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S&P500 Market Values

Standard & Poor's 500
Market Values
In $U.S. Millions
Ordered Alphabetically: Market Value: Second Quintile Alphabetically (101 - 200) Sample of 12 from the S&P 500 Sample of 60 from S&P 500 Sample of 12 from Second Quintile
1 3M Company 120,052 ACTIVISION BLIZZARD INC 46,561 1 O'REILLY AUTOMOTIVE INC 17,978 ABBOTT LABORATORIES 85,341 1 ALLSTATE CORP 33,171
2 ABBOTT LABORATORIES 85,341 AETNA INC 51,185 2 VERIZON COMMUNICATIONS INC 197,424 AES CORP 7,380 2 BB&T CORP 38,239
3 ABBVIE INC 111,265 AFLAC INC 31,581 3 PG&E CORP 34,713 ALBEMARLE CORP 12,825 3 COGNIZANT TECH SOLUTIONS 40,829
4 ACCENTURE PLC 79,609 AIR PRODUCTS & CHEMICALS INC 30,949 4 JUNIPER NETWORKS INC 10,680 ALLSTATE CORP 33,171 4 DOMINION ENERGY INC 48,545
5 ACTIVISION BLIZZARD INC 46,561 ALEXION PHARMACEUTICALS INC 30,652 5 CARDINAL HEALTH INC 24,405 AMERICAN INTERNATIONAL GROUP 60,353 5 EXPRESS SCRIPTS HOLDING CO 36,178
6 ACUITY BRANDS INC 8,530 ALLSTATE CORP 33,171 6 SL GREEN REALTY CORP 10,158 ANADARKO PETROLEUM CORP 25,592 6 HEWLETT PACKARD ENTERPRISE 28,762
7 ADOBE SYSTEMS INC 72,279 AMERICAN ELECTRIC POWER CO 34,694 7 APPLE INC 775,454 APPLIED MATERIALS INC 47,617 7 JOHNSON CONTROLS INTL PLC 36,317
8 ADVANCE AUTO PARTS INC 8,271 ANALOG DEVICES 28,998 8 U S BANCORP 88,618 AUTONATION INC 4,291 8 MCKESSON CORP 34,035
9 ADVANCED MICRO DEVICES 12,862 ANTHEM INC 49,823 9 UNITED TECHNOLOGIES CORP 94,711 BARD (C.R.) INC 23,299 9 PHILLIPS 66 43,223
10 AES CORP 7,380 AON PLC 36,210 10 STANLEY BLACK & DECKER INC 21,547 BLOCK H & R INC 6,319 10 PUBLIC STORAGE 35,770
11 AETNA INC 51,185 APPLIED MATERIALS INC 47,617 11 SYMANTEC CORP 18,935 C H ROBINSON WORLDWIDE INC 9,242 11 SOUTHWEST AIRLINES 33,565
12 AFFILIATED MANAGERS GRP INC 10,518 AVALONBAY COMMUNITIES INC 26,503 12 ALLERGAN PLC 84,749 CATERPILLAR INC 67,127 12 VALERO ENERGY CORP 30,846
13 AFLAC INC 31,581 BAXTER INTERNATIONAL INC 32,881 Mean 114,948 CERNER CORP 21,344 Mean 36,623
14 AGILENT TECHNOLOGIES INC 19,213 BB&T CORP 38,239 Median 29,559 CIGNA CORP 44,434 Median 35,974
15 AIR PRODUCTS & CHEMICALS INC 30,949 BECTON DICKINSON & CO 45,789 Standard Deviation 215,093.99 CLOROX CO/DE 17,193 Standard Deviation 5,478.53
16 AKAMAI TECHNOLOGIES INC 8,149 BOSTON SCIENTIFIC CORP 36,453 COMERICA INC 12,727
17 ALASKA AIR GROUP INC 10,528 CAPITAL ONE FINANCIAL CORP 41,685 COSTCO WHOLESALE CORP 69,521
18 ALBEMARLE CORP 12,825 CARNIVAL CORP/PLC (USA) 35,783 DANAHER CORP 56,610
19 ALEXANDRIA R E EQUITIES INC 11,043 CBS CORP 26,734 DOVER CORP 13,082
20 ALEXION PHARMACEUTICALS INC 30,652 CIGNA CORP 44,434 E TRADE FINANCIAL CORP 11,277
21 ALIGN TECHNOLOGY INC 13,434 CME GROUP INC 41,665 EMERSON ELECTRIC CO 38,417
22 ALLEGION PLC 7,717 COGNIZANT TECH SOLUTIONS 40,829 ESSEX PROPERTY TRUST 17,167
23 ALLERGAN PLC 84,749 CONSTELLATION BRANDS -CL A 37,862 EXXON MOBIL CORP 339,129
24 ALLIANCE DATA SYSTEMS CORP 13,442 CORNING INC 26,319 FIRSTENERGY CORP 14,178
25 ALLIANT ENERGY CORP 9,234 CROWN CASTLE INTL CORP 40,862 FORTIVE CORP 22,471
26 ALLSTATE CORP 33,171 CSX CORP 45,063 GENERAL ELECTRIC CO 221,730
27 ALPHABET INC 605,366 CUMMINS INC 28,203 GOODYEAR TIRE & RUBBER CO 7,933
28 ALTRIA GROUP INC 124,649 DEERE & CO 41,032 HCA HEALTHCARE INC 29,477
29 AMAZON.COM INC 474,510 DELTA AIR LINES INC 35,738 HOME DEPOT INC 178,855
30 AMEREN CORP 13,612 DOMINION ENERGY INC 48,545 IDEXX LABS INC 14,669
31 AMERICAN AIRLINES GROUP INC 24,565 EATON CORP PLC 35,001 INTERPUBLIC GROUP OF COS 8,499
32 AMERICAN ELECTRIC POWER CO 34,694 EBAY INC 38,242 JACOBS ENGINEERING GROUP INC 6,349
33 AMERICAN EXPRESS CO 75,342 ECOLAB INC 38,190 KIMBERLY-CLARK CORP 43,513
34 AMERICAN INTERNATIONAL GROUP 60,353 ELECTRONIC ARTS INC 36,141 L3 TECHNOLOGIES INC 13,686
35 AMERICAN TOWER CORP 58,510 EMERSON ELECTRIC CO 38,417 LILLY (ELI) & CO 91,008
36 AMERICAN WATER WORKS CO INC 14,451 EQUINIX INC 35,117 MACERICH CO 8,137
37 AMERIPRISE FINANCIAL INC 21,775 EXELON CORP 35,507 MASTERCARD INC 134,085
38 AMERISOURCEBERGEN CORP 20,486 EXPRESS SCRIPTS HOLDING CO 36,178 METTLER-TOLEDO INTL INC 14,710
39 AMETEK INC 14,170 FIDELITY NATIONAL INFO SVCS 30,156 MOLSON COORS BREWING CO 17,559
40 AMGEN INC 127,336 FISERV INC 27,291 MYLAN NV 20,899
41 AMPHENOL CORP 23,416 FORD MOTOR CO 43,768 NORFOLK SOUTHERN CORP 32,444
42 ANADARKO PETROLEUM CORP 25,592 GENERAL MILLS INC 32,121 OMNICOM GROUP 18,170
43 ANALOG DEVICES 28,998 GENERAL MOTORS CO 52,430 PAYPAL HOLDINGS INC 70,400
44 ANSYS INC 11,076 HALLIBURTON CO 36,990 PHILIP MORRIS INTERNATIONAL 181,273
45 ANTHEM INC 49,823 HCA HEALTHCARE INC 29,477 PRICE (T. ROWE) GROUP 19,882
46 AON PLC 36,210 HEWLETT PACKARD ENTERPRISE 28,762 PUBLIC STORAGE 35,770
47 APACHE CORP 18,824 HP INC 32,152 RANGE RESOURCES CORP 5,227
48 APARTMENT INVST & MGMT CO 7,120 HUMANA INC 33,366 REPUBLIC SERVICES INC 21,627
49 APPLE INC 775,454 ILLINOIS TOOL WORKS 48,608 S&P GLOBAL INC 39,473
50 APPLIED MATERIALS INC 47,617 INCYTE CORP 27,294 SEALED AIR CORP 8,520
51 ARCHER-DANIELS-MIDLAND CO 23,973 INTERCONTINENTAL EXCHANGE 39,497 SMUCKER (JM) CO 13,824
52 ARCONIC INC 10,867 INTUIT INC 35,156 STERICYCLE INC 6,572
53 ARTHUR J GALLAGHER & CO 10,592 INTUITIVE SURGICAL INC 34,857 TE CONNECTIVITY LTD 28,409
54 ASSURANT INC 5,768 JOHNSON CONTROLS INTL PLC 36,317 TORCHMARK CORP 9,238
55 AT&T INC 239,460 KIMBERLY-CLARK CORP 43,513 U S BANCORP 88,618
56 AUTODESK INC 24,417 KINDER MORGAN INC 45,625 UNITED TECHNOLOGIES CORP 94,711
57 AUTOMATIC DATA PROCESSING 53,202 LAM RESEARCH CORP 25,788 VERISK ANALYTICS INC 14,428
58 AUTONATION INC 4,291 LYONDELLBASELL INDUSTRIES NV 35,658 WESTERN UNION CO 9,307
59 AUTOZONE INC 15,132 MARATHON PETROLEUM CORP 28,331 WYNN RESORTS LTD 13,240
60 AVALONBAY COMMUNITIES INC 26,503 MARRIOTT INTL INC 39,477 ZIONS BANCORPORATION 9,161
61 AVERY DENNISON CORP 8,230 MARSH & MCLENNAN COS 39,962
62 BAKER HUGHES A GE CO 15,790 MCKESSON CORP 34,035
63 BALL CORP 14,713 MICRON TECHNOLOGY INC 31,328
64 BANK OF AMERICA CORP 238,260 MONSANTO CO 51,322
65 BANK OF NEW YORK MELLON CORP 54,788 MONSTER BEVERAGE CORP 29,952
66 BARD (C.R.) INC 23,299 NORFOLK SOUTHERN CORP 32,444
67 BAXTER INTERNATIONAL INC 32,881 NORTHROP GRUMMAN CORP 45,824
68 BB&T CORP 38,239 OCCIDENTAL PETROLEUM CORP 47,351
69 BECTON DICKINSON & CO 45,789 PG&E CORP 34,713
70 BERKSHIRE HATHAWAY 431,906 PHILLIPS 66 43,223
71 BEST BUY CO INC 17,791 PIONEER NATURAL RESOURCES CO 27,742
72 BIOGEN INC 61,229 PPG INDUSTRIES INC 26,995
73 BLACKROCK INC 69,186 PPL CORP 26,186
74 BLOCK H & R INC 6,319 PRAXAIR INC 37,229
75 BOEING CO 143,314 PROGRESSIVE CORP-OHIO 27,383
76 BORGWARNER INC 9,865 PROLOGIS INC 32,343
77 BOSTON PROPERTIES INC 18,603 PRUDENTIAL FINANCIAL INC 48,576
78 BOSTON SCIENTIFIC CORP 36,453 PUBLIC STORAGE 35,770
79 BRISTOL-MYERS SQUIBB CO 93,312 RAYTHEON CO 49,856
80 BROADCOM LTD 100,290 REGENERON PHARMACEUTICALS 51,315
81 BROWN FORMAN CORP 19,344 S&P GLOBAL INC 39,473
82 C H ROBINSON WORLDWIDE INC 9,242 SEMPRA ENERGY 28,346
83 CA INC 13,075 SHERWIN-WILLIAMS CO 31,504
84 CABOT OIL & GAS CORP 11,502 SIMON PROPERTY GROUP INC 49,483
85 CAMPBELL SOUP CO 16,011 SOUTHERN CO 47,671
86 CAPITAL ONE FINANCIAL CORP 41,685 SOUTHWEST AIRLINES 33,565
87 CARDINAL HEALTH INC 24,405 STATE STREET CORP 35,076
88 CARMAX INC 12,140 SUNTRUST BANKS INC 27,593
89 CARNIVAL CORP/PLC (USA) 35,783 SYSCO CORP 28,162
90 CATERPILLAR INC 67,127 TARGET CORP 31,265
91 CBOE HOLDINGS INC 10,591 TE CONNECTIVITY LTD 28,409
92 CBRE GROUP INC 12,836 TJX COMPANIES INC 45,229
93 CBS CORP 26,734 TRAVELERS COS INC 35,346
94 CELGENE CORP 105,938 VALERO ENERGY CORP 30,846
95 CENTENE CORP 13,698 VERTEX PHARMACEUTICALS INC 38,277
96 CENTERPOINT ENERGY INC 12,149 WASTE MANAGEMENT INC 33,069
97 CENTURYLINK INC 12,771 WELLTOWER INC 27,072
98 CERNER CORP 21,344 WILLIAMS COS INC 26,259
99 CF INDUSTRIES HOLDINGS INC 6,844 YUM BRANDS INC 26,284
100 CHARTER COMMUNICATIONS INC 101,067 ZOETIS INC 30,685
101 CHESAPEAKE ENERGY CORP 4,504
102 CHEVRON CORP 206,867
103 CHIPOTLE MEXICAN GRILL INC 9,802
104 CHUBB LTD 68,386
105 CHURCH & DWIGHT INC 13,298
106 CIGNA CORP 44,434
107 CIMAREX ENERGY CO 9,419
108 CINCINNATI FINANCIAL CORP 12,543
109 CINTAS CORP 14,213
110 CISCO SYSTEMS INC 157,252
111 CITIGROUP INC 186,526
112 CITIZENS FINANCIAL GROUP INC 17,770
113 CITRIX SYSTEMS INC 11,938
114 CLOROX CO/DE 17,193
115 CME GROUP INC 41,665
116 CMS ENERGY CORP 13,040
117 COACH INC 13,252
118 COCA-COLA CO 195,522
119 COGNIZANT TECH SOLUTIONS 40,829
120 COLGATE-PALMOLIVE CO 63,597
121 COMCAST CORP 190,274
122 COMERICA INC 12,727
123 CONAGRA BRANDS INC 14,264
124 CONCHO RESOURCES INC 19,333
125 CONOCOPHILLIPS 55,213
126 CONSOLIDATED EDISON INC 25,304
127 CONSTELLATION BRANDS -CL A 37,862
128 COOPER COMPANIES INC 11,915
129 CORNING INC 26,319
130 COSTCO WHOLESALE CORP 69,521
131 COTY INC 15,312
132 CROWN CASTLE INTL CORP 40,862
133 CSRA INC 5,339
134 CSX CORP 45,063
135 CUMMINS INC 28,203
136 CVS HEALTH CORP 81,432
137 D R HORTON INC 13,358
138 DANAHER CORP 56,610
139 DARDEN RESTAURANTS INC 10,520
140 DAVITA INC 12,606
141 DEERE & CO 41,032
142 DELPHI AUTOMOTIVE PLC 24,223
143 DELTA AIR LINES INC 35,738
144 DENTSPLY SIRONA INC 14,219
145 DEVON ENERGY CORP 17,511
146 DIGITAL REALTY TRUST INC 18,706
147 DISCOVER FINANCIAL SVCS 22,853
148 DISCOVERY COMMUNICATIONS INC 9,152
149 DISH NETWORK CORP 14,565
150 DISNEY (WALT) CO 172,027
151 DOLLAR GENERAL CORP 20,611
152 DOLLAR TREE INC 17,061
153 DOMINION ENERGY INC 48,545
154 DOVER CORP 13,082
155 DOW CHEMICAL 78,594
156 DR PEPPER SNAPPLE GROUP INC 16,566
157 DTE ENERGY CO 19,206
158 DU PONT (E I) DE NEMOURS 71,342
159 DUKE ENERGY CORP 59,574
160 DUKE REALTY CORP 10,166
161 DXC TECHNOLOGY COMPANY 22,243
162 E TRADE FINANCIAL CORP 11,277
163 EASTMAN CHEMICAL CO 12,050
164 EATON CORP PLC 35,001
165 EBAY INC 38,242
166 ECOLAB INC 38,190
167 EDISON INTERNATIONAL 25,635
168 EDWARDS LIFESCIENCES CORP 24,321
169 ELECTRONIC ARTS INC 36,141
170 EMERSON ELECTRIC CO 38,417
171 ENTERGY CORP 13,769
172 ENVISION HEALTHCARE CORP 6,632
173 EOG RESOURCES INC 54,921
174 EQT CORP 11,041
175 EQUIFAX INC 17,507
176 EQUINIX INC 35,117
177 EQUITY RESIDENTIAL 24,998
178 ESSEX PROPERTY TRUST 17,167
179 EVEREST RE GROUP LTD 10,775
180 EVERSOURCE ENERGY 19,264
181 EXELON CORP 35,507
182 EXPEDIA INC 21,724
183 EXPEDITORS INTL WASH INC 10,608
184 EXPRESS SCRIPTS HOLDING CO 36,178
185 EXTRA SPACE STORAGE INC 10,014
186 EXXON MOBIL CORP 339,129
187 F5 NETWORKS INC 7,673
188 FACEBOOK INC 401,179
189 FASTENAL CO 12,372
190 FEDERAL REALTY INVESTMENT TR 9,581
191 FEDEX CORP 55,806
192 FIDELITY NATIONAL INFO SVCS 30,156
193 FIFTH THIRD BANCORP 20,032
194 FIRSTENERGY CORP 14,178
195 FISERV INC 27,291
196 FLIR SYSTEMS INC 5,120
197 FLOWSERVE CORP 5,372
198 FLUOR CORP 6,071
199 FMC CORP 10,235
200 FOOT LOCKER INC 6,196
201 FORD MOTOR CO 43,768
202 FORTIVE CORP 22,471
203 FORTUNE BRANDS HOME & SECUR 10,103
204 FRANKLIN RESOURCES INC 24,974
205 FREEPORT-MCMORAN INC 21,152
206 GAP INC 9,431
207 GARMIN LTD 9,440
208 GARTNER INC 11,604
209 GENERAL DYNAMICS CORP 58,793
210 GENERAL ELECTRIC CO 221,730
211 GENERAL MILLS INC 32,121
212 GENERAL MOTORS CO 52,430
213 GENUINE PARTS CO 12,470
214 GGP INC 19,956
215 GILEAD SCIENCES INC 99,374
216 GLOBAL PAYMENTS INC 14,390
217 GOLDMAN SACHS GROUP INC 88,697
218 GOODYEAR TIRE & RUBBER CO 7,933
219 GRAINGER (W W) INC 9,619
220 HALLIBURTON CO 36,990
221 HANESBRANDS INC 8,346
222 HARLEY-DAVIDSON INC 8,519
223 HARRIS CORP 13,955
224 HARTFORD FINANCIAL SERVICES 20,037
225 HASBRO INC 13,235
226 HCA HEALTHCARE INC 29,477
227 HCP INC 14,830
228 HELMERICH & PAYNE 5,496
229 HERSHEY CO 15,989
230 HESS CORP 14,160
231 HEWLETT PACKARD ENTERPRISE 28,762
232 HILTON WORLDWIDE HOLDINGS 20,273
233 HOLOGIC INC 12,379
234 HOME DEPOT INC 178,855
235 HONEYWELL INTERNATIONAL INC 103,529
236 HORMEL FOODS CORP 18,061
237 HOST HOTELS & RESORTS INC 13,809
238 HP INC 32,152
239 HUMANA INC 33,366
240 HUNT (JB) TRANSPRT SVCS INC 9,927
241 HUNTINGTON BANCSHARES 14,443
242 IDEXX LABS INC 14,669
243 IHS MARKIT LTD 18,623
244 ILLINOIS TOOL WORKS 48,608
245 ILLUMINA INC 25,382
246 INCYTE CORP 27,294
247 INGERSOLL-RAND PLC 22,292
248 INTEL CORP 166,674
249 INTERCONTINENTAL EXCHANGE 39,497
250 INTERPUBLIC GROUP OF COS 8,499
251 INTL BUSINESS MACHINES CORP 134,824
252 INTL FLAVORS & FRAGRANCES 10,517
253 INTL PAPER CO 22,701
254 INTUIT INC 35,156
255 INTUITIVE SURGICAL INC 34,857
256 INVESCO LTD 14,148
257 IRON MOUNTAIN INC 9,632
258 JACOBS ENGINEERING GROUP INC 6,349
259 JOHNSON & JOHNSON 357,526
260 JOHNSON CONTROLS INTL PLC 36,317
261 JPMORGAN CHASE & CO 326,147
262 JUNIPER NETWORKS INC 10,680
263 KANSAS CITY SOUTHERN 10,878
264 KELLOGG CO 23,819
265 KEYCORP 19,713
266 KIMBERLY-CLARK CORP 43,513
267 KIMCO REALTY CORP 8,589
268 KINDER MORGAN INC 45,625
269 KLA-TENCOR CORP 14,521
270 KOHL'S CORP 7,049
271 KRAFT HEINZ CO 106,494
272 KROGER CO 22,003
273 L BRANDS INC 13,307
274 L3 TECHNOLOGIES INC 13,686
275 LABORATORY CP OF AMER HLDGS 16,256
276 LAM RESEARCH CORP 25,788
277 LAUDER (ESTEE) COS INC -CL A 22,161
278 LEGGETT & PLATT INC 6,374
279 LENNAR CORP 12,047
280 LEUCADIA NATIONAL CORP 9,366
281 LEVEL 3 COMMUNICATIONS INC 21,206
282 LILLY (ELI) & CO 91,008
283 LINCOLN NATIONAL CORP 16,386
284 LKQ CORP 10,666
285 LOCKHEED MARTIN CORP 84,131
286 LOEWS CORP 16,386
287 LOWE'S COMPANIES INC 65,343
288 LYONDELLBASELL INDUSTRIES NV 35,658
289 M & T BANK CORP 24,887
290 MACERICH CO 8,137
291 MACY'S INC 7,232
292 MARATHON OIL CORP 10,395
293 MARATHON PETROLEUM CORP 28,331
294 MARRIOTT INTL INC 39,477
295 MARSH & MCLENNAN COS 39,962
296 MARTIN MARIETTA MATERIALS 14,182
297 MASCO CORP 12,148
298 MASTERCARD INC 134,085
299 MATTEL INC 6,858
300 MCCORMICK & CO INC 11,873
301 MCDONALD'S CORP 126,450
302 MCKESSON CORP 34,035
303 MEDTRONIC PLC 114,118
304 MERCK & CO 174,722
305 METLIFE INC 59,176
306 METTLER-TOLEDO INTL INC 14,710
307 MGM RESORTS INTERNATIONAL 18,935
308 MICHAEL KORS HOLDINGS LTD 5,678
309 MICROCHIP TECHNOLOGY INC 18,627
310 MICRON TECHNOLOGY INC 31,328
311 MICROSOFT CORP 560,372
312 MID-AMERICA APT CMNTYS INC 11,762
313 MOHAWK INDUSTRIES INC 18,505
314 MOLSON COORS BREWING CO 17,559
315 MONDELEZ INTERNATIONAL INC 66,791
316 MONSANTO CO 51,322
317 MONSTER BEVERAGE CORP 29,952
318 MOODY'S CORP 25,141
319 MORGAN STANLEY 86,296
320 MOSAIC CO 8,474
321 MOTOROLA SOLUTIONS INC 14,823
322 MYLAN NV 20,899
323 NASDAQ INC 12,284
324 NATIONAL OILWELL VARCO INC 12,431
325 NAVIENT CORP 4,112
326 NETAPP INC 11,799
327 NETFLIX INC 78,432
328 NEWELL BRANDS INC 25,469
329 NEWFIELD EXPLORATION CO 5,723
330 NEWMONT MINING CORP 19,822
331 NEWS CORP 8,404
332 NEXTERA ENERGY INC 68,550
333 NIELSEN HOLDINGS PLC 15,336
334 NIKE INC -CL B 77,482
335 NISOURCE INC 8,429
336 NOBLE ENERGY INC 12,472
337 NORDSTROM INC 8,065
338 NORFOLK SOUTHERN CORP 32,444
339 NORTHERN TRUST CORP 19,995
340 NORTHROP GRUMMAN CORP 45,824
341 NRG ENERGY INC 7,782
342 NUCOR CORP 18,400
343 NVIDIA CORP 96,693
344 O'REILLY AUTOMOTIVE INC 17,978
345 OCCIDENTAL PETROLEUM CORP 47,351
346 OMNICOM GROUP 18,170
347 ONEOK INC 21,493
348 ORACLE CORP 206,545
349 PACCAR INC 24,046
350 PACKAGING CORP OF AMERICA 10,314
351 PARKER-HANNIFIN CORP 22,106
352 PATTERSON COMPANIES INC 4,019
353 PAYCHEX INC 20,791
354 PAYPAL HOLDINGS INC 70,400
355 PENTAIR PLC 11,446
356 PEOPLE'S UNITED FINL INC 6,005
357 PEPSICO INC 166,229
358 PERKINELMER INC 7,244
359 PERRIGO CO PLC 10,743
360 PFIZER INC 197,894
361 PG&E CORP 34,713
362 PHILIP MORRIS INTERNATIONAL 181,273
363 PHILLIPS 66 43,223
364 PINNACLE WEST CAPITAL CORP 9,676
365 PIONEER NATURAL RESOURCES CO 27,742
366 PNC FINANCIAL SVCS GROUP INC 61,824
367 PPG INDUSTRIES INC 26,995
368 PPL CORP 26,186
369 PRAXAIR INC 37,229
370 PRICE (T. ROWE) GROUP 19,882
371 PRICELINE GROUP INC 99,685
372 PRINCIPAL FINANCIAL GRP INC 19,264
373 PROCTER & GAMBLE CO 232,283
374 PROGRESSIVE CORP-OHIO 27,383
375 PROLOGIS INC 32,343
376 PRUDENTIAL FINANCIAL INC 48,576
377 PUBLIC SERVICE ENTRP GRP INC 22,750
378 PUBLIC STORAGE 35,770
379 PULTEGROUP INC 7,368
380 PVH CORP 9,286
381 QORVO INC 8,720
382 QUALCOMM INC 78,512
383 QUANTA SERVICES INC 4,996
384 QUEST DIAGNOSTICS INC 14,774
385 RALPH LAUREN CORP 4,189
386 RANGE RESOURCES CORP 5,227
387 RAYMOND JAMES FINANCIAL CORP 11,981
388 RAYTHEON CO 49,856
389 REALTY INCOME CORP 15,639
390 RED HAT INC 17,545
391 REGENCY CENTERS CORP 11,263
392 REGENERON PHARMACEUTICALS 51,315
393 REGIONS FINANCIAL CORP 17,505
394 REPUBLIC SERVICES INC 21,627
395 RESMED INC 10,937
396 ROBERT HALF INTL INC 5,755
397 ROCKWELL AUTOMATION 21,262
398 ROCKWELL COLLINS INC 17,309
399 ROPER TECHNOLOGIES INC 23,718
400 ROSS STORES INC 21,530
401 ROYAL CARIBBEAN CRUISES LTD 24,319
402 S&P GLOBAL INC 39,473
403 SALESFORCE.COM INC 64,589
404 SCANA CORP 9,200
405 SCHEIN (HENRY) INC 14,455
406 SCHLUMBERGER LTD 95,318
407 SCHWAB (CHARLES) CORP 57,363
408 SCRIPPS NETWORKS INTERACTIVE 8,383
409 SEAGATE TECHNOLOGY PLC 9,624
410 SEALED AIR CORP 8,520
411 SEMPRA ENERGY 28,346
412 SHERWIN-WILLIAMS CO 31,504
413 SIGNET JEWELERS LTD 4,184
414 SIMON PROPERTY GROUP INC 49,483
415 SKYWORKS SOLUTIONS INC 19,269
416 SL GREEN REALTY CORP 10,158
417 SMITH (A O) CORP 7,866
418 SMUCKER (JM) CO 13,824
419 SNAP-ON INC 8,879
420 SOUTHERN CO 47,671
421 SOUTHWEST AIRLINES 33,565
422 STANLEY BLACK & DECKER INC 21,547
423 STAPLES INC 6,666
424 STARBUCKS CORP 78,168
425 STATE STREET CORP 35,076
426 STERICYCLE INC 6,572
427 STRYKER CORP 55,025
428 SUNTRUST BANKS INC 27,593
429 SYMANTEC CORP 18,935
430 SYNCHRONY FINANCIAL 24,115
431 SYNOPSYS INC 11,509
432 SYSCO CORP 28,162
433 TARGET CORP 31,265
434 TE CONNECTIVITY LTD 28,409
435 TECHNIPFMC PLC 13,317
436 TEXAS INSTRUMENTS INC 81,081
437 TEXTRON INC 13,005
438 THERMO FISHER SCIENTIFIC INC 68,671
439 TIFFANY & CO 11,905
440 TIME WARNER INC 79,431
441 TJX COMPANIES INC 45,229
442 TORCHMARK CORP 9,238
443 TOTAL SYSTEM SERVICES INC 11,679
444 TRACTOR SUPPLY CO 7,215
445 TRANSDIGM GROUP INC 14,678
446 TRAVELERS COS INC 35,346
447 TRIPADVISOR INC 5,011
448 TWENTY-FIRST CENTURY FOX INC 53,533
449 TYSON FOODS INC -CL A 18,260
450 U S BANCORP 88,618
451 UDR INC 10,459
452 ULTA BEAUTY INC 15,583
453 UNDER ARMOUR INC 7,703
454 UNION PACIFIC CORP 82,408
455 UNITED CONTINENTAL HLDGS INC 20,590
456 UNITED PARCEL SERVICE INC 75,880
457 UNITED RENTALS INC 10,057
458 UNITED TECHNOLOGIES CORP 94,711
459 UNITEDHEALTH GROUP INC 184,840
460 UNIVERSAL HEALTH SVCS INC 9,912
461 UNUM GROUP 11,313
462 VALERO ENERGY CORP 30,846
463 VARIAN MEDICAL SYSTEMS INC 8,927
464 VENTAS INC 23,987
465 VERISIGN INC 10,109
466 VERISK ANALYTICS INC 14,428
467 VERIZON COMMUNICATIONS INC 197,424
468 VERTEX PHARMACEUTICALS INC 38,277
469 VF CORP 24,904
470 VIACOM INC 14,304
471 VISA INC 219,217
472 VORNADO REALTY TRUST 15,024
473 VULCAN MATERIALS CO 16,271
474 WAL-MART STORES INC 241,130
475 WALGREENS BOOTS ALLIANCE INC 86,325
476 WASTE MANAGEMENT INC 33,069
477 WATERS CORP 13,882
478 WEC ENERGY GROUP INC 19,872
479 WELLS FARGO & CO 267,920
480 WELLTOWER INC 27,072
481 WESTERN DIGITAL CORP 24,791
482 WESTERN UNION CO 9,307
483 WESTROCK CO 14,418
484 WEYERHAEUSER CO 24,862
485 WHIRLPOOL CORP 12,982
486 WHOLE FOODS MARKET INC 13,372
487 WILLIAMS COS INC 26,259
488 WILLIS TOWERS WATSON PLC 20,127
489 WYNDHAM WORLDWIDE CORP 10,891
490 WYNN RESORTS LTD 13,240
491 XCEL ENERGY INC 24,022
492 XEROX CORP 7,795
493 XILINX INC 15,727
494 XL GROUP LTD 11,477
495 XYLEM INC 10,195
496 YUM BRANDS INC 26,284
497 ZIMMER BIOMET HOLDINGS INC 24,463
498 ZIONS BANCORPORATION 9,161
499 ZOETIS INC 30,685

Big Stacks

Big Stacks
Required Sample Size

BowlingData

Mr. Consistency Mr. Unpredictable
1 219 177
2 223 269
3 217 188
4 231 256
5 202 200
6 227 277
7 204 171
8 248 256
9 202 229
10 222 169
11 231 226
12 228 183
13 221 222
14 207 255
15 216 177
16 225 222
17 227 171
18 209 279
19 227 200
20 219 175
21 224 165
22 209 228
23 227 265
24 221 171
25 212 199
26 217 255
27 224 228
28 228 219
29 228 265
30 217 187
31 203 277
32 216 235
33 228 188
34 191 255
35 221 209
36 218 220
37 235 255
38 192 216
39 217 179
40 219 209
41 219 288
42 225 177
43 232 239
44 212 205
45 211 217
46 232 255
47 231 213
48 231 245
49 198 199
50 217 198
51 221 175
52 216 233
53 221 279
54 234 216
55 218 210
56 231 244
57 233 222
58 209 237
59 238 223
60 219 198

Coat Sales

Sales by Sales Person
Name Region Sales
Ali Chicago 1560
Bailey Chicago 1280
Camden LA 690
Denton Chicago 1840
Erickson LA 770
Foreman LA 800
Gehrig Chicago 1330
Hacker LA 620
Ippolito Chicago 950
Jackson Chicago 1370
Kemp LA 860
Louis LA 660
McGwire LA 410
Norton Chicago 1750
Owens Chicago 1900
Pujols LA 770
Quisenberry LA 490
Robinson Chicago 2000
Sosa LA 250
Thomas Chicago 1400
Utley Chicago 1550
Van Slyke LA 790
Wainwright Chicago 1880
Xavier LA 680
Young Chicago 1490
Zimmerman LA 880

Two Variables (GPA and ACT +)

The Impact of ACT on GPA
The Impact of Absences on GPA
Average
Name ACT GPA ACT Absences GPA
Allen 25 3.21 25 0 3.21
Berglund 28 3.00 28 5 3.00
Crowder 19 3.05 19 0 3.05
Dennison 35 3.87 35 1 3.87
Edmundson 22 2.49 22 10 2.49
Fabbri 20 2.91 20 3 2.91
Granderson 31 3.81 31 0 3.81
Hunter 22 1.89 22 17 1.89
Isner 33 2.78 33 8 2.78
Jones 21 2.89 21 2 2.89
Klein 24 2.54 24 9 2.54
Langston 28 3.51 28 0 3.51
Monday 35 3.68 35 1 3.68
Nolan 20 3.01 20 0 3.01
O'Reilly 29 3.18 29 3 3.18
Prendergast 30 3.55 30 0 3.55
Quick 22 1.76 22 21 1.76
Racine 32 3.32 32 9 3.32
Steen 23 2.77 23 7 2.77
Unger 25 1.78 25 24 1.78
Victor 29 3.78 29 0 3.78
Wilson 22 2.61 22 9 2.61
Xavier 32 3.66 32 0 3.66
Young 26 3.28 26 1 3.28
Zander 29 3.19 29 3 3.19
Mean 26.48 3.02 26.48 5.32 3.02
Standard Deviation 4.91 0.60 4.91 6.81 0.60
Median 26 3.05 26 3 3.05
Mode 22 22 0

Bat Production Costs

Louistown Masher
Bat Production
Number of Total Cost per
Bats Produced Cost Bat
1 January 21,000 236,000 11.24
2 February 30,000 330,000 11.00
3 March 18,000 203,000 11.28
4 April 17,500 202,600 11.58
5 May 17,000 201,000 11.82
6 June 16,000 193,000 12.06
7 July 15,000 188,000 12.53
8 August 14,000 182,000 13.00
9 September 10,000 131,000 13.10
10 October 17,000 200,500 11.79
11 November 13,000 170,000 13.08
12 December 14,000 177,000 12.64
Total 202,500 2,414,100 11.92
Average per month 16,875 201,175 11.92
Median per month 16,500 196,750 11.92

CPA Exam Scores

Self Study In class
Student Control Student Treatment
1 76 A 81
2 80 B 79
3 70 C 86
4 83 D 91
5 77 E 75
6 84 F 84
7 81 G 79
8 75 H 87
9 88 I 94
10 86 J 79
11 77 K 70
12 79 L 92

Student Test Scores

Total Points (Out of 100)
Exam One Exam Two Exam One Exam Two Change Change
Top Top Bottom Bottom Top Bottom
Student Quartile Quartile Student Quartile Quartile Quartile Quartile
1 96 91 1 44 58 -5 14
2 93 94 2 38 58 1 20
3 88 92 3 38 70 4 32
4 90 99 4 50 53 9 3
5 90 100 5 35 60 10 25
6 88 99 6 47 57 11 10
7 97 95 7 39 60 -2 21
8 90 88 8 49 71 -2 22
9 99 99 9 37 53 0 16
10 93 94 10 40 63 1 23
11 88 97 11 45 71 9 26
12 95 100 12 46 54 5 8
13 92 93 13 41 69 1 28
14 90 93 14 42 62 3 20
15 95 100 15 46 61 5 15
16 93 98 16 48 58 5 10
17 98 91 17 46 66 -7 20
18 94 99 18 49 67 5 18
19 96 96 19 42 52 0 10
20 99 92 20 37 69 -7 32
21 99 90 21 35 66 -9 31
22 93 97 22 37 69 4 32
23 91 94 23 45 64 3 19
24 92 88 24 42 55 -4 13
25 92 96 25 42 66 4 24
26 96 97 26 49 60 1 11
27 93 92 27 42 51 -1 9
28 89 98 28 43 60 9 17
29 90 90 29 40 54 0 14
30 95 98 30 43 50 3 7
31 97 88 31 45 64 -9 19
32 100 88 32 46 60 -12 14
33 92 99 33 41 60 7 19
34 94 90 34 38 60 -4 22
35 98 88 35 36 65 -10 29
36 95 88 36 37 69 -7 32
37 89 89 37 35 74 0 39
38 93 95 38 48 73 2 25
39 97 93 39 36 63 -4 27
40 97 89 40 42 58 -8 16

Pharmacy

Pharmaceutical Experiment
Pills Taken (out of 28)
Sample Sample
Control Treatment
Without With
Subject Message Subject Message
1 20 31 26
2 21 32 25
3 25 33 28
4 20 34 22
5 19 35 28
6 22 36 28
7 25 37 26
8 28 38 27
9 20 39 25
10 18 40 26
11 25 41 28
12 28 42 28
13 17 43 28
14 16 44 26
15 21 45 25
16 26 46 27
17 20 47 28
18 19 48 23
19 22 49 26
20 25 50 28
21 25 51 27
22 21 52 28
23 22 53 28
24 20 54 27
25 22 55 25
26 21 56 28
27 20 57 27
28 20 58 28
29 28 59 28
30 23 60 28

ANOVA_Bowling

ANOVA Illustration Cardinal Northtown ShowTime
Ross 550 650 750 650.00
Tony 600 680 790 690.00
Pete 520 600 640 586.67
Mean 556.67 643.33 726.67 642.22

Regression (Celcius_Fahrenheit)

0 32
3 37
4 39
8 46
14 57
21 70
27 81
33 91
38 100
41 106
44 111
48 118
51 124
56 133
59 138
66 151
71 160
78 172
87 189
93 199
100 212

Regression_SalaryMonths

Name Salary Months of Service
Anderson 50,000 15
Bailey 87,000 55
Connor 91,000 57
Dalton 55,000 25
Eagan 79,000 49
Falstaff 94,000 62
Graham 62,000 35
Hemingway 105,000 77
Italiano 110,000 46
Justice 67,000 36
Kent 81,000 49
Lerner 67,000 36
Montgomery 53,000 21
Nelson 88,000 56
O'Hara 72,000 46
Pujols 134,000 108
Quentin 101,000 71
Roosevelt 75,000 46
Sanders 48,000 13
Thompson 131,000 59
Unnerstall 123,000 34
Vancil 100,000 70
Washington 91,000 57
Xavier 112,000 55
Yetton 107,000 66
Zoltek 66,000 36

Regression_Salary_Months_Degree

Name Salary Months of Service Master's Degree
Anderson 50,000 15 0
Bailey 87,000 55 0
Connor 91,000 57 0
Dalton 55,000 25 0
Eagan 79,000 49 0
Falstaff 94,000 62 0
Graham 62,000 35 0
Hemingway 105,000 77 0
Italiano 110,000 46 1
Justice 67,000 36 0
Kent 81,000 49 0
Lerner 67,000 36 0
Montgomery 53,000 21 0
Nelson 88,000 56 0
O'Hara 72,000 46 0
Pujols 134,000 108 1
Quentin 101,000 71 0
Roosevelt 75,000 46 0
Sanders 48,000 13 0
Thompson 131,000 59 1
Unnerstall 123,000 34 1
Vancil 100,000 70 0
Washington 91,000 57 0
Xavier 112,000 55 1
Yetton 107,000 66 1
Zoltek 66,000 36 0

MACC Starting Salaries

Student Starting
Name GPA Salary
Aronson 3.06 49,000
Boston 2.99 41,000
Cantoni 2.75 38,500
Denton 3.39 54,000
Evans 3.59 57,000
Fenton 2.46 31,000
Gardner 3.88 62,500
Houston 2.07 35,000
Islip 3.83 51,500
Jackson 4 58,000
Kirkwood 3.01 38,000
Louis 2.44 37,000
Montana 2.59 40,000
Nobles 2.77 45,000
Owens 2.3 32,000
Pratt 3.95 63,000
Quinton 3.25 55,000
Russell 3.1 41,000
Sanger 3.5 53,000

Decision Analysis

Student Hours Score Hours Worked on Harvard Course
1 2 18 Score on Exam 0 - 10 11 - 20 21 - 30 31 - 40 Over 40 Total
2 4 51 80 - 100 0 0 1 23 13 37
3 5 55 60 - 79 0 1 14 24 2 41
4 7 33 40 - 59 3 6 4 3 0 16
5 9 52 20 - 39 1 3 1 0 0 5
6 11 34 0 - 19 1 0 0 0 0 1
7 11 44 Total 5 10 20 50 15 100
8 12 48
9 14 47
10 16 29
11 18 58
12 18 77
13 18 57
14 19 57
15 20 38
16 22 37
17 22 58
18 23 56
19 23 57
20 23 57
21 24 71
22 25 78
23 25 66
24 25 65
25 25 88
26 26 71
27 26 77
28 26 70
29 26 74
30 27 75
31 27 78
32 27 72
33 27 77
34 28 67
35 28 78
36 32 79
37 32 58
38 32 77
39 33 71
40 33 75
41 33 57
42 33 77
43 34 58
44 34 88
45 34 86
46 34 77
47 35 79
48 35 78
49 35 92
50 35 75
51 35 93
52 35 76
53 35 72
54 35 73
55 35 99
56 36 97
57 36 78
58 36 71
59 36 70
60 37 78
61 37 74
62 37 90
63 37 94
64 37 88
65 37 75
66 37 77
67 37 71
68 37 91
69 38 92
70 38 79
71 38 96
72 38 83
73 38 94
74 38 100
75 38 96
76 38 77
77 38 78
78 38 78
79 38 100
80 38 87
81 39 89
82 39 95
83 39 98
84 39 97
85 39 99
86 42 79
87 42 95
88 43 100
89 44 99
90 45 81
91 46 89
92 46 86
93 47 83
94 48 97
95 49 100
96 51 92
97 52 88
98 54 93
99 66 97
100 85 78
Average 32.56 75.59
Median 35 77.5
Mode 38 78
Sorted on Score
Student Hours Score
74 38 100
79 38 100
88 43 100
95 49 100
55 35 99
85 39 99
89 44 99
83 39 98
56 36 97
84 39 97
94 48 97
99 66 97
71 38 96
75 38 96
82 39 95
87 42 95
63 37 94
73 38 94
51 35 93
98 54 93
49 35 92
69 38 92
96 51 92
68 37 91
62 37 90
81 39 89
91 46 89
25 25 88
44 34 88
64 37 88
97 52 88
80 38 87
45 34 86
92 46 86
72 38 83
93 47 83
90 45 81
36 32 79
47 35 79
70 38 79
86 42 79
22 25 78
31 27 78
35 28 78
48 35 78
57 36 78
60 37 78
77 38 78
78 38 78
100 85 78
12 18 77
27 26 77
33 27 77
38 32 77
42 33 77
46 34 77
66 37 77
76 38 77
52 35 76
30 27 75
40 33 75
50 35 75
65 37 75
29 26 74
61 37 74
54 35 73
32 27 72
53 35 72
21 24 71
26 26 71
39 33 71
58 36 71
67 37 71
28 26 70
59 36 70
34 28 67
23 25 66
24 25 65
11 18 58
17 22 58
37 32 58
43 34 58
13 18 57
14 19 57
19 23 57
20 23 57
41 33 57
18 23 56
3 5 55
5 9 52
2 4 51
8 12 48
9 14 47
7 11 44
15 20 38
16 22 37
6 11 34
4 7 33
10 16 29
1 2 18

LP_Max_Gateway

Gateway City Chair Company
(Allocation Problem)
Parameters
Time Requirements
Hours Required per Unit Time Available
Club Chairs (C) Recliners (R) Gliders (G) (Hours)
Fabrication 2000
Assembly 2400
Shipping 1600
Demand/Output Potential
C R G
Profit per Unit Sold 17 27 24
Decision Variables
Product Mix: C R G
Objective Function
Total Profit:
Constraints
i. Constraints on Time
Hours Used per Unit Hours Used Hours Available
Club Chairs (C) Recliners (R) Gliders (G)
Fabrication <=
Assembly <=
Shipping <=
ii. Constraints on Demand/Outputs
Total Sales/Ouptut Cap on Sales/Output
Club Chairs (C): <=
Recliners (R): <=
Gliders (G): <=

LP_Min_XY

X&Y Industries
(Cost Minimization Problem — X&Y Under Contract)
Parameters
Time Requirements
Time Available
X Y (Hours)
Processing Time (per unit)
Minimum Production Requirements
Minimum Production of Y:
Minimum Total Production:
Minimum Production of X:
Cost per Unit Produced 4 3
Decision Variables
Product Mix: X Y
Objective Function
Total Cost:
Constraints
i. Constraints on Production
Production Production Requirement
Product Y >=
Total >=
Output of X Produced
Non-negative Production of X >=
ii. Constraint on Time
Hours Used Hours Available
<=

Optimization

Linear Programming

Profit Maximization

Gateway Chair Company

Club Chairs (C)

Recliners (R)

Gliders (G)

Total Profit

→ $17 profit per unit

→ $27 profit per unit

→ $24 profit per unit

Profit Maximization

3 departments

Only variable input is time

Time is limited in each department

Upper bound on demand for each product

Profit Maximization

Hours Required per Unit Time Available
Department Chairs (C) Recliners (R) Gliders (G) (Hours)
Fabrication 4 6 5 2000
Assembly 3 7 7 2400
Shipping 2 5 2 1600
C Production R Production G Production
Demand Potential (Production Limit) 300 280 250

Profit Maximization

Which combination (,,) profit?

optimization problem.

maximizes

Constrained

Profit Maximization

Constrained optimization: find the best set of decisions, according to some performance measure, subject to satisfying the constraints of the problem.

Structure:

Decision variables

Objective function

Constraints

Solution:

Optimal decisions

Sensitivity analysis (shadow prices)

Setup

Most of the work is in setting up the problem.

Decision variables: Controllable inputs in an optimization model.

This Example: and and

Setup

Objective function: A function of the decision variables which is to be maximized or minimized.

This Example: total profit from selling , and

Setup

Constraints: Restrictions which limit the values of the decision variables.

This Example: we have time constraints and demand (or production) constraints

Setup

Constraints: Restrictions which limit the values of the decision variables.

Time constraints:

Hours Available
Fabrication:
Assembly:
Shipping:

Setup

Constraints: Restrictions which limit the values of the decision variables.

Demand potential constraints:

Upper Bound
Club Chairs:
Recliners:
Gliders:

Solution

Use Excel Solver to get solution (see beginning of Course Packet).

In this example the objective and all constraints are linear functions of the decisions variables (,,).

→ Linear Programming Problem

Solution

Solve for (). Get the maximized profit.

But we also want to perform a sensitivity analysis: study of how changes in the parameters of the model affect the optimal solution.

→ Shadow Price (for a constraint): the change in the optimal objective function value for a one unit increase in the right-hand side of a constraint.

Shadow Price

Say we have solved for (). Profit is maximized.

Will total maximized profit ↑ ?

Upper Bound
Club Chairs:
Recliners:
Gliders:

RHS

+ 1

By how much?

Shadow Price

A non-binding constraint has a shadow price = 0.

Make a binding constraint less restrictive → makes objective no worse. Usually makes it better.

Make a binding constraint more restrictive → makes objective no better. Usually makes it worse.

Optimization Contd.

Minimization Problem

Cost Minimization

X&Y Industries

Contract

Output

Output

Produce at least 226

Produce at least 325 of either or

1150 hours available

Total Cost

→ 2 hours per unit, cost per unit

→ 4 hours per unit, cost per unit

Cost Minimization

Minimize cost of fulfilling contract and satisfying time constraint. Find ().

Non-negativity constraints: .

Can’t minimize cost by .

Structure: decision variables, objective function, constraints.

Another constrained optimization problem.

Setup

Decision Variables:

and

Setup

Objective function:

Total cost from producing and

Setup

Constraints on production levels, time and non-negativity constraints on .

Production constraints:

Production
Minimum : 0
Minimum total: Y

Setup

Constraints on production levels, time and non-negativity constraints on .

Time constraint:

Time
Time Constraint:

Setup

Constraints on production levels, time and non-negativity constraints on .

Non-negativity constraints:

Production
Minimum : 0
Minimum : Y

superfluous

Solution

Again, we use Excel Solver to get solution.

Select min this time in the solver.

Enter the correct inequalities in the constraints list.

Recall

Solve for (). Get the minimized cost.

But we also want to perform a sensitivity analysis.

→ Shadow Prices for the constraints

Shadow Prices

A non-binding constraint has a shadow price = 0.

Make a binding constraint less restrictive → makes objective no worse. Usually makes it better.

Make a binding constraint more restrictive → makes objective no better. Usually makes it worse.

Shadow Prices

Say we have solved for (). Cost is minimized.

Will total minimized cost change?

Min :
Min output:
Time:

RHS

+ 1

By how much?

Non-neg :

+ 1