate predecessors</li>
<li>From the most recently drawn circles draw the start of arcs directed from left to right to represent the activities starting from those circles</li>
<li>To the right of any previous drawing, draw circles to represent the start of all those activities for which all immediate predecessors have already been drawn. Two activities which have precisely the same immediate predecessors can share a start circle</li>
<li>Join all loose arcs to relevant right hand circles, introducing dummy activities as necessary</li>
<li>Repeat steps 2, 3, and 4 until all activities have been considered</li>
<li>Join all loose arcs to a circle at the right hand side of the diagram, to represent the finish</li>
</ol><h4 id="earliest-and-latest-start-times"><a class="anchor hidden-xs" href="#earliest-and-latest-start-times" title="earliest-and-latest-start-times"><span class="octicon octicon-link"></span></a>Earliest and latest start times</h4><p>The earliest time after the start of a project that an activity can be started is called the earliest start time.</p><p>The earliest start times are calculated by performing a forward pass, starting from the left.</p><ol>
<li>Label the initial node with earliest start time 0</li>
<li>Choose any node, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2"><span class="mi" id="MathJax-Span-3" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-1">X</script></span>, such that all nodes to its left have already been given an early time</li>
<li>Consider all activities leading into <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-5"><span class="mi" id="MathJax-Span-6" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-2">X</script></span>, and the nodes from which they come. Label <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-7" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-8"><span class="mi" id="MathJax-Span-9" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-3">X</script></span> with an earliest start time equal to the maximum value of the earliest start time of the preceding node plus the duration of the preceding activity for all these activities</li>
<li>If all nodes have been labelled then stop. Otherwise, return to step 2</li>
</ol><p>The latest start time for a node is the latest that activities to the left can finish and not delay the completion of the project.<br>
Once the earliest start times have been determined, the latest start times can be found by performing a reverse pass from the right to the left.</p><ol>
<li>Label the final node with a latest start time equal to its earliest start time</li>
<li>Choose any node, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-10" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-11"><span class="mi" id="MathJax-Span-12" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-4">X</script></span>, such that all nodes to its right have already been given a latest start time</li>
<li>Consider all activities leading out of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-13" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-14"><span class="mi" id="MathJax-Span-15" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-5">X</script></span>. Label <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-16" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-17"><span class="mi" id="MathJax-Span-18" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-6">X</script></span> with a late time equal to the least value of the latest start time of the following node minus the duration of the following activity for all these activities</li>
<li>If all nodes have been labelled then stop. Otherwise, return to step 2</li>
</ol><h3 id="critical-paths"><a class="anchor hidden-xs" href="#critical-paths" title="critical-paths"><span class="octicon octicon-link"></span></a>Critical paths</h3><p><strong>Critical activity</strong><br>
Critical activities are such that they connect two nodes where both nodes have the same earliest start times and latest start times<br>
Their duration is the difference between the labels of their left and right nodes<br>
A critical activity has no scheduling flexibility</p><p><strong>Critical path</strong> A path of critical activities leading from the start node to the final node</p><p><strong>Float</strong> The amount of time by which an activity can be delayed or extended</p><p><strong>Independent float</strong> A float which does not affect other activities</p><h3 id="cascade-diagrams"><a class="anchor hidden-xs" href="#cascade-diagrams" title="cascade-diagrams"><span class="octicon octicon-link"></span></a>Cascade diagrams</h3><p>Once the earliest start times and latest end times have been calculated, a cascade chart provides a method of displaying information in a way which is easy to understand.</p><p>It is convention to place an activity at its earliest possible start time, and use dashed lines to display the possible range.</p><h4 id="resource-leveling"><a class="anchor hidden-xs" href="#resource-leveling" title="resource-leveling"><span class="octicon octicon-link"></span></a>Resource leveling</h4><p>The smoothing out of the usage of resources is called resource leveling.</p><p>Draw the cascade chart, starting with any critical paths along the bottom.<br>
Then insert other activities to produce a chart of minimum height.</p><h2 id="allocation-with-the-hungarian-algorithm"><a class="anchor hidden-xs" href="#allocation-with-the-hungarian-algorithm" title="allocation-with-the-hungarian-algorithm"><span class="octicon octicon-link"></span></a>Allocation with the Hungarian algorithm</h2><p><strong>Assignment problem</strong> Finding a maximum weight matching or minimum weight perfect matching in a weighted bipartite graph</p><p>If the matrix is not a square, add a dummy row or column filled with values which are equal to the maximum value in the matrix.</p><p>If the objective is to establish a maximal matching, subtract each of the values from the maximum value in the matrix.</p><ol>
<li>Reduce the array of costs by both row and column subtraction. This is subtracting the smallest value from every value in each row and column</li>
<li>Cover the zero elements with the minimum number of lines. If this minimum number is the same as the size of the array, go to step 4.</li>
<li>Let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-19" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-20"><span class="mi" id="MathJax-Span-21" style="font-family: STIXGeneral; font-style: italic;">m</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script type="math/tex" id="MathJax-Element-7">m</script></span> be the minimum uncovered element. Augment the array by reducing all uncovered elements by m and increasing all elements covered by two lines by <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-22" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-23"><span class="mi" id="MathJax-Span-24" style="font-family: STIXGeneral; font-style: italic;">m</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script type="math/tex" id="MathJax-Element-8">m</script></span>. Return to step 2.</li>
<li>There is a maximal matching using only zeros. Apply this pattern to the original array.</li>
</ol><h2 id="dynamic-programming"><a class="anchor hidden-xs" href="#dynamic-programming" title="dynamic-programming"><span class="octicon octicon-link"></span></a>Dynamic programming</h2><p><strong>States</strong> Nodes are called states</p><p><strong>Stage</strong> The transition from states with a stage variable <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-25" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-26"><span class="mi" id="MathJax-Span-27" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-9">k</script></span> to those with <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&#x2212;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-28" style="width: 2.666em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.05em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-29"><span class="mi" id="MathJax-Span-30" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-31" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-32" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-10">k-1</script></span> is called stage <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-33" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-34"><span class="mi" id="MathJax-Span-35" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-11">k</script></span></p><p><strong>Actions</strong> Arcs are called actions. an action transforms a situation from one state to another</p><p><strong>Costs</strong> The weight on an arc is called the cost</p><p>A general strategy can be used to solve optimisation problems. The principle of this strategy is to work backwards through the decisions, at each stage working out the best strategy from that point.</p><p><strong>Sub-optimal strategy</strong> The best strategy from each point is called the sub-optimal strategy</p><h3 id="maximin-and-minimax-problems"><a class="anchor hidden-xs" href="#maximin-and-minimax-problems" title="maximin-and-minimax-problems"><span class="octicon octicon-link"></span></a>Maximin and minimax problems</h3><p><strong>Maximin</strong> A problem where the objective is to maximise the minimum value</p><p><strong>Minimax</strong> A problem where the objective is to minimise the maximum value</p><h4 id="minimum"><a class="anchor hidden-xs" href="#minimum" title="minimum"><span class="octicon octicon-link"></span></a>Minimum</h4><table>
<thead>
<tr>
<th>Stage</th>
<th>State</th>
<th>From</th>
<th>Calculation</th>
<th>Value</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>I</td>
<td>K</td>
<td>12</td>
<td>12</td>
</tr>
<tr>
<td></td>
<td>J</td>
<td>K</td>
<td>14</td>
<td>14</td>
</tr>
<tr>
<td>2</td>
<td>G</td>
<td>I</td>
<td>15 + 12</td>
<td>27</td>
</tr>
<tr>
<td></td>
<td></td>
<td>J</td>
<td>15 + 14</td>
<td>(28)</td>
</tr>
<tr>
<td></td>
<td>H</td>
<td>I</td>
<td>12 + 13</td>
<td>25</td>
</tr>
<tr>
<td></td>
<td></td>
<td>J</td>
<td>14 + 12</td>
<td>(26)</td>
</tr>
<tr>
<td>3</td>
<td>D</td>
<td>G</td>
<td>27 + 5</td>
<td>32</td>
</tr>
<tr>
<td></td>
<td>E</td>
<td>G</td>
<td>27 + 9</td>
<td>36</td>
</tr>
<tr>
<td></td>
<td></td>
<td>H</td>
<td>25 + 12</td>
<td>(37)</td>
</tr>
<tr>
<td></td>
<td>F</td>
<td>H</td>
<td>25 + 13</td>
<td>38</td>
</tr>
<tr>
<td>4</td>
<td>B</td>
<td>D</td>
<td>32 + 4</td>
<td>36</td>
</tr>
<tr>
<td></td>
<td></td>
<td>E</td>
<td>36 + 4</td>
<td>40</td>
</tr>
<tr>
<td></td>
<td>C</td>
<td>E</td>
<td>36 + 9</td>
<td>(45)</td>
</tr>
<tr>
<td></td>
<td></td>
<td>F</td>
<td>38 + 6</td>
<td>44</td>
</tr>
<tr>
<td>5</td>
<td>A</td>
<td>B</td>
<td>36 + 8</td>
<td>44</td>
</tr>
<tr>
<td></td>
<td></td>
<td>B</td>
<td>30 + 8</td>
<td>(48)</td>
</tr>
<tr>
<td></td>
<td>A</td>
<td>C</td>
<td>44 + 4</td>
<td>48</td>
</tr>
</tbody>
</table><p>To find the minimum we keep track of the total for each path.<br>
The largest value from each set is removed. E.g. there is no reason to pass through G from J as it can be reach faster from I, and there is no reason to pass through E from H as it can be reached faster from G.</p><h4 id="maximum"><a class="anchor hidden-xs" href="#maximum" title="maximum"><span class="octicon octicon-link"></span></a>Maximum</h4><p>To find the maximum from above we would eliminate the smaller values instead.</p><h4 id="maximin"><a class="anchor hidden-xs" href="#maximin" title="maximin"><span class="octicon octicon-link"></span></a>Maximin</h4><table>
<thead>
<tr>
<th>Stage</th>
<th>State</th>
<th>From</th>
<th>Value</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>H</td>
<td>K</td>
<td>18</td>
</tr>
<tr>
<td></td>
<td>I</td>
<td>K</td>
<td>15</td>
</tr>
<tr>
<td></td>
<td>J</td>
<td>K</td>
<td>12</td>
</tr>
<tr>
<td>2</td>
<td>E</td>
<td>H</td>
<td>(17)</td>
</tr>
<tr>
<td></td>
<td></td>
<td>I</td>
<td>15</td>
</tr>
<tr>
<td></td>
<td>F</td>
<td>H</td>
<td>(15)</td>
</tr>
<tr>
<td></td>
<td></td>
<td>I</td>
<td>14</td>
</tr>
<tr>
<td></td>
<td></td>
<td>J</td>
<td>12</td>
</tr>
<tr>
<td></td>
<td>G</td>
<td>I</td>
<td>(14)</td>
</tr>
<tr>
<td></td>
<td></td>
<td>J</td>
<td>12</td>
</tr>
<tr>
<td>3</td>
<td>B</td>
<td>E</td>
<td>11</td>
</tr>
<tr>
<td></td>
<td></td>
<td>F</td>
<td>(13)</td>
</tr>
<tr>
<td></td>
<td>C</td>
<td>E</td>
<td>12</td>
</tr>
<tr>
<td></td>
<td></td>
<td>F</td>
<td>13</td>
</tr>
<tr>
<td></td>
<td></td>
<td>G</td>
<td>(14)</td>
</tr>
<tr>
<td></td>
<td>D</td>
<td>F</td>
<td>(15)</td>
</tr>
<tr>
<td></td>
<td></td>
<td>G</td>
<td>14</td>
</tr>
<tr>
<td>4</td>
<td>A</td>
<td>B</td>
<td>12</td>
</tr>
<tr>
<td></td>
<td></td>
<td>C</td>
<td>(14)</td>
</tr>
<tr>
<td></td>
<td></td>
<td>D</td>
<td>13</td>
</tr>
</tbody>
</table><p>ACGIK</p><p>In each case the highest value in getting to the new state is selected.<br>
When tracing back we move from A to C (14), from C to G (14), G to I (14), and finally I to K (15).<br>
The minimum value is 14.</p><h4 id="minimax"><a class="anchor hidden-xs" href="#minimax" title="minimax"><span class="octicon octicon-link"></span></a>Minimax</h4><p>For each stage from each node, choose the minimum value.</p><table>
<thead>
<tr>
<th>Stage</th>
<th>State</th>
<th>From</th>
<th>Value</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>H</td>
<td>K</td>
<td>2.7</td>
</tr>
<tr>
<td></td>
<td>I</td>
<td>K</td>
<td>2.3</td>
</tr>
<tr>
<td></td>
<td>J</td>
<td>K</td>
<td>2.5</td>
</tr>
<tr>
<td>2</td>
<td>E</td>
<td>H</td>
<td>2.7</td>
</tr>
<tr>
<td></td>
<td></td>
<td>I</td>
<td>(2.4)</td>
</tr>
<tr>
<td></td>
<td>F</td>
<td>H</td>
<td>2.7</td>
</tr>
<tr>
<td></td>
<td></td>
<td>I</td>
<td>2.6</td>
</tr>
<tr>
<td></td>
<td></td>
<td>J</td>
<td>(2.5)</td>
</tr>
<tr>
<td></td>
<td>G</td>
<td>I</td>
<td>(2.6)</td>
</tr>
<tr>
<td></td>
<td></td>
<td>J</td>
<td>2.9</td>
</tr>
<tr>
<td>3</td>
<td>B</td>
<td>E</td>
<td>2.8</td>
</tr>
<tr>
<td></td>
<td></td>
<td>F</td>
<td>(2.7)</td>
</tr>
<tr>
<td></td>
<td>C</td>
<td>E</td>
<td>2.8</td>
</tr>
<tr>
<td></td>
<td></td>
<td>F</td>
<td>(2.5)</td>
</tr>
<tr>
<td></td>
<td></td>
<td>G</td>
<td>2.6</td>
</tr>
<tr>
<td></td>
<td>D</td>
<td>F</td>
<td>2.8</td>
</tr>
<tr>
<td></td>
<td></td>
<td>G</td>
<td>(2.6)</td>
</tr>
<tr>
<td>4</td>
<td>A</td>
<td>B</td>
<td>2.7</td>
</tr>
<tr>
<td></td>
<td></td>
<td>C</td>
<td>(2.5)</td>
</tr>
<tr>
<td></td>
<td></td>
<td>D</td>
<td>2.7</td>
</tr>
</tbody>
</table><p>Now move backwards, picking the starred lowest values.</p><p>From A to C with 2.5, then C to F with 2.5, then F to J with 2.5, then J to K with 2.5.</p><p>ACFJK</p><h3 id="negative-edge-lengths"><a class="anchor hidden-xs" href="#negative-edge-lengths" title="negative-edge-lengths"><span class="octicon octicon-link"></span></a>Negative edge lengths</h3><h2 id="network-flows"><a class="anchor hidden-xs" href="#network-flows" title="network-flows"><span class="octicon octicon-link"></span></a>Network flows</h2><p><strong>Source</strong> The start of a network</p><p><strong>Sink</strong> The end of a network</p><p><strong>Capacity</strong> The weight on an arc</p><p><strong>Saturated</strong> An arc is said to be saturated if the flow through it is equal to its capacity</p><p><strong>Cut</strong> A continuous line which separates the source from the sink. A cut must not pass through any nodes</p><h3 id="maximum-flow-minimum-cut-theorem"><a class="anchor hidden-xs" href="#maximum-flow-minimum-cut-theorem" title="maximum-flow-minimum-cut-theorem"><span class="octicon octicon-link"></span></a>Maximum flow minimum cut theorem</h3><p>The maximum flow is less than or equal to the value of the minimum cut.</p><p>If a flow and cut with the same value can be found, the maximum flow has been found.</p><h3 id="labelling-procedure-for-flow-augmentation"><a class="anchor hidden-xs" href="#labelling-procedure-for-flow-augmentation" title="labelling-procedure-for-flow-augmentation"><span class="octicon octicon-link"></span></a>Labelling procedure for flow augmentation</h3><ol>
<li>Begin with any initial flow, or zero flow</li>
<li>Replace each arc with two arcs, the forward arc showing excess capacity, and the backwards arc showing potential backflow (The current flow)</li>
<li>If the source is still connected to the sink, find a new flow and alter the excess capacities and potential backflows as necessary</li>
<li>Repeat step 3 until the source is disconnected from the sink. If the source is disconnected from the sink, then the maximum flow is the sum of all the flows of steps 1 and 3</li>
</ol><p>The maximum flow can also be determined by tracing the flow.</p><h3 id="super-sources-and-sinks"><a class="anchor hidden-xs" href="#super-sources-and-sinks" title="super-sources-and-sinks"><span class="octicon octicon-link"></span></a>Super sources and sinks</h3><p>Some problems may have multiple sources or sinks.<br>
These problems can be solved by combining the sources and sinks into super sources and super sinks respectively.</p><p>In order to apply the labelling procedure, create the super source which feeds the existing sources with capacity sufficiently large so as not to affect the solution. Create a super sink in the same manner.</p><p>The maximum flow through the network can then be found by removing the super sources and sinks once the process is complete.</p><h3 id="restricted-capacity"><a class="anchor hidden-xs" href="#restricted-capacity" title="restricted-capacity"><span class="octicon octicon-link"></span></a>Restricted capacity</h3><p><strong>Restricted capacity</strong> If a node has restricted capacity, there is a limit to the flow which can pass through it</p><p>In order to solve a problem involving restricted capacity, the restricted node must be represented by two unrestricted nodes, with an arc between of the same capacity as the restricted node.</p><p>If a node has multiple entrances, each of them must flow through the first unrestricted node.</p><h3 id="minimum-capacities"><a class="anchor hidden-xs" href="#minimum-capacities" title="minimum-capacities"><span class="octicon octicon-link"></span></a>Minimum capacities</h3><p><strong>Minimum capacity</strong> An arc with a minimum is required to have a minimum flow through it</p><p>It is convention to label each directed arc with two numbers, an upper capacity and a lower capacity.</p><p>For such a network the value of a cut is defined as follows</p><blockquote>
<p>The value of a cut is the sum of the upper capacities for arcs that cross the cut line in the direction from the source to the sink minus the sum of the lower capacities for arcs that cross the cut line in the direction from the sink to the source</p>
</blockquote><h2 id="linear-programming"><a class="anchor hidden-xs" href="#linear-programming" title="linear-programming"><span class="octicon octicon-link"></span></a>Linear programming</h2><h3 id="the-simplex-method"><a class="anchor hidden-xs" href="#the-simplex-method" title="the-simplex-method"><span class="octicon octicon-link"></span></a>The Simplex method</h3><p>The simplex algorithm simultaneously solves the equations of the constraints to find the nodes. It then works out whether that node maximises the objective function.</p><h4 id="slack-variables"><a class="anchor hidden-xs" href="#slack-variables" title="slack-variables"><span class="octicon octicon-link"></span></a>Slack variables</h4><p>Slack variables are used in order to remove inequalities from each equation.</p><h4 id="the-tableau"><a class="anchor hidden-xs" href="#the-tableau" title="the-tableau"><span class="octicon octicon-link"></span></a>The tableau</h4><p>The top row of the tableau is the objective function.<br>
There is one row for each further equation.</p><p>Suppose we have <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>&#x2264;</mo><mn>6</mn><mo stretchy="false">&#x2192;</mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>s</mi><mo>=</mo><mn>6</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-36" style="width: 14.501em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.888em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1011.84em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-37"><span class="mi" id="MathJax-Span-38" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-39" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mn" id="MathJax-Span-40" style="font-family: STIXGeneral; padding-left: 0.259em;">2</span><span class="mi" id="MathJax-Span-41" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-42" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mn" id="MathJax-Span-43" style="font-family: STIXGeneral; padding-left: 0.31em;">6</span><span class="mo" id="MathJax-Span-44" style="font-family: STIXGeneral; padding-left: 0.31em;">→</span><span class="mi" id="MathJax-Span-45" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-46" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mn" id="MathJax-Span-47" style="font-family: STIXGeneral; padding-left: 0.259em;">2</span><span class="mi" id="MathJax-Span-48" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-49" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-50" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">s</span><span class="mo" id="MathJax-Span-51" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-52" style="font-family: STIXGeneral; padding-left: 0.31em;">6</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>≤</mo><mn>6</mn><mo stretchy="false">→</mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>s</mi><mo>=</mo><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-12">x + 2y \leq 6 \to x + 2y + s = 6</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>&#x2264;</mo><mn>5</mn><mo stretchy="false">&#x2192;</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>t</mi><mo>=</mo><mn>5</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-53" style="width: 13.22em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.812em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1010.76em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-54"><span class="mi" id="MathJax-Span-55" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-56" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-57" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">y</span><span class="mo" id="MathJax-Span-58" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mn" id="MathJax-Span-59" style="font-family: STIXGeneral; padding-left: 0.31em;">5</span><span class="mo" id="MathJax-Span-60" style="font-family: STIXGeneral; padding-left: 0.31em;">→</span><span class="mi" id="MathJax-Span-61" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-62" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-63" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">y</span><span class="mo" id="MathJax-Span-64" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-65" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-66" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-67" style="font-family: STIXGeneral; padding-left: 0.31em;">5</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>≤</mo><mn>5</mn><mo stretchy="false">→</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>t</mi><mo>=</mo><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-13">x + y \leq 5 \to x + y + t = 5</script></span> the tableau would be:</p><table>
<thead>
<tr>
<th><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-68" style="width: 0.771em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.62em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-69"><span class="mi" id="MathJax-Span-70" style="font-family: STIXGeneral; font-style: italic;">P</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></span></span><script type="math/tex" id="MathJax-Element-14">P</script></span></th>
<th><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-71" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-72"><span class="mi" id="MathJax-Span-73" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-15">x</script></span></th>
<th><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-74" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-75"><span class="mi" id="MathJax-Span-76" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span><script type="math/tex" id="MathJax-Element-16">y</script></span></th>
<th><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-77" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-78"><span class="mi" id="MathJax-Span-79" style="font-family: STIXGeneral; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math></span></span><script type="math/tex" id="MathJax-Element-17">s</script></span></th>
<th><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-80" style="width: 0.31em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.259em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.847em 1000.26em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-81"><span class="mi" id="MathJax-Span-82" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.816em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span><script type="math/tex" id="MathJax-Element-18">t</script></span></th>
<th>Value</th>
<th>Equation</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>-2</td>
<td>-1</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>1</td>
</tr>
<tr>
<td>0</td>
<td>1</td>
<td>2</td>
<td>1</td>
<td>0</td>
<td>6</td>
<td>2</td>
</tr>
<tr>
<td>0</td>
<td>1</td>
<td>1</td>
<td>0</td>
<td>1</td>
<td>5</td>
<td>3</td>
</tr>
</tbody>
</table><h4 id="method"><a class="anchor hidden-xs" href="#method" title="method"><span class="octicon octicon-link"></span></a>Method</h4><p>Once the tableau has been created, choose any column which has a negative number in the top row.<br>
The aim is now to manipulate the equations to eliminate this variable from the other equations.</p><p>To do this a pivot value is chosen. The pivot is the value in the selected column which has not already been a pivot and provides the smallest value when equation value is divided by it.</p><p>In the case above the column chosen would be <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-83" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-84"><span class="mi" id="MathJax-Span-85" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-19">x</script></span>, and the pivot value would be <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-86" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-87"><span class="mn" id="MathJax-Span-88" style="font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-20">1</script></span> in equation 3 as <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>1</mn></mfrac><mo>&lt;</mo><mfrac><mn>6</mn><mn>2</mn></mfrac><mo>&lt;</mo><mfrac><mn>0</mn><mn>2</mn></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-89" style="width: 5.638em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.613em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.488em 1004.61em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-90"><span class="mfrac" id="MathJax-Span-91"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.31em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-92" style="font-size: 70.7%; font-family: STIXGeneral;">5</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-93" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-94" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mfrac" id="MathJax-Span-95" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.31em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-96" style="font-size: 70.7%; font-family: STIXGeneral;">6</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-97" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-98" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mfrac" id="MathJax-Span-99" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-100" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-101" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>1</mn></mfrac><mo><</mo><mfrac><mn>6</mn><mn>2</mn></mfrac><mo><</mo><mfrac><mn>0</mn><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-21">\frac 5 1 < \frac 6 2 < \frac 0 2</script></span>.</p><p>Divide the pivot row to get <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-102" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-103"><span class="mn" id="MathJax-Span-104" style="font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-22">1</script></span> in the select column.<br>
Then add a multiple of the new pivot equation to each of the other equations.</p><p>If the top row has no negative elements, stop.<br>
If it still contains negative elements, find a new pivot from the new equations and repeat.</p><h4 id="reading-the-solutions"><a class="anchor hidden-xs" href="#reading-the-solutions" title="reading-the-solutions"><span class="octicon octicon-link"></span></a>Reading the solutions</h4><p>Find the maximum by interpreting the tableau as described below:</p><ul>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-105" style="width: 0.771em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.62em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-106"><span class="mi" id="MathJax-Span-107" style="font-family: STIXGeneral; font-style: italic;">P</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></span></span><script type="math/tex" id="MathJax-Element-23">P</script></span> equals the value in the top row</li>
<li>Any non zero elements in the top row equal zero</li>
<li>The other equations are then used to find the remaining variables</li>
</ul><h2 id="game-theory-for-zero-sum-games"><a class="anchor hidden-xs" href="#game-theory-for-zero-sum-games" title="game-theory-for-zero-sum-games"><span class="octicon octicon-link"></span></a>Game theory for zero sum games</h2><p><strong>Zero sum game</strong> A game such that the gain for one party is exactly equal to the loss for the other party</p><h3 id="pay-off-matrix"><a class="anchor hidden-xs" href="#pay-off-matrix" title="pay-off-matrix"><span class="octicon octicon-link"></span></a>Pay off matrix</h3><p>The pay off matrix represents the outcomes of each pair of decisions.<br>
The outcomes are the pay offs to the player on the left side of the matrix, and as a result the losses for the player on the top of the matrix.</p><h3 id="play-safe-strategies"><a class="anchor hidden-xs" href="#play-safe-strategies" title="play-safe-strategies"><span class="octicon octicon-link"></span></a>Play safe strategies</h3><p>A play safe strategy is to choose the option whose word outcome is as good as possible</p><p>Method</p><ul>
<li>Add an extra column of the worst values from each row (The worst outcomes for the player on the left)</li>
<li>Add an extra row of the worst values from each column (The worst outcomes for the player on the top)</li>
</ul><h4 id="stable-solutions"><a class="anchor hidden-xs" href="#stable-solutions" title="stable-solutions"><span class="octicon octicon-link"></span></a>Stable solutions</h4><p>If MAX(MIN(row)) = MIN(MAX(column)) there is a stable solution.</p><p>If a game has a stable solution, then both players should adopt a play safe strategy and the pay off from these strategies is called the value of the game.</p><h3 id="dominance"><a class="anchor hidden-xs" href="#dominance" title="dominance"><span class="octicon octicon-link"></span></a>Dominance</h3><p>A row of a pay off matrix can be ignored if each element is less than or equal to the corresponding element of another row.<br>
A column of a pay off matrix can be ignored if each element is greater than or equal to the corresponding element of another column,</p><h3 id="2-times-n-games"><a class="anchor hidden-xs" href="#2-times-n-games" title="2-times-n-games"><span class="octicon octicon-link"></span></a><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>&#x00D7;</mo><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-108" style="width: 2.625em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.133em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.723em 1002.09em 2.666em -999.998em); top: -2.498em; left: 0em;"><span class="mrow" id="MathJax-Span-109"><span class="mn" id="MathJax-Span-110" style="font-family: STIXGeneral;">2</span><span class="mo" id="MathJax-Span-111" style="font-family: STIXGeneral; padding-left: 0.248em;">×</span><span class="mi" id="MathJax-Span-112" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.248em;">n</span></span><span style="display: inline-block; width: 0px; height: 2.502em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.098em; border-left: 0px solid; width: 0px; height: 0.953em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-24">2 \times n</script></span> games</h3><p>For each column there is a line on the graph.<br>
The optimal value is at the lowest intersection on the graph</p><p>To solve a <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>&#x00D7;</mo><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-113" style="width: 2.666em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-114"><span class="mn" id="MathJax-Span-115" style="font-family: STIXGeneral;">2</span><span class="mo" id="MathJax-Span-116" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mi" id="MathJax-Span-117" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-25">2 \times n</script></span> game between A and B</p><ol>
<li>Assign probabilities of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-118" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-119"><span class="mi" id="MathJax-Span-120" style="font-family: STIXGeneral; font-style: italic;">p</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-26">p</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-27-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>&#x2212;</mo><mi>p</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-121" style="width: 2.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.15em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-122"><span class="mn" id="MathJax-Span-123" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-124" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-125" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">p</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>−</mo><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-27">1 - p</script></span> to the two options for A</li>
<li>Plot A’s expected pay off under each of B’s options as lines on a graph of pay off against <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-28-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-126" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-127"><span class="mi" id="MathJax-Span-128" style="font-family: STIXGeneral; font-style: italic;">p</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-28">p</script></span></li>
<li>Shade the region which is below every line</li>
<li>Find the value of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-29-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-129" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-130"><span class="mi" id="MathJax-Span-131" style="font-family: STIXGeneral; font-style: italic;">p</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-29">p</script></span> at the highest point of the shaded region</li>
<li>Use this value of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-30-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-132" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-133"><span class="mi" id="MathJax-Span-134" style="font-family: STIXGeneral; font-style: italic;">p</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-30">p</script></span> to find the value, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-31-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-135" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-136"><span class="mi" id="MathJax-Span-137" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math></span></span><script type="math/tex" id="MathJax-Element-31">V</script></span>, of the game</li>
<li>Assign probability <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-32-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-138" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.51em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-139"><span class="mi" id="MathJax-Span-140" style="font-family: STIXGeneral; font-style: italic;">q</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math></span></span><script type="math/tex" id="MathJax-Element-32">q</script></span> to one of those options for B which determine the highest point of the shaded region</li>
<li>Find the value of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-33-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-141" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.51em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-142"><span class="mi" id="MathJax-Span-143" style="font-family: STIXGeneral; font-style: italic;">q</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math></span></span><script type="math/tex" id="MathJax-Element-33">q</script></span> which gives B an expected pay off of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-34-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2212;</mo><mi>V</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-144" style="width: 1.693em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.386em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.39em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-145"><span class="mo" id="MathJax-Span-146" style="font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-147" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mi>V</mi></math></span></span><script type="math/tex" id="MathJax-Element-34">-V</script></span> under either of A’s options</li>
</ol><h3 id="finding-the-mixed-strategy-with-simplex"><a class="anchor hidden-xs" href="#finding-the-mixed-strategy-with-simplex" title="finding-the-mixed-strategy-with-simplex"><span class="octicon octicon-link"></span></a>Finding the mixed strategy with simplex</h3><p>If there are more than two rows, the mixed strategy can be found with simplex.</p><ol>
<li>Let the probabilities of each of the options for A be <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-35-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><msub><mi>p</mi><mn>2</mn></msub><mo>,</mo><msub><mi>p</mi><mn>3</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-148" style="width: 6.457em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.279em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1005.23em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-149"><span class="msubsup" id="MathJax-Span-150"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-151" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-152" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-153" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-154" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-155" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-156" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-157" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-158" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-159" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-160" style="font-size: 70.7%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-161" style="font-family: STIXGeneral;">,</span><span class="mo" id="MathJax-Span-162" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-163" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-164" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><msub><mi>p</mi><mn>2</mn></msub><mo>,</mo><msub><mi>p</mi><mn>3</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-35">p_1, p_2, p_3, ...</script></span></li>
<li>If necessary, add a constant value <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-36-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-165" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-166"><span class="mi" id="MathJax-Span-167" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-36">k</script></span> to eliminate any negative pay-offs from the matrix</li>
<li>The objective is then to maximise <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-37-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>&#x2212;</mo><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-168" style="width: 2.871em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.359em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.36em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-169"><span class="mi" id="MathJax-Span-170" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-171" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-172" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>−</mo><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-37">V-k</script></span> subject to the following constraints
<ol>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-38-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>&#x2264;</mo><mtext>expected pay off from 1</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-173" style="width: 14.091em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.529em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1011.43em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-174"><span class="mi" id="MathJax-Span-175" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-176" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mtext" id="MathJax-Span-177" style="font-family: STIXGeneral; padding-left: 0.31em;">expected pay off from 1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>≤</mo><mtext>expected pay off from 1</mtext></math></span></span><script type="math/tex" id="MathJax-Element-38">V \leq \text{expected pay off from 1}</script></span></li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-39-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>&#x2264;</mo><mtext>expected pay off from 2</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-178" style="width: 14.091em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.529em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1011.48em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-179"><span class="mi" id="MathJax-Span-180" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-181" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mtext" id="MathJax-Span-182" style="font-family: STIXGeneral; padding-left: 0.31em;">expected pay off from 2</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>≤</mo><mtext>expected pay off from 2</mtext></math></span></span><script type="math/tex" id="MathJax-Element-39">V \leq \text{expected pay off from 2}</script></span> …</li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-40-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub><mo>+</mo><msub><mi>p</mi><mn>2</mn></msub><mo>+</mo><msub><mi>p</mi><mn>3</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>&#x2264;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-183" style="width: 10.095em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.251em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1008.15em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-184"><span class="msubsup" id="MathJax-Span-185"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-186" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-187" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-188" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-189" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-190" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-191" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-192" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-193" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-194" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-195" style="font-size: 70.7%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-196" style="font-family: STIXGeneral;">+</span><span class="mo" id="MathJax-Span-197" style="font-family: STIXGeneral;">.</span><span class="mo" id="MathJax-Span-198" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-199" style="font-family: STIXGeneral; padding-left: 0.207em;">≤</span><span class="mn" id="MathJax-Span-200" style="font-family: STIXGeneral; padding-left: 0.31em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub><mo>+</mo><msub><mi>p</mi><mn>2</mn></msub><mo>+</mo><msub><mi>p</mi><mn>3</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>≤</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-40">p_1 + p_2 + p_3 + .. \leq 1</script></span></li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-41-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><msub><mi>p</mi><mn>2</mn></msub><mo>,</mo><msub><mi>p</mi><mn>3</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>&#x2265;</mo><mn>0</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-201" style="width: 8.558em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.021em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1007.02em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-202"><span class="msubsup" id="MathJax-Span-203"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-204" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-205" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-206" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-207" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-208" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-209" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-210" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-211" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-212" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-213" style="font-size: 70.7%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-214" style="font-family: STIXGeneral;">,</span><span class="mo" id="MathJax-Span-215" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-216" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-217" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-218" style="font-family: STIXGeneral; padding-left: 0.207em;">≥</span><span class="mn" id="MathJax-Span-219" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><msub><mi>p</mi><mn>2</mn></msub><mo>,</mo><msub><mi>p</mi><mn>3</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>≥</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-41">p_1, p_2, p_3, ... \geq 0</script></span></li>
</ol>
</li>
</ol></div>
<div class="ui-toc dropup unselectable hidden-print" style="display:none;">
<div class="pull-right dropdown">
<a id="tocLabel" class="ui-toc-label btn btn-default" data-toggle="dropdown" href="#" role="button" aria-haspopup="true" aria-expanded="false" title="Table of content">
<i class="fa fa-bars"></i>
</a>
<ul id="ui-toc" class="ui-toc-dropdown dropdown-menu" aria-labelledby="tocLabel">
<div class="toc"><ul class="nav"><li class=""><a href="#decision-2-notes">Decision 2 Notes</a><ul class="nav"><li class=""><a href="#critical-path-analysis">Critical path analysis</a><ul class="nav"><li class=""><a href="#activity-networks">Activity networks</a></li><li><a href="#critical-paths">Critical paths</a></li><li><a href="#cascade-diagrams">Cascade diagrams</a></li></ul></li><li><a href="#allocation-with-the-hungarian-algorithm">Allocation with the Hungarian algorithm</a></li><li><a href="#dynamic-programming">Dynamic programming</a><ul class="nav"><li><a href="#maximin-and-minimax-problems">Maximin and minimax problems</a></li><li><a href="#negative-edge-lengths">Negative edge lengths</a></li></ul></li><li><a href="#network-flows">Network flows</a><ul class="nav"><li><a href="#maximum-flow-minimum-cut-theorem">Maximum flow minimum cut theorem</a></li><li><a href="#labelling-procedure-for-flow-augmentation">Labelling procedure for flow augmentation</a></li><li><a href="#super-sources-and-sinks">Super sources and sinks</a></li><li><a href="#restricted-capacity">Restricted capacity</a></li><li><a href="#minimum-capacities">Minimum capacities</a></li></ul></li><li><a href="#linear-programming">Linear programming</a><ul class="nav"><li><a href="#the-simplex-method">The Simplex method</a></li></ul></li><li><a href="#game-theory-for-zero-sum-games">Game theory for zero sum games</a><ul class="nav"><li><a href="#pay-off-matrix">Pay off matrix</a></li><li><a href="#play-safe-strategies">Play safe strategies</a></li><li><a href="#dominance">Dominance</a></li><li><a href="#2-times-n-games">2 \times n games</a></li><li><a href="#finding-the-mixed-strategy-with-simplex">Finding the mixed strategy with simplex</a></li></ul></li></ul></li></ul></div><div class="toc-menu"><a class="expand-toggle" href="#">Expand all</a><a class="back-to-top" href="#">Back to top</a><a class="go-to-bottom" href="#">Go to bottom</a></div>
</ul>
</div>
</div>
<div id="ui-toc-affix" class="ui-affix-toc ui-toc-dropdown unselectable hidden-print" data-spy="affix" style="top:17px;display:none;" >
<div class="toc"><ul class="nav"><li class=""><a href="#decision-2-notes">Decision 2 Notes</a><ul class="nav"><li class=""><a href="#critical-path-analysis">Critical path analysis</a><ul class="nav"><li class=""><a href="#activity-networks">Activity networks</a></li><li><a href="#critical-paths">Critical paths</a></li><li><a href="#cascade-diagrams">Cascade diagrams</a></li></ul></li><li><a href="#allocation-with-the-hungarian-algorithm">Allocation with the Hungarian algorithm</a></li><li><a href="#dynamic-programming">Dynamic programming</a><ul class="nav"><li><a href="#maximin-and-minimax-problems">Maximin and minimax problems</a></li><li><a href="#negative-edge-lengths">Negative edge lengths</a></li></ul></li><li><a href="#network-flows">Network flows</a><ul class="nav"><li><a href="#maximum-flow-minimum-cut-theorem">Maximum flow minimum cut theorem</a></li><li><a href="#labelling-procedure-for-flow-augmentation">Labelling procedure for flow augmentation</a></li><li><a href="#super-sources-and-sinks">Super sources and sinks</a></li><li><a href="#restricted-capacity">Restricted capacity</a></li><li><a href="#minimum-capacities">Minimum capacities</a></li></ul></li><li><a href="#linear-programming">Linear programming</a><ul class="nav"><li><a href="#the-simplex-method">The Simplex method</a></li></ul></li><li><a href="#game-theory-for-zero-sum-games">Game theory for zero sum games</a><ul class="nav"><li><a href="#pay-off-matrix">Pay off matrix</a></li><li><a href="#play-safe-strategies">Play safe strategies</a></li><li><a href="#dominance">Dominance</a></li><li><a href="#2-times-n-games">2 \times n games</a></li><li><a href="#finding-the-mixed-strategy-with-simplex">Finding the mixed strategy with simplex</a></li></ul></li></ul></li></ul></div><div class="toc-menu"><a class="expand-toggle" href="#">Expand all</a><a class="back-to-top" href="#">Back to top</a><a class="go-to-bottom" href="#">Go to bottom</a></div>
</div>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.1.1/jquery.min.js" integrity="sha256-hVVnYaiADRTO2PzUGmuLJr8BLUSjGIZsDYGmIJLv2b8=" crossorigin="anonymous"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.7/js/bootstrap.min.js" integrity="sha256-U5ZEeKfGNOja007MMD3YBI0A3OSZOQbeG6z2f2Y0hu8=" crossorigin="anonymous" defer></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/gist-embed/2.6.0/gist-embed.min.js" integrity="sha256-KyF2D6xPIJUW5sUDSs93vWyZm+1RzIpKCexxElmxl8g=" crossorigin="anonymous" defer></script>
<script>
var markdown = $(".markdown-body");
//smooth all hash trigger scrolling
function smoothHashScroll() {
var hashElements = $("a[href^='#']").toArray();
for (var i = 0; i < hashElements.length; i++) {
var element = hashElements[i];
var $element = $(element);
var hash = element.hash;
if (hash) {
$element.on('click', function (e) {
// store hash
var hash = this.hash;
if ($(hash).length <= 0) return;
// prevent default anchor click behavior
e.preventDefault();
// animate
$('body, html').stop(true, true).animate({
scrollTop: $(hash).offset().top
}, 100, "linear", function () {
// when done, add hash to url
// (default click behaviour)
window.location.hash = hash;
});
});
}
}
}
smoothHashScroll();
var toc = $('.ui-toc');
var tocAffix = $('.ui-affix-toc');
var tocDropdown = $('.ui-toc-dropdown');
//toc
tocDropdown.click(function (e) {
e.stopPropagation();
});
var enoughForAffixToc = true;
function generateScrollspy() {
$(document.body).scrollspy({
target: ''
});
$(document.body).scrollspy('refresh');
if (enoughForAffixToc) {
toc.hide();
tocAffix.show();
} else {
tocAffix.hide();
toc.show();
}
$(document.body).scroll();
}
function windowResize() {
//toc right
var paddingRight = parseFloat(markdown.css('padding-right'));
var right = ($(window).width() - (markdown.offset().left + markdown.outerWidth() - paddingRight));
toc.css('right', right + 'px');
//affix toc left
var newbool;
var rightMargin = (markdown.parent().outerWidth() - markdown.outerWidth()) / 2;
//for ipad or wider device
if (rightMargin >= 133) {
newbool = true;
var affixLeftMargin = (tocAffix.outerWidth() - tocAffix.width()) / 2;
var left = markdown.offset().left + markdown.outerWidth() - affixLeftMargin;
tocAffix.css('left', left + 'px');
} else {
newbool = false;
}
if (newbool != enoughForAffixToc) {
enoughForAffixToc = newbool;
generateScrollspy();
}
}
$(window).resize(function () {
windowResize();
});
$(document).ready(function () {
windowResize();
generateScrollspy();
});
//remove hash
function removeHash() {
window.location.hash = '';
}
var backtotop = $('.back-to-top');
var gotobottom = $('.go-to-bottom');
backtotop.click(function (e) {
e.preventDefault();
e.stopPropagation();
if (scrollToTop)
scrollToTop();
removeHash();
});
gotobottom.click(function (e) {
e.preventDefault();
e.stopPropagation();
if (scrollToBottom)
scrollToBottom();
removeHash();
});
function scrollToTop() {
$('body, html').stop(true, true).animate({
scrollTop: 0
}, 100, "linear");
}
function scrollToBottom() {
$('body, html').stop(true, true).animate({
scrollTop: $(document.body)[0].scrollHeight
}, 100, "linear");
}
</script>
</body>
</html>