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Chapter Nine: Multicriteria Decision Making
PROBLEM SUMMARY
1. Model formulation, product mix 2. Model formulation, transportation, computer solution 3. Model formulation, urban recreation facility allocation 4. Model formulation, crop determination, computer solution 5. Model formulation, product mix, computer solution 6. Model formulation, OSHA safety compliance, computer solution 7. Computer solution; graphical solution 8. Computer solution; graphical solution 9. Computer solution 10. Model formulation, computer solution 11. Model formulation, product mix, computer solution 12. Model formulation, product mix, computer solution 13. Model formulation, clinic personnel selection, computer solution 14. Model formulation, production scheduling, computer solution, sensitivity analysis 15. Model formulation, employee scheduling, computer solution 16. Model formulation, R&D project selection 17. AHP, company takeover 18. Pairwise comparison (9–17) 19. AHP, faculty raises 20. Pairwise comparisons (9–19) 21. AHP, mutual funds 22. AHP (9–21) 23. AHP, utility vehicles 24. AHP, anchor persons 25. AHP, hotel selection 26. AHP, college selection 27. AHP, dating service 28. AHP, R&D projects 29. AHP, student selection 30. AHP, athletic facilities 136 2. a) 31. AHP, vacation locations 32. Pairwise comparisons (9–31) 33. AHP, major options 34. AHP, basketball players 35. AHP, school facilities 36. Student’s pairwise comparisons 37. AHP, emergency rooms 38. AHP, class sections 39. Student car selection 40. Scoring model, plant site selection 41. Scoring model, site selection 42. Scoring model, Asian distribution center 43. Scoring model, campus student center 44. Scoring model, college selection 45. Scoring model, time-share condominium 46. Scoring model, condominium selection 47. Scoring model, soccer tournament 48. Scoring model, student’s restaurant selection model

PROBLEM SOLUTIONS
1.

b)

The noninteger solution values were rounded to integer values (making sure no rim requirements were violated.) 3. a)

– – b) The goal constraint d3 + d7 d8 = 100 must be – + converted to 10x + 12x2 + 18x3 + d7 d7 = 900 for QM for Windows input. The solution is x1 = 25, d 1– = 1,975, d 3– = 550, d 4– = 775, d 5– = 900, – d 6– = 1,100, d 7 = 650. P 1 and P 2 are achieved; P3 (6,000), P4 (1,975), and P5 (550) are not achieved.

+

6. a)

b) x1 d2 d7 4. a)

6, x2 5, x3 x4 0, 4,000, d3 14, d4 4, d5 8, d8 12

1, d6

5, b)

7.

b)

5. a)

x1 20, d2 satisfied

10, d3

50

137

8.

b) x1 = 30 x2 = 15 Goal achievement: P1: achieved P2: achieved – P3: not achieved, d3 = 5 – P4: not achieved, d4 = 6
+ + + + – – – – 14. a) minimize P1(d1 , d2 , d3 , d4 ), P2(d5 , d6 , d7 , d8 ), – + P3d9 , P3d10 subject to
– + .06x1 + .17x2 + .10x3 + .14x4 + d1 – d1 = 700 – + .18x1 + .20x2 + .14x4 + d2 – d2 = 700 – + .07x1 + .20x2 + .08x3 + .12x4 + d3 – d3 = 800 – + .09x1 + .12x2 + .07x3 + .15x4 + d4 – d4 = 600 – + x1 + d5 – d5 = 2,600 – + x2 + d6 – d6 = 1,800 – + x3 + d7 – d7 = 4,100 – + x4 + d8 – d8 = 1,200 – + 90x1 + 100x2 + 80x3 + 120x4 + d9 – d9 = 700,000 – + 2.6x1 + 1.4x2 + 2.5x3 + 3.2x4 + d10– d10 = 15,000

x1 500, x2 satisfied 9. 10. a) x1 15, d1 satisfied

300, d3 12, d2

500 10, d4 6

b) x1 40, x2 satisfied 11.a)

50, d1

140, d4

120

b) x1 60, x2 satisfied 12. a)

30, d1

10, d3

5

b) x1 = 2,600 x2 = 0 x3 = 0 x4 = 1657.14 Goal achievement: P1: achieved; no overtime P2: not achieved; demand not met for parts 2 – (d6 = 1800) and 4 (d7– = 4100) P3: not achieved; profit goal not met by – $72,166.67 (d9 = 267,143) c) x1 = 2,600 x2 = 1,800 x3 = 4,100 x4 = 1,200 Goal achievement: P1: achieved, demand met P2: not achieved; overtime required for all + + + processes (d1 = 340, d2 = 296, d3 = + 214, d4 = 317) P3: achieved, profit goal met + P4: not achieved; 8,370 ft2 (d10 = 8,370) extra material required.

b)

The solution is x1 = 500,x2 = 317.5, + – – d1 = 480, d2 = 113.125, d4 = 52.5.

13. a)

138

+ – – – – – 15. a) minimize P1d1 , P2(d2 , d8 ), P3(d7 ), P4(d3 , d4 , – – d5 , d6 )

+ + + – – – + 16. a) minimize P1d1 , P2 d2 , P3 d3 , P4(d4 , d5 ), P5 d6 , – P6 d7

subject to – + x1 + x2 + x3 + x4 + x5 + x6 + x7 + d1 – d1 = 60 – + x1 + x4 + x5 + x6 + x7 + d2 – d2 = 47 – x1 + x2 + x5 + x6 + x7 + d3 – d3+ = 22 – + x1 + x2 + x3 + x6 + x7 + d4 – d4 = 28 – + x1 + x2 + x3 + x4 + x7 + d5 – d5 = 35 + x1 + x2 + x3 + x4 + x5 + d6– – d6 = 34 – + x2 + x3 + x4 + x5 + x6 + d7 – d7 = 43 – + x3 + x4 + x5 + x6 + x7 + d8 – d8 = 53 – xi, d i , di+ ≥ 0 b) Employees beginning work on day i = xi x1 = 0 – Sunday x2 = 7 – Monday x3 = 0 – Tuesday x4 = 27 – Wednesday x5 = 0 – Thursday x6 = 25 – Friday x7 = 1 – Saturday Goal achievement: P1: achieved P2: achieved P3: achieved P4: achieved 17. Normalized Matrices:

subject to .675x1 + 1.05x2+ .725x3 + .430x4 + 1.24x5 + – + .890x6 + 1.620x7 + 1.20x8 + d1 – d1 = 5.0 6x1 + 5x2+ 7x3 + 8x4 + 10x5 + 6x6 + 7x7 + 6x8 – + + d2 – d2 = 27 .820x1 + 1.75x2 + 1.60x3 + 1.90x4 + .930x5 + – + 1.70x6 + 1.30x7 + 1.80x8 + d3 – d3 = 6.5 – + x1 + x3 + x4 + x6 + d4 – d4 = 2 – + x2 + x5 + x7 + x8 + d5 – d5 = 2 – + x2 + x3 + x5 + x6 + x7 + d6 – d6 = 3 – + x5 + x6 + d7 – d7 = 2 b) Solving sequentially using Excel: x2 = 1 x4 = 1 x6 = 1 x7 = 1
– d1 = 1.01 – d2 = 1 d3+ = 0.15 – d7 = 2

All goals are achieved except the priority 6 – goal; d1 = 1, meaning only one of the two projects was selected.

Profitability Company A B C A .2414 .7241 .0345 B .2308 .6923 .0769 C .4118 .5294 .0588 Row Averages .2946 .6486 .0567 1.0000

Growth Potential Company A B C A .1250 .2500 .6250 B .1111 .2222 .6667 C .1304 .2174 .6522 Row Averages .1222 .2299 .6479 1.0000

Criteria Profit Growth

Profit .2500 .7500

Growth .2500 .7500

Row Averages .2500 .7500 1.0000

139

A B C

Profit Growth Company .2946 .1222 A .6486 .2299 × Profit .25 = B .0567 .6479 Growth .75     C

Score .1653 .3346 .5001 1.0000

18. Profitability: (1) (2) (3)

 1 1/ 3 7 .2946  .9077  3 1 9 × .6486 = 2.0427      1/ 7 1/ 9 1  .0567  .1709 (3)/(2) 3.0811 3.1494   3.0133 9.2438 9.2438 = 3.081, CI = 3.081− 3 = 0.41 3 2 CI / RI = .041 = .07 .58

Since .07 < .10 this preference comparison is consistent. Growth: (1) (2) (3)

1 1/ 2 1/ 5 .1222  .3667 2 1 1/ 3 × .2299 =  .6003       1 5 3 .6479 1.9486 (3)/(2) 3.0011 3.0025 3.0076   9.0111 9.0111 = 3.004, CI = 3.004 − 3 = .0018 3 2 CI / RI = .0018 /.58 = .003

Since .003 < .10 this preference is consistent. Since there are only two preferences for the criteria matrix and they are exact reciprocals this preference comparison is perfectly consistent. 19. Normalized matrices: Teaching Faculty A B C A .2222 .1111 .6667 B .2500 .1250 .6250 C .2174 .1304 .6522 Row Averages .2299 .1222 .6479

Research Faculty A B C A .3000 .1000 .6000 B .6000 .2000 .2000 140 C .2000 .4000 .4000 Row Averages .3667 .2333 .4000

Service Faculty A B C A .6667 .2222 .1111 B .6667 .2222 .1111 C .6667 .2222 .1111 Row Averages .6667 .2222 .1111

Criteria Teaching Research Service

Teaching .6522 .2174 .1304
Criteria

Research .6667 .2222 .1111
Faculty Score

Service .6250 .2500 .1250

Row Averages .6479 .2299 .1222

Teaching Research Service

A B C

.2299 .1222 .6479 

.3667 .2333 .4000

.6667 Abbott .3417 .6479 .2222 × .2299 = Bates .1600 .1222 .1111 Cook .5253    1.0000

20. Criteria: (1) (2) (3) (3)/(2) 9.0111 = 3.004, CI = 3.004 − 3 = .0018 3 2 CI / RI = .0018 = .003 .58

1.9486  3.0076 .6479  1 3 5 1/ 3 1 2 × .2299 =  .6903 3.0025        .3667 3.0011 .1222 1/ 5 1/ 2 1   9.0111 Since .003

.10 this preference comparison is consistent.

21. Normalized matrices: Return Fund Global Blue Chip Bond Global .1818 .7273 .0909 Blue Chip .1765 .7059 .1176 Bond .2222 .6667 .1111 Row Averages .1935 .6999 .1066

Risk Fund Global Blue Chip Bond Global .222 .111 .667 Blue Chip .2500 .1250 .6250 Bond .23 .12 .65 Row Averages 0.23 0.12 0.65

141

Load Fund Global Blue Chip Bond Global .2000 .2000 .6000 Blue Chip .2000 .2000 .6000 Bond .2000 .2000 .6000 Row Averages .2000 .2000 .6000

Criteria Return Risk Load
Return Global Blue Chip Bond Risk

Return .6522 .2174 .1304
Load

Risk .6667 .2222 .1111
Fund Global .6479  .2299  = Blue Chip .1222  Bond   Criteria Score

Load .6250 .2500 .1250

Row Averages .6479 .2299 .1222

.1935 .6999 .1066 

.23 .12 .65

.2000 .2000 × .6000 

.203 .506 .291 1.0000

Megan should invest in the Blue Chip Fund. 22. Global Blue Chip Bond 85,000(.203) 85,000(.506) 85,000(.291) $17,225.50 $43,014.50 $24,760.00 $85,000.00

23.

Normalized matrices: Consumer Digest Vehicle Explorer Trooper Passport Explorer .6316 .1579 .2105 Trooper .5714 .1429 .2857 Passport .6667 .1111 .2222 Row Averages .6232 .1373 .2395

Price Vehicle Explorer Trooper Passport Explorer .0909 .3636 .5455 Trooper .1429 .5714 .2857 Passport .0526 .6316 .3158 Row Averages .0955 .5222 .3823

Appearance Vehicle Explorer Trooper Passport Explorer .6316 .1579 .2105 Trooper .5714 .1429 .2857 Passport .6667 .1111 .2222 Row Averages .6232 .1373 .2395

142

Criteria Consumer Digest Price Appearance Consumer Digest Explorer  .6232 Trooper  .1373  Passport  .2395

Consumer Digest .5714 .2857 .1429

Price .6000 .3000 .1000

Appearance .5000 .3750 .1250 Score .4542 .2605 .2852 1.0000

Row Averages .5571 .3202 .1226

Vehicle Criteria Price Appearance .0955 .6232  Explorer .5571  .5222 .1373  × .3202  = Trooper  .3823 .2395  Passport  .1226  

Alex should purchase the Explorer. 24. Normalized matrices: Appearance Anchor Pawlie Cooric Brokenaw Pawlie .6087 .3043 .0870 Cooric .6250 .3125 .0625 Intelligence Anchor Pawlie Cooric Brokenaw Pawlie .1250 .3750 .5000 Cooric .1000 .3000 .6000 Speech Anchor Pawlie Cooric Brokenaw Pawlie .2222 .6667 .1111 Cooric .2222 .6667 .1111 Brokenaw .2222 .6667 .1111 Row Averages .2222 .6667 .1111 Row Averages .6571 .0683 .2746 Brokenaw .1429 .2857 .5714 Row Averages .1226 .3202 .5571 Brokenaw .5385 .3846 .0769 Row Averages .5907 .3338 .0755

Criteria Appearance Intelligence Speech

Appearance .6857 .0857 .2286

Intelligence .5714 .0714 .3571

Speech .7143 .0476 .2381

Appearance Intelligence Speech

Criteria

Anchor

Score

Pawlie Cooric Brokenaw

.5907 .3338 .0755 

.1226 .3202 .5571

.2222 Pawlie .4575 .6571 Cooric .4243 .6667 × .0683 = .2746 Brokenaw .1182 .1111    1.0000

143

25.

Normalized matrices: Ambiance Hotel Cheraton Milton Harriott Cheraton .1250 .2500 .6250 Milton .1111 .2222 .6667 Harriott .1304 .2174 .6522 Row Averages .1222 .2229 .6479

Location Hotel Cheraton Milton Harriott Cheraton .6522 .1304 .2174 Milton .5000 .1000 .4000 Harriott .7059 .0588 .2353 Row Averages .6194 .0964 .2842

Cost Hotel Cheraton Milton Harriott Cheraton .5882 .2941 .1176 Milton .5714 .2857 .1429 Harriott .6250 .2500 .1250 Row Averages .5949 .2766 .1285

Criteria Ambiance Location Cost Ambiance  .1222  .2299  .6479 

Ambiance .5714 .2857 .1429 Location .6194 .0964 .2842

Location .6000 .3000 .1000 Score .3393 .1929 .4677 1.0000

Cost .5000 .3750 .1250

Row Averages .5571 .3202 .1226

Cheraton Milton Harriott

Hotel Cost Criteria .5949  Cheraton  .5571  .2766  ×  .3202  = Milton  .1226  Harriott .1285    

The Harriott Hotel should be selected for the meeting. 26. Normalized matrices: Academic College A B C A .3000 .6000 .1000 B .2857 .5714 .1429 C .3750 .5000 .1250 Row Averages .3202 .5571 .1226

144

Location College A B C A .4286 .4286 .1429 B .4545 .4545 .0909 Cost College A B C A .1429 .2857 .5714 B .1429 .2857 .5714 C .1429 .2857 .5714 Row Averages .1429 .2857 .5714 C .3333 .5556 .1111 Row Averages .4055 .4796 .1150

Social College A B C A .6667 .2222 .1111 B .6923 .2308 .0769 C .6000 .3000 .1000 Row Averages .6530 .2510 .0960 Row Averages .3146 .0795 .4667 .1392

Criteria Academic Location Cost Social

Academic .2791 .0698 .5581 .0930 Cost .1429 .2857 .5714

Location .3333 .0833 .4167 .1667

Cost .2459 .0984 .4918 .1639

Social .4000 .0667 .4000 .1333

Academic Location A .3202 .4055 B  .5571 .4796 C .1226 .1150 

Criteria Social College Score .3146 .6530  Arrington .2906 .2510  × .0795 = Barton .3817 .4667 .0960  6  Claiborne .3277 .1392   1.0000

Aaron should select Barton (by a small margin). Consistency: (1)  1 1/ 4  2  1/ 3 (2) (3)

4 1/ 2 3 .3146 1.2836  1 1/ 5 1/ 2  × .0795 =  .3211 .4667 1.9110  5 1 3 .1392   2 1/ 3 1     .5586

(3)/(2)  4.0799  4.0389  4.0947 4.40132    16.2297

16.2267 = 4.0567, CI = 4.0567 − 4 = .019 4 3 .019 CI / RI = = .021 .90 .10 this preference comparison for criteria is consistent. 145

Since .021

27.

Normalized matrices: Attractiveness Client Robin Terry Kelly Robin .6522 .2174 .1304 Terry .6667 .2222 .1111 Intelligence Client Robin Terry Kelly Robin .2857 .1429 .5714 Terry .2857 .1429 .5714 Personality Client Robin Terry Kelly Robin .2222 .1111 .6667 Terry .4000 .2000 .4000 Kelly .1818 .2727 .5455 Row Averages .2680 .1946 .5374 Row Averages .4429 .3873 .1698 Kelly .2857 .1429 .5714 Row Averages .2857 .1429 .5714 Kelly .6250 .2500 .1250 Row Averages .6479 .2299 .1222

Criteria Attractiveness Intelligence Personality

Attractiveness .4286 .4286 .1429

Intelligence .4000 .4000 .2000 Personality Criteria Client .2680  Robin .4429 .1946  × .3873 = Terry .1698 .5374  Kelly   

Personality .5000 .3333 .1667 Score .4431 .1902 .3667 1.0000

Robin Terry Kelly

Attractiveness Intelligence .2857  .6479 .1429  .2299  .1222 .5714 

The best match for Chris is Robin. 28. Normalized matrices: Profit Project A B C A .7059 .1765 .1176 B .7273 .1818 .0909 P (Success) Project A B C A .2222 .1111 .6667 146 B .2222 .1111 .6667 C .2222 .1111 .6667 Row Averages .2222 .1111 .6667 C .6667 .2222 .1111 Row Averages .6999 .1935 .1066

Cost Project A B C A .1250 .3750 .5000 B .1000 .3000 .6000 C .1429 .2857 .5714 Row Averages .1226 .3202 .5571 Row Averages .5869 .3238 .0893

Criteria Profit P (Success) Cost Profit P (Success) .2222 .6999 .1111 .1935 .1066 .6667 

Profit .6000 .3000 .1000

P (Success) .6154 .3077 .0769 Score .4937 .1781 .3282 1.0000

Cost .5455 .3636 .0909

A B C 29. 30.

Cost Criteria Project .1226 A .5869 .3202 × .3238 = B .0893 .5571 C   

The solution to this problem depends on how the students set up the preference comparisons for the three degree disciplines. Normalized matrices: Usage Facility Gym Field Tennis Pool Gym .2069 .6207 .0690 .1034 Field .1869 .5607 .1121 .1402 Cost Facility Gym Field Tennis Pool Gym .1364 .5455 .2727 .0455 Field .1448 .5793 .1931 .0828 Tennis .1053 .6316 .2105 .0526 Pool .2000 .4667 .2667 .0667 Row Averages .1466 .5558 .2358 .0619 Row Averages .8333 .1667 Tennis .2500 .4167 .0833 .2500 Pool .2727 .5455 .0455 .1364 Row Averages .2291 .5359 .0775 .1575

Criteria Usage Cost Usage .2291 .5359 .0775 .1575 

Usage .8333 .1667 Cost Criteria .1466 .5558 .8333 .2358 × .1667 =   .0619  Facility Gym Field Tennis Pool Score .215 .539 .104 .142 1.0000

Cost .8333 .1667

Gym Field Tennis Pool

Rank: (1) Field; (2) Gym; (3) Pool; (4) Tennis 147

Usage: (1) 1 3 1/ 3 1/ 2  1/ 3 1 1/ 5 1/ 4 3 5 1 3 (2) (3)

2 .2291  .9552 4  × .5359 = 2.2407 .0775  .3135 1/ 3 .1575  .6385 1     

(3)/(2) 4.1695 16.4506 = 4.1127, CI = 4.1127 − 4 = .038 4.1812 4 4.0458 3   .038 4.0541 CI /CR = = .042 .90 16.4506 Since .042 Cost: (1) 1/ 4 1/ 2 1 3 1/ 3 1 1/ 7 1/ 4 (2) (3) 3 .1446  .5872 7 .5558 2.2749 4 × .2358 =  .9579 .0619  .2485 1      .10 this pairwise comparison is consistent.

1 4 2 1/ 3 

(3)/(2) 4.0605  4.0930 16.2295 = 4.0573, CI = 4.0573 − 4 = .019 4 4.0622 3 .019   4.0137 CI /CR = = .021 .90 16.2295

Since .021 .10 this pairwise comparison is consistent. The pairwise comparison matrix for criteria is consistent since it includes only one comparison and the preference for one is an exact reciprocal of the other. 31. Normalized matrices: Weather Location MB DB FL MB .1429 .4286 .4286 DB .1429 .4286 .4286 FL .1429 .4286 .4286 Row Averages .1429 .4286 .4286

Cost Location MB DB FL MB .6522 .2174 .1304 DB .6667 .2222 .1111 FL .6250 .2500 .1250 Row Averages .6479 .2299 .1222

148

Fun Location MB DB FL MB .3125 .6250 .0625 DB .2729 .5455 .1818 FL .5556 .3333 .1111 Row Averages .3803 .5013 .1185 Row Averages .2543 .0952 .6505

Criteria Weather Cost Fun Weather Cost .1429 .6479 .4286 .2299 .4286 .1222 

Weather .1905 .0476 .7619

Cost .4000 .1000 .5000 Score .3454 .4570 .1977 1.0000

Fun .1724 .1379 .6897

MB DB FL

Fun Criteria Location .3803 MB .2543 .5013 × .0952 = DB   .1185 FL .6505 

They will select Daytona Beach. 32. Weather: (1) (2) (3)

1 1/ 3 1/ 3 .1429  .4286  3 1 1  × .4286 = 1.2859 3 1 1  .4286 1.2859       (3)/(2) 2.9995 3.0002 3.0002 9.0000/3 = 3.0000   9.0000 This set of pairwise comparisons is perfectly consistent. Cost: (1) (2) (3)

1.9486  .6479  1 3 5 1/ 3 1 2 × .2299 =  .6903       .3667 .1222 1/ 5 1/ 2 1  (3)/(2) 3.0076 3.0025 9.0111 = 3.0037, CI = 3.0037 − 3 = .0019 3 3.0011 2   9.0111 CI / RI = .0019 = .003 .58 Since .003 Fun. (1) (2) (3)  1 1/ 2 5 .3803 1.2235  1 3 × .5013 = 1.6174  2 1/ 5 1/ 3 1 .1185  .3617       149 .10 this pairwise comparison for cost is consistent.

(3)/(2) 3.2171 9.4955 = 3.1652, CI = 3.1652 − 3 = .0825 3.2264 3 2 3.0520   9.4955 CI / RI = .0825 = .14 .58 Since .14 Criteria: (1) (2) (3)  .7977  1 4 1/ 4 .2543 1/ 4 1 1/ 5 × .0952 =  .2889 2.1437 4 5 .6505 1       (3)/(2) 3.1369 9.4668 = 3.1556, CI = 3.1556 − 3 = .0778 3.0344 3 2 3.2955   CI /RI = .0778 = .13 9.4668 .58 Since .13 33. .10 this pairwise comparison for the criteria also appears inconsistent. .10 this pairwise comparison for fun is inconsistent and the comparisons should be made again.

Normalized matrices: Aptitude Option DSS OM DSS .75 .25 Jobs Option DSS OM DSS .8000 .2000 Faculty Option DSS OM DSS .1667 .8333 OM .1667 .8333 Row Averages .1667 .8333 Row Averages .1373 .2395 .6232 OM .8000 .2000 Row Averages .8000 .2000 OM .75 .25 Row Averages .75 .25

Criteria Aptitude Faculty Jobs Aptitude Faculty DSS .75 .1667 OM .25 .8333 

Aptitude .1429 .2857 .5714 Criteria Jobs .1373 .8000 × .2395 = .2000  .6232  

Faculty .1111 .2222 .6667 Option DSS OM Score .6415 .3585 1.0000

Jobs .1579 .2105 .6316

The student should select the DSS option. 150

34. 35.

The solution to this problem depends on how the student develops the pairwise comparisons for the individual criteria and between the criteria. Normalized matrices: PTA Option Renovate New Renovate .25 .75 Students Proposal Renovate New Renovate .6667 .3333 Teachers Option Renovate New Renovate .10 .90 New .10 .90 Row Averages .10 .90 New .6667 .3333 Row Averages .6667 .3333 New .25 .75 Row Averages .25 .75

Town Council Proposal Renovate New Renovate .8333 .1667 New .8333 .1667 Row Averages .8333 .1667 Town Council .1570 .0897 .1256 .6278 Row Averages .2172 .0535 .1424 .5868

Group PTA Teachers Students Town Council

PTA .1754 .0351 .0877 .7018

Teachers .2941 .0588 .2353 .4118

Students .2424 .0303 .1212 .6061

Criteria Town Proposal PTA Teachers Students Council .2172 Renovate .25 Renovate .10 .6667 .8333 × .0535 = New .75 New .1424 .90 .3333 .1667   .5868  

Score .6436 .3564 1.0000

(a) The old middle school should be renovated. (b)  1 1/ 5 1/ 2  4  (1) (2) (3)

5 2 1/ 4 .2172  .9162 1 1/ 4 1/ 7  .2164 .0535 4 1 1/ 5 × .1424 =  .5824 .5868 2.5421 7 5 1     

151

(3)/(2) 4.2182 4.0443 4.0896   4.3321 16.6843 Since .063 36. 37.

16.6843 = 4.1711, CI = 4.1711− 4 = .057 4 3 .057 CI / RI = = .063 .90 .10 the pairwise comparison for the group is consistent.

Answers depend on student preferences. Preference matrix: Hospital County Memorial General Normalized matrix: Criteria Criteria Medical Care Medical Care .6154 Distance .0769 Speed of attention .1026 Cost .2051 Hospital Memorial County General Score .4633 .3203 .2163 Distance .4706 .0588 .1176 .3529 Speed of Attention .5217 .0435 .0870 .3478 Cost .6792 .0377 .0566 .2264 Row Averages .5717 .0542 .0909 .2831 1.000 Medical Care .0960 .6530 .2510 Criteria Distance .7043 .1686 .1271 Speed of Attention .3202 .5571 .1226 Cost .6999 .1066 .1935

38. Criteria Section 1 2 3 4 Time/Day .5681 .2410 .1333 .0576 Grading .0800 .5017 .3548 .0635 Atmosphere .5287 .2198 .1231 .1284 Homework .0613 .2677 .5698 .1012 Jokes .2103 .6173 .1145 .0579

Criteria Time/Day Grading Atmosphere Homework Jokes .4220 .3157 .0552 .1723 .0347

X

152

Section 1 2 3 4 39. 40.

Score .3120 .3398 .2772 .0709 1.000

46.

S (Tysons) S (Fairfax)

85.27 80.94 80.55 77.00 76.39 84.74

S (Alexandria) S (Manassas) S (Dupont) 47. S (Roanoke) S (Richmond) S (Greensboro) S (Charlotte) S (Bristol) S (Knoxville) 48.

Solution depends on student selections and preferences. S (Abbeton) S (Bayside) 76.65 79.80 74.15 77.85

83.79

S (Cane Creek) S (Dunnville) Select Bayside 41. S (1) S (2) S (3) 77.50 80.80 82.05

81.10 79.02

t Enter 82.61

76.93

Depends on the student’s selection of restaurants, decision criteria and preferences

Site 3 is recommended 42. S (Shanghai) S (Hong Kong) S (Singapore) 64.70 71.90 80.55

CASE SOLUTION: OAKDALE COUNTY SCHOOL BUSSING

Recommended Singapore 43. S (South) S (West A) S (West B) S (East) 73.80 74.50 67.25 76.75

Recommended the East campus site. 44. S (Central) S (A&M) S (Western) S (Tech) S (State) 45. S (C) S (E) S (D) S (B) S (A) 76 75 74 70 69 77.74 76.69 76.65 71.62 69.89

153

20x1 + 18x2 + 22x3 + 24x4 + 17x5 + 19x6 + – + d1 – d1 = 450 – .27x1 + .21x2 + .28x3 + .19x4 + .23x5 + .33x6 + d2 – + d2 = 5 18x1 + 26x2 + 10x3 + 34x4 + 25x5 + 17x6 + – d3 – d + =350 3 1700x1 + 900x2 + 650x3 + 230x4 + 1600x5 + 520x6 – + + d4 – d4 = 30,000 – .32x1 + .65x2 + .43x3 + .87x4 + .55x5 + .49x6 + d5 – + d5 = 15 This model was solved iteratively using Excel. Following is the model formulation and solution output. The first 3 goals were achieved however, d4 16340 sight contacts and d5 2.67 minutes. (Thus, the actual response time is approximately 16 minutes.) x1 4, x2 5, x3 2, x4 2, x5 330, d4 5, x6 5

d1 8, d2 2.67

0.87, d3

6,340, d5

CASE SOLUTION: KATHERINE MILLER’S JOB SELECTION
The solution depends on the student’s preferences and the criteria the student uses.

CASE SOLUTION: CATAWBA VALLEY HIGHWAY PATROL
The goal programming model formulation is as follows. + – – – – minimize P1d1 , P2d2 , P3d3 , P4d4 , P5d5 subject to x1 + x2 + x3 + x4 + x5 + x6 = 23 x1 2 x2 2 x3 2 x4 2 x5 2 x6 2 x1 5 x2 5 x3 5 x4 5 x5 5 x6 5 154

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