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TASK 3 UNCERTAINTY ENVIRONMENTS A

LORAINYS ESTEFANYS DURAN

JESUS FABIAN ARENAS SAAV RAY LEONARC ATENCIO

CARLOS EDUBER CASTRO LA JOSÉ ALBERTO VEGA

DEYANIRA PEREZ BRAV

TEORIA DE LAS DECISION

NATIONAL OPEN AND DISTANCE U

SCHOOL OF BASIC SCIENCES TECHONOLOG

Nov-20

ERTAINTY ENVIRONMENTS AND GAME THEORY Activity

ORAINYS ESTEFANYS DURAN

1,065,806,565

ESUS FABIAN ARENAS SAAVEDRA

1,100,894,455

RAY LEONARC ATENCIO

1,065,986,590 CARLOS EDUBER CASTRO LAGOS 1,007,835,954 JOSÉ ALBERTO VEGA Tutora DEYANIRA PEREZ BRAVO TEORIA DE LAS DECISIONES CURSO 212066_47

IONAL OPEN AND DISTANCE UNIVERSITY

ASIC SCIENCES TECHONOLOGY AND ENGINEERING

Nov-20

INTRODUCTION

### Through the development of this activity, ways to solve production and service problems will be shown by applying game

### Also through the application of game theory, analytically seeking decision-making on process problems in production and

will be shown by applying game theory.

ess problems in production and services.

Exercise 1. Laplace, Wald or pessimistic, optimistic, Hurwicz and Savage criteria (Profit Matrix):

In the company ABC several alternatives are presented to choose the best technology of four possible, wh performance depends on the adaptation of the workers who will manipulate the equipments that compris expected benefits of each alternative and degree of adaptation of the workers are given in the table, in mi ($).

EVENT ALTERNATIVE TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

DOES NOT FIT FITS ACCEPTABLY 530 670 750 670 550

FITS SUCCESSFULLY FITS WELL

585 525 650 590 610

615 575 615 610 710

LAPLACE CRITERIA TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

618 608 669.6 650 609

669.6

WALD CRITERIA TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

530 525 615 590 550

615

HURWICZ CRITERIA

TECHNOLOGY 1

MAX 710

MIN 530

620

650 580 623 650 550

TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

690 750 730 710

525 615 590 550

607.5 682.5 660 630

SAVAGE CRITERIA

TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

750

650

710

650

220 80 0 80 200

65 125 0 60 40

95 135 95 100 0

0 70 27 0 100

95

TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

MAX 220 135 95 100 200

OPTIMIST CRITERIA TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

710 690 750 730 710

750

Matrix):

echnology of four possible, whose e the equipments that comprise it. The ers are given in the table, in millions of pesos

FITS VER WELL 710 690 710 730 625

ALPHA 0.5

The best technology to be applied is technology 3, depending on the application of the criteria.

682.5

730 20 40 20 0 105

POINT 1 Decision Tables Enter Enter the the profits profits in in the the main main body body of of the the data data table. table. Enter Enter probabilities probabilities in in the the first first row row ifif you you want want to to compute compute the the expected expected value. value.

Data Profit Probability Decision 1 Decision 2 Decision 3 Decision 4 Decision 5

Results EMV

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 0.2 0.2 0.2 0.2 0.2 530 585 615 650 710 670 525 575 580 690 750 650 615 623 710 670 590 610 650 730 550 610 710 550 625 Maximum

618 608 669.6 650 609 669.6

Minimum

Maximum 530 525 615 590 550 615

710 690 750 730 710 750

Hurwicz alpha 0.5 620 607.5 682.5 660 630

Exercise 2. Laplace, Wald or pessimistic, optimistic, Hurwicz and Savage criteria (Cost Matrix):

Fabcom, a company that manufactures electronic components for the introduction in its product catalo whether to manufacture a new product in its main plant, subcontract it with company supervision or if external supplier. The profits depend on the demand of the product. The table shows projected costs, i dollars.

EVENT ALTERNATIVE TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

DOES NOT FIT FITS ACCEPTABLY 415 318 650 540 497

FITS SUCCESSFULLY FITS WELL

435 535 575 718 535

510 575 556 560 435

LAPLACE CRITERIA TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

509 530.4 587.6 561.8 488.4

587.6

WALD CRITERIA TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

415 318 546 459 415

546

HURWICZ CRITERIA

TECHNOLOGY 1

MAX 610

MIN 415

512.5

575 603 546 459 415

TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

621 650 718 560

318 546 459 415

469.5 598 588.5 487.5

SAVAGE CRITERIA

TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

650

718

575

603

235 332 0 110 153

283 183 143 0 183

65 0 19 15 140

28 0 57 144 188

143

TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

MAX 283 332 143 144 188

OPTIMIST CRITERIA TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

610 621 650 718 560

718

st Matrix):

ntroduction in its product catalog, must decide with company supervision or if it buys it from an e table shows projected costs, in millions of

FITS VER WELL 610 621 611 532 560

ALPHA 0.5

The best technology to be applied is technology 3, depending on the application of the criteria.

598

621 11 0 10 89 61

POINT 2 Decision Tables Enter Enter the the profits profits in in the the main main body body of of the the data data table. table. Enter Enter probabilities probabilities in in the the first first row row ifif you you want want to to compute compute the the expected expected value. value.

Data Profit Probability Decision 1 Decision 2 Decision 3 Decision 4 Decision 5

Results EMV

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 0.2 0.2 0.2 0.2 0.2 415 435 510 575 610 318 535 575 603 621 650 575 556 546 611 540 718 560 459 532 497 535 435 415 560 Maximum

509 530.4 587.6 561.8 488.4 587.6

Minimum

Maximum 415 318 546 459 415 546

610 621 650 718 560 718

Hurwicz alpha 0.5 512.5 469.5 598 588.5 487.5

Exercise 3. Laplace, Wald or pessimistic, optimistic, Hurwicz and Savage criteria (Cost Matrix):

Fabricater company that has a productive experience in the foreign market of 20 years, must decide if i new product in its main plant, or if on the contrary the purchase from an external supplier. The profits demand of the product. The table shows projected costs, in millions of dollars.projected costs, in millio

EVENT ALTERNATIVE TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

DOES NOT FIT FITS ACCEPTABLY 519 457 560 670 542

FITS SUCCESSFULLY FITS WELL

585 525 650 574 610

615 560 605 600 710

LAPLACE CRITERIA TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

615.8 561.8 629.6 644.8 607.4

644.8

WALD CRITERIA TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

519 457 560 574 542

574

HURWICZ CRITERIA

TECHNOLOGY 1

MAX 710

MIN 519

614.5

650 580 623 650 550

TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

687 710 730 710

457 560 574 542

572 635 652 626

SAVAGE CRITERIA

TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

670

650

710

650

151 213 110 0 128

65 125 0 76 40

95 150 105 110 0

0 70 27 0 100

110

TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

MAX 151 213 110 110 128

OPTIMIST CRITERIA TECHNOLOGY 1 TECHNOLOGY 2 TECHNOLOGY 3 TECHNOLOGY 4 TECHNOLOGY 5

710 687 710 730 710

730

st Matrix):

ket of 20 years, must decide if it manufactures a n external supplier. The profits depend on the dollars.projected costs, in millions of dollars.

FITS VER WELL 710 687 710 730 625

ALPHA 0.5

The best technology to be applied is technology 4, depending on the application of the criteria.

652

730 20 43 20 0 105

POINT 3 Decision Tables Enter Enter the the profits profits in in the the main main body body of of the the data data table. table. Enter Enter probabilities probabilities in in the the first first row row ifif you you want want to to compute compute the the expected expected value. value.

Data Profit Probability Decision 1 Decision 2 Decision 3 Decision 4 Decision 5

Results EMV

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 0.2 0.2 0.2 0.2 0.2 519 585 615 650 710 457 525 560 580 687 560 650 605 623 710 670 574 600 650 730 542 610 710 550 625 Maximum

615.8 561.8 629.6 644.8 607.4 644.8

Minimum

Maximum 519 457 560 574 542 574

710 687 710 730 710 730

Hurwicz alpha 0.5 614.5 572 635 652 626

Exercise 4. Game Theory method:

In exercise 4 you must find the game strategy of the players so that through game theory a fair game i according to the objective of the method, one player wins as much as the other loses and vice versa. P corresponding method according to the nature of the exercise proposed and answer the questions. Gr only applicable to games in which at least one of the players has only two strategies. Consider the follo

PLAYER 1

STRATEGY

PLAYER 2 B 17 7 17

I II MAX-MIN

A 24 29 29

C 13 33 33

MINI-MAX 13 7

X1 X2 X3

0 0 0

X1 X2 X3

1 1 1

PE1 PE2 PE3

29 7 33

PE1 PE2 PE3

24 17 13

3533 29 30 24

PE

25 20

17

15

13

10 7 5 0

0

0.2

0.4

0.6

X

0.8

1

1.2

0

0

0.2

0.4

0.6

X

0.8

1

1.2

gh game theory a fair game is carried out e other loses and vice versa. Proceed to use the and answer the questions. Graphical solutions are o strategies. Consider the following 2 x n game:

SOLVER

PLAYER I 0.866666667 0.133333333 1

24.66666667 15.66666667 15.66666667 -9 0 0

MAX

15.66666667

There is an equilibrium point that is the strategy, because the MINI-MAX and the MAX-MIN is the same, player 1 must select strategy I and player 2 strategy B, with a gain of 17 monetary units for player 1 and a loss of 13 monetary units for player 2.

24

17 13

REGIÓN COMÚN 1

1.2

1

1.2

MIN is the y units for

POINT 4

POINT 4 Zero sum games

Enter Enter the the values values in in the the shaded shaded area area then then use use the the Run Run Excel's Excel's Solver Solver button. button.Alternatively, Alternatively, or or to to view view the the sensitivity sensitivity results, results, open open Solver Solver by by go g to to the the Data Data Tab Tab (Excel (Excel 2007, 2007, 2010, 2010, 2013, 2013, 2016) 2016) or or the theTools Tools menu menu (Exc (Exc 2003, 2003, 2011). 2011).

Game value 15.66667 Data Col strat 1 Col strat 2 Col strat 3 row mix wtd avg Row min 24 17 13 Row strat 1 0.8667 15.66667 13 29 7 33 Row strat 2 0.1333 15.66667 7 col mix 0 0.6667 0.3333 1 wtd avg 24.66667 15.66667 15.66667 1 maximin 13 Col max

29

17

33 0.866667 minimax 17

Page 28

POINT 4

nn use use the the Run Run Excel's Excel's Solver Solver tivity itivity results, results, open open Solver Solver by by going going 3, 3, 2016) 2016) or or the theTools Tools menu menu (Excel (Excel

Page 29

Exercise 5. Game Theory method:

Graphical solutions are only applicable to games in which at least one of the players has only two strate Consider the following game m x 2:

PLAYER 1

STRATEGY

PLAYER 2

I II III MAX-MIN

A 24 29 19 29

B 17 7 11 17

MINI-MAX 17 7 11

Y1 Y2 Y3

0 0 0

Y1 Y2 Y3

1 1 1

PE1 PE2 PE3

17 7 11

PE1 PE2 PE3

24 29 19

There is an equilib the same, player 1 units for player 1 a the value of the ga      Because it is a pur method.

35 29

30

24

25

19

PE

2017 15 11 10 7

REGIÓN COMÚN

5 0

0

0.2

0.4

0.6

X

0.8

1

1.2

5 0

0

0.2

0.4

0.6

X

0.8

1

1.2

ayers has only two strategies.

SOLVER

PLAYER I 1 0 0 1

24 -7 MAX

17 0 17

There is an equilibrium point that is the strategy, because the MINI-MAX and the MAX-MIN is the same, player 1 must select strategy I and player 2 strategy B, with a gain of 17 monetary units for player 1 and a loss of 17 monetary units for player 2, thus having a pure strategy and the value of the game is 17      Because it is a pure strategy, that is, it has a saddle point, it is not necessary to apply the graphic method.

REGIÓN COMÚN

POINT 5

POINT 5 Zero sum games

Game value Data

Enter Enter the the values values in in the the shaded shaded area area then then use use the the Run Run Excel's Excel's Solver Solver button. button.Alternatively, Alternatively, or or to to view view the the sensitivity sensitivity results, results, open open Solver Solver by by go g to to the the Data Data Tab Tab (Excel (Excel 2007, 2007, 2010, 2010, 2013, 2013, 2016) 2016) or or the theTools Tools menu menu (Exc (Exc 2003, 2003, 2011). 2011).

17

Col strat 1 Col strat 2 row mix wtd avg Row min 24 17 Row strat 1 1 17 17 29 7 Row strat 2 0 7 7 19 11 Row strat 3 0 11 11 col mix 0 1 1 wtd avg 24 17 1 maximin 17 Col max

29

17 minimax

1 17

Page 33

POINT 5

nn use use the the Run Run Excel's Excel's Solver Solver tivity itivity results, results, open open Solver Solver by by going going 3, 3, 2016) 2016) or or the theTools Tools menu menu (Excel (Excel

Page 34

Exercise 6. Optimum solution of two-person games:

The games represent the latest case of lack of information where intelligent opponents are working environment. The result is that a very conservative criterion is generally proposed to solve sets of tw zero, called minimax - maximin criterion. To determine a fair game, the minimax = maximin, it is nec stable strategy through the Solver

MINI-MAX AND MAX-MIN

PLAYER A

PLAYER B

MAX-MIN

66 35 76 48 76

73 65 64 77 77

47 51 52 65 65

58 27 69 38 69

There is an equilibrium point that is the strategy, because the SOLVER is the same, player 1 mus strategy III and player 2 strategy C with a gain of 52 monetary units for player 1 and a loss of 65 monetary units for player 2, thus having a pure strategy and the value of the game is 52   

t opponents are working in a conflicting oposed to solve sets of two people and sum imax = maximin, it is necessary to solve the

SOLVER

PLAYER B PLAYER A

MINI-MAX 47 27 52 38

66 35 76 48

73 65 64 77

47 51 52 65

65.18181818 69.02272727 -8.15909091 -12

R is the same, player 1 must select or player 1 and a loss of 65 ue of the game is 52 PLAYER A 0 0 0.613636364 0.386363636 1

SOLVER

B 58 27 69 38

57.02272727 57.02272727 0 0

MAX

57.02272727

POINT 6

POINT 6 Zero sum games

Enter Enter the the values values in in the the shaded shaded area area then then use use the the Run Run Excel's Excel's Solver Solver button. button.Alternatively, Alternatively, or or to to view view the the sensitivity sensitivity results, results, open open Solver Solver by by go g to to the the Data Data Tab Tab (Excel (Excel 2007, 2007, 2010, 2010, 2013, 2013, 2016) 2016) or or the theTools Tools menu menu (Exc (Exc 2003, 2003, 2011). 2011).

Game value 57.02273 Data Col strat 1 Col strat 2 Col strat 3 Col strat 4 row mix wtd avg Row min 66 73 47 58 Row strat 1 0 50.25 47 35 65 51 27 Row strat 2 0 43.90909 27 76 64 52 69 Row strat 3 0.6136 57.02273 52 48 77 65 38 Row strat 4 0.3864 57.02273 38 col mix 0 0 0.7045 0.2955 1 wtd avg 65.18182 69.02273 57.02273 57.02273 1 maximin 52 Col max

76

77

65

69 0.613636 minimax 65

Page 38

POINT 6

nn use use the the Run Run Excel's Excel's Solver Solver tivity itivity results, results, open open Solver Solver by by going going 3, 3, 2016) 2016) or or the theTools Tools menu menu (Excel (Excel

Page 39

bibliography

- Sharma, J. (2016). Operations Research: Theory and Applications. (pp. 341- 391), New Delhi: Laxmi Publications Pvt Ltd, v. Sixth edition. - Kelly, A. (2003). Decision Making Using Game Theory (pp. 28-51): An Introduction for Managers: Cambridge, UK: Cambridge University Press Editorial.

341- 391), New Delhi: Laxmi

ntroduction for Managers: