decision making criterion

OPERATION RESEARCH

GAURAV SONKAR

OPERATION RESEARCH

of

An operation may be defined as the set of acts required for the achievements of a desired

outcomes.

DEFNITIONS OF O.R.

I

OPERATION RESEARCH IS SYSTEMATIC, METHOD ORIENTED STUDY OF THE BASIC STRUCTURE, CHARACTERISTICS, FUNCTION & RELATIONSHIP OF AN ORGANISATION TO PROIDE THE EXECUTIVE WITH A SOUND, SCIENTIFIC AND QUANITATIVE BASIS FOR THE DECESION MAKING. ------------------------------BY E.L.ARNOFF & M.J.NETZORG

O.R. IS AN AID FOR THE EXECUTIVE IN MAKING HIS DECISIONS BY PROVIDING HIM WITH THE NEEDED QUANTITATIVE INFORMATION BASED ON THE SCIENTIFIC METHOD OF ANALYSIS ---------------BY C. KITTEL

O.R. IS THE APPLICATION OF SCIENTIFIC METHODS TO THE PROBLEM ARISING FROM OPERATIONS INVOLVING INTEGRATED SYSTEMS OF MEN, MACHINE AND MATERIALS. IT NORMALLY UTILIZES THE KNOWLEDGE AND SKILLS OF INTERDISCIPLINARY RESEARCH TEAM TO PROVIDE THE MANAGERS OF SUCH SYSTEMS WITH OPTIMUM OPERATING SOLUTIONS. --------------------BY FABRYCKY & TORGERSEN

CHARACTERISTICS OF OR:

ITS SYSTEM ORIENTED

USE OF INTERDISCIPLINARY TEAM

APPLICATION OF SCIENTIFIC METHODS

UNCOVERING OF NEW PROBLEMS

IMPROVE THE QUALITY OF DECESIONS

USE OF COMPUTERS

QUANTITATIVE SOLUTION

HUMAN FACTORS

SCOPE OF OPERATION RESEARCH: I. ALLOCATION AND DISTRIBUTION:

Optimal allocation of limited resources such as men, machine, and material.Location and size of warehouses, distribution centre, retail depot etc.Distribution policy.

II. PRODUCTION AND FACILITY PLANNING:Selection, location and design of production plant.Project scheduling & allocation of resources.Forecasting.Maintenance policy.Scheduling & sequencing.

III. PROCUREMENT: What, when and how to purchase at minimum procurement costBidding and replacement policies.

IV. MARKETING:Product selection, timing & competitive actionSelection of advertising media.Demand forecast and stock level.Customer’s preference for size, colour & packaging of various products.

V. FINANCE:Capital requirement, cash flow analysis.Credit policies, credit risks etc.Profit plan of the company.Determination of optimum replacement policies.

VI. PERSONNEL:Selection of personnel, determination of retirement age and skillsRecruitment of policies & assignments of jobs.

VII. RESEARCH AND DEVELOPMENT:

Determination of areas of Research and DevelopmentReliability & control of development of projects.Selection of projects & preparation of their budgets.

METHODOLOGY OF OR:FORMULATE THE PROBLEM

CONSTRUCT A MATHEMATICAL MODEL

SOLVE THE MODEL

TEST THE MODEL

ANALYSE THE RESULT

IMPLEMENTATION OF SELECTED STRATEGY

DIFFICULTY IN O.R.

PROBLEM FORMULATION

DATA COLLECTION

STUDY BASED ON OBSERVATION OR OLD LAWS

TIME FACTOR

HUMAN FACTOR

DECESION THEORY:

DECESION THEORY PROVIDES A RATIONAL APPROACH IN DEALING PROBLEMS CONFRONTED WITH THE PARTIAL , IMPERFECT OR UNCERTAIN FUTURE CONDITION

STEPS IN DECESION THEORY APPROACH:

LIST ALL THE VIABLE ALTERNATIVES

CONSTRUCT A PAY-OFF TABLE

SELECT OPTIMUM DECESION CRITERION

DECESION MAKING ENVIRONMENT:DECESIONS ARE MADE UNDER THREE TYPES OF

ENVIRONMENT:

D.M.E.

CERTAINITY UNCERTAINITY RISK

IN THIS , ONLY ONE STATE OF NATURE EXISTS i.e. THERE IS COMPLETE CERTAINITY ABOUT THE FUTURE

HERE MORE THAN ONE S.O.N. EXISTS BUT D. MAKER LACKS SUFFICIENT KNOWLEDGE TO ALLOW HIM ASSIGN PROB TO VARIOUS S.O.N.

HERE ALSO MORE THAN ONE S.O.N. EXISTS BUT THE D. MAKER HAS SUFFICIENT INFO TO ALLOW HIM ASSIGN PROB TO EACH OF THESE STATES

D.M. UNDER UNCERTAINITY: Under condition of uncertainty, the decision maker

has knowledge about states of nature that happens but lacks the knowledge about the probabilities of their occurrence.

Under conditions of uncertainty, a few decision criterions are available which could be of help to the decision maker.

D.M. UNDER UNCERTAINITY

Maximax Criterion

or Criterion

of optimism

Maximin Criterion

or Criterion

of pessimism

Minimax Criterion

or Regret

Criterion

Hurwicz Criterion

or Criterion

of Realism

Laplace Criterion

or Criterion

of Rationality

ILLUSTRATION: CONSIDERING A MANUFACTURING COMPANY THAT IS THINKING OF VARIOUS ALTERNATIVES TO INCREASE ITS PRODUCTION TO MEET THE INCREASING MARKET DEMAND.

WHICH STRATEGY OR ALTERNATIVE WILL THE CO. EMPLOY ON THE BASIS OF VARIOUS METHODS.

ALTERNATIVESSTATE OF NATURE (PRODUCT DEMAND)

HIGH MODERATE LOW NIL

EXPAND 50,000 25,000 - 25,000 - 45,000

CONSTRUCT 70,000 30,000 - 40,000 - 80,000

SUBCONTRACT 30,000 15,000 - 1,000 - 10,000

(I) MAXIMAX CRITERION OR CRITERION OF OPTIMISM: This criterion provides the decision maker with

optimistic criterion. The working method is summarizing as follow.

Locate the maximum payoff values corresponding to each alternative (or course of action or strategy), thenSelect an alternative with maximum payoff value.

THUS THE MAXIMAX PAYOFF IS Rs. 70,000 CORRESPONDING TO THE ALTERNATIVE “CONSTRUCT”.

ALTERNATIVES

STATE OF NATURE (PRODUCT DEMAND)

HIGH MODERATE

LOW NIL

EXPAND 50,000 25,000 - 25,000 - 45,000

CONSTRUCT

70,000 30,000 - 40,000 - 80,000

SUBCONTRACT

30,000 15,000 - 1,000 - 10,000

MAXIMUM OF

ROW

50,000

70,000

30,000

(II) MAXIMIN CRITERION OR CRITERION OF PESSIMISM:This criterion provides the decision maker with

pessimistic criterion. The working method is summarizing as follow.

Locate the minimum payoff values corresponding to each alternative (or course of action or strategy), then Select an alternative with maximum payoff value.

THUS THE MINIMAX PAYOFF IS Rs. – 10,000 CORRESPONDING TO THE ALTERNATIVE - “SUBCONTRACT”

ALTERNATIVES

STATE OF NATURE (PRODUCT DEMAND)

HIGH MODERATE

LOW NIL

EXPAND 50,000 25,000 - 25,000 - 45,000

CONSTRUCT

70,000 30,000 - 40,000 - 80,000

SUBCONTRACT

30,000 15,000 - 1,000 - 10,000

MINIMUM OF

ROW

-45,000

-80,000

-10,000

(III) MINIMAX CRITERION OR MINIMUM REGRET CRITERION: This criterion is also known as opportunity loss

decision criterion or minimax regret criterion. The working method is summarizing as follow.

Determine the amount of regret corresponding to each alternative for each state of nature. The regret for jth event corresponding to ith

alternative is given byith regret = (maximum payoff – ith payoff) for the jth event Determine the maximum regret amount for each alternative.

Choose the alternative which corresponds to the minimum of the maximum regrets.

ALTERNATIVESSTATE OF NATURE (PRODUCT DEMAND)

HIGH MODERATE LOW NIL

EXPAND 50,000 25,000 - 25,000 - 45,000

CONSTRUCT 70,000 30,000 - 40,000 - 80,000

SUBCONTRACT 30,000 15,000 - 1,000 - 10,000

ALTERNATIVES

CALCULATION OF REGRET

HIGH MODERATE

LOW NIL

EXPAND 20000 5000 24000 35000

CONSTRUCT 0 0 39000 70000

SUBCONTRACT

40000 15,000 0 0

MAXIMUM OF

ROW

35,000

70,000

40,000

THIS TABLE SHOWS THAT THE COMPANY WILL MINIMIZE ITS REGRET TO RS 35,000 BY SELECTING ALTERNATIVE-

“EXPANSION”

(IV) HURWICZ CRITERION OR CRITERION OF REALISM: Also called weighted average criterion, it is a

compromise between the maximax (optimistic) and minimax (pessimistic) decision criterion. This concept allows the decision maker to take into account both maximum and minimum for each alternative and assign them weights according to his degree of optimism (or pessimism). The working method is summarizing as follow:Choose an appropriate degree of optimism, α so that (1-α) represents degree of pessimism.Determine the maximum as well as minimum of each alternative and obtain

P = α. Maximum + (1-α). Minimum for each alternative.

Choose the alternative that yields the maximum value of P.

HERE LET α = 0.8

WORKING NOTES:

H1 = 0.8 * 50000 + 0.2 * -45000 = 31000

H2 = 0.8 * 70000 + 0.2 * -80000 = 40000

H3 = 0.8 * 30000 + 0.2 * -10000 = 22000THUS ACCORDING TO HURWICZ CRITERION , COMPANY WILL

CHOOSE ALTERNATIVE – “CONSTRUCT”

ALTERNATIVES

STATE OF NATURE

HIGH MOD LOW NIL

EXP 50,000 25,000 - 25,000

- 45,000

CONST 70,000 30,000 - 40,000

- 80,000

SUBCONTRACT

30,000 15,000 - 1,000 - 10,000

H

-80000

-10000

40000

MAXOF

ROW

-45000

70,000

31000

MIN OF

ROW

50000

30000 22000

(V) LAPLACE CRITERION OR CRITERION OF RATIONALITY: Also known as equal probabilities criterion or

criterion of rationality. Since the probability of states of nature are not known, it is assumed that all states of nature will occur with equal probability, i.e. assign an equal probability. The working method is summarizing as follow:Determine expected value for each alternative; if n denotes the number of events and P’s denote the payoffs, then expected value is given by 1\n[P1+P2+….+Pn]Choose the alternative that yields the maximum value of P.

ALTERNATIVES

STATE OF NATURE EXPECTED PAYOFFHIGH MODERAT

ELOW NIL

EXPAND 50,000 25,000 - 25,000 - 45,000

CONSTRUCT

70,000 30,000 - 40,000 - 80,000

SUBCONTRACT

30,000 15,000 - 1,000 - 10,000

WORKING NOTES:

(E.P.)1 = ¼(50000 + 25000 - 25000 – 45000) = 1250

(E.P.)2 = ¼(70000 + 30000 – 40000 – 80000)= - 5000

(E.P.)3 = ¼(30000 + 15000 – 1000 – 10000 ) = 8500

THUS ACCORDING TO LAPLACE CRITERION , COMPANY WILL CHOOSE ALTERNATIVE – “SUBCONTRACT”

- 5000

1250

8500

ILLUSTRATION: THE FOLLOWING MATRIX GIVES THE PAYOFF OF DIFFERENT STRATEGIES S1, S2, S3 AGAINST CONDITIONS N1, N2, N3 AND N4.

INDICATE THE DECESION TAKEN UNDER THE FOLLOWING APPROACH:OPTIMISTICPESSIMISTICREGRETHURWICZ, THE DEGREE OF OPTIMISM BEING 0.7EQUAL PROBABILITY

STRATEGY CONDITIONS

N1 (Rs)

N2

(Rs)N3

(Rs)N4

(Rs)

S1 4000 - 100 6000 18000

S2 20000 5000 400 0

S3 20000 15000 - 2000 1000

SOLUTION(I) OPTIMISTIC CRITERION:

SO, ACCORDING TO O.C., MAXIMAX PAYOFF IS Rs. 20000 CORRESPONDING TO THE STRATEGY – “S2” AND “S3” .

(II) PESSIMISTIC CRITERION:

SO, ACCORDING TO P.C., MAXIMIN PAYOFF IS Rs. 0 CORRESPONDING TO THE STRATEGY – “S2”

STRATEGYCONDITIONS

N1 (Rs)

N2

(Rs)N3

(Rs)N4

(Rs)

S1 4000 - 100 6000 18000

S2 20000 5000 400 0

S3 20000 15000 - 2000 1000

MAX OF

ROW

2000020000

18000

STRATEGYCONDITIONS

N1 (Rs)

N2

(Rs)N3

(Rs)N4

(Rs)

S1 4000 - 100 6000 18000

S2 20000 5000 400 0

S3 20000 15000 - 2000 1000

MINOF

ROW

- 20000

- 100

(III) REGRET (SAVAGE CRITERION): THE BEST PAY OFFS FOR EACH STATE OF NATURE

N1, N2, N3 & N4 ARE Rs. 20000, Rs. 15000, Rs. 6000 & RS. 18000 RESPECTIVELY.

SUBSTRACTING FROM THESE THE PAYOFFS OF CORRESPONDING COLUMN WE GET

THE MINIMAX REGRET CORRESPONDS TO STRATEGY “S1”.

STRATEGYCONDITION

N1 (Rs)

N2

(Rs)N3

(Rs)N4

(Rs)

S1 16000 15100 0 0

S2 0 10000 5600 18000

S3 0 0 8000 17000

MAX OF

ROW

1700018000

16000

(IV) HURWICZ CRITERION:

STRATEGY

CONDITION H= .max + (1-) min

N1 (Rs)

N2

(Rs)N3

(Rs)N4

(Rs)

S1 4000 - 100 6000 18000

S2 20000 5000 400 0

S3 20000 15000 - 2000 1000

MAX OF

ROW

2000020000

18000

MINOF

ROW

- 20000

- 100

1340014000

12570

HERE = 0.7 WORKING NOTES:

H1 = 0.7 *18000 + 0.3 * (- 100) = 12570H2 = 0.7 * 20000+ 0.3 * 0 = 14000H3 = 0.7 * 20000 + 0.3 * (- 2000) = 13400

THE MAXIMUM VALUE OF H = Rs. 14000 WHICH CORRESPONDS TO

STRATEGY “S2”.

(V) EQUAL PROBABILITY CRITERION:

WORKING NOTES:(E.P.)1 = ¼(4000 - 100 + 6000 + 18000) = 6975.

(E.P.)2 = ¼(20000 + 5000 + 400 + 0 ) = 6350.

(E.P.)3 = ¼(20000 + 15000 – 2000 + 1000 ) = 8500.

THE MAXIMUM PAYOFF IS Rs. 8500 WHICH CORRESPONDS TO THE STRATEGY – “S3”.

STRATEGYCONDITION

N1 (Rs)

N2

(Rs)N3

(Rs)N4

(Rs)

S1 4000 - 100 6000 18000

S2 20000 5000 400 0

S3 20000 15000 - 2000 1000

EXPECTED PAYOFF

8500

6975

6350

DECESION MAKING UNDER RISK:

Here more than one state of nature exists and the decision maker has sufficient information to assign probabilities to each of these states.

These probabilities could be obtained from the past records or simply the subjective judgment of the decision maker.

Under conditions of risk, knowing the probability distribution of the state of nature, the best decision is to select the course of action which has the largest expected pay off value.

DECESION MAKING UNDER RISK:

Expected Value Criterion

or Expected

Monetary Value Criterion

Expected Opportunity Loss

Criterion or

Expected Value of Regret

Expected Value for Perfect

Information

Conditional Profit Table

Expected Profit Table

Conditional Profit Table

Conditional Loss table

Expected Loss Table

Conditional Profit Table with P.I.

Expected Profit Table with P.I.

ILLUSTRATIONA newspaper boy has the following probabilities of

selling a magazine:No. of copies sold Probability

10 0.1011 0.1512 0.2013 0.2514 0.30

Cost of the copy is 30 paisa and sale price is 50 paisa. He cannot return the unsold copies. How many should he order?

EXPECTED VALUE CRITERION:The expected monetary value for a given course

of action is the weighted sum of possible payoffs for each alternative. It is obtained by summing the payoffs for each course of action multiplied by the probabilities associated with state of nature. It consists of following steps:Construct a payoff table listing the alternative decisions and the various state of nature. Enter the conditional profit for each decision event combination along with the associated probabilities. (Construct Conditional profit table).

Calculate the EMV for each decision alternative by multiplying the conditional profits by assigned probabilities and adding the resulting conditional values. (Construct expected profit table).

Select the alternative that yields the highest EMV.

SOLUTIONCost Price = 30 paisa.Selling Price = 50 paisa.Profit = Selling price – Cost price = 20 paisa.STEP I : CONSTRUCT CONDITIONAL PROFIT TABLE

Possible Demand(No. of Copies)

Probability

Possible Stock Action10

Copies11

Copies12

Copies13

Copies14

Copies

10 0.10

11 0.15

12 0.20

13 0.25

14 0.30

Conditional Profit =

Profit * S.P. = 20 S.P. ;When D ≥ S

S.P. * D – C.P. * S = 50D – 30S; When D < S

200

200 220

240

260

280

200

200

200 220220

220

240

240 260

110 80130160190

210 180

230

140170

STEP II: CONSTRUCT EXPECTED PROFIT TABLE:Possible Demand(No. of Copies)

Probability


Copies11

Copies12

Copies13

Copies14

Copies

10 0.10

11 0.15

12 0.20

13 0.25

14 0.30

20 17 14 11 83040

33 28.5 24 19.544 48 42 36

50

55

60

65

57.560

66

72

78

84

200

222.5

220

205TOTAL EXPECTED PROFIT 215

THE NEWS BOY MUST, THEREFORE, ORDER 12 COPIES TO EARN THE HIGHEST POSSIBLE AVERAGE

DAILY PROFIT OF 222.5 PAISE

EXPECTED OPPORTUNITY LOSS CRITERION:EOL represents the amount by which maximum possible

profit will be reduced under various possible stock actions. The course of action that minimizes these losses or reductions is the optimal decision alternative. The procedure to calculate expected opportunity losses is as follows: Prepare the conditional profit table for each decision-event combination and write associated probabilities. (Construct Conditional profit table). For each event, determine the conditional opportunity loss (COL) by subtracting the payoff from the maximum payoff for that event. (Construct Conditional loss table).Calculate the expected opportunity loss for each decision alternative by multiplying the COL’s by the associated probabilities and then adding the values. (Construct Expected loss table).Select the alternative that yields the lowest EOL.

SOLUTIONCost Price = 30 paisa.Selling Price = 50 paisa.Profit = Selling price – Cost price = 20 paisa.STEP I : CONSTRUCT CONDITIONAL PROFIT TABLE

Conditional Profit =

Profit * S.P. = 20 S.P. ;When D ≥ S

S.P. * D – C.P. * S = 50D – 30S; When D < S


Probability


Copies11

Copies12

Copies13

Copies14

Copies

10 0.10

11 0.15

12 0.20

13 0.25

14 0.30

200

200 220

240

260

280

200

200

200 220220

220

240

240 260

110 80130160190

210

230

140170

180

STEP II: CONSTRUCT CONDITIONAL LOSS TABLE


Probability


Copies11

Copies12

Copies13

Copies14

Copies

10 0.10

11 0.15

12 0.20

13 0.25

14 0.30

020 0

00

06080

40 204060

2040 20

90 120906030

3030

6030

60

ROW-WISE SUBSTRACTION

ROW MAX – OTHER ELEMENTS OF ROW

STEP III: CONSTRUCT EXPECTED LOSS TABLE:

THE OPTIMUM STOCK ACTION IS THE ONE WHICH WILL MINIMIZE EXPECTED OPPORTUNITY LOSS; THIS ACTION CALLS FOR THE STOCKING OF 12 COPIES EACH DAY AT WHICH POINT THERE IS MINIMUM EXPECTED LOSS OF 27.5 PAISE.


Probability


Copies11

Copies12

Copies13

Copies14

Copies

10 0.10

11 0.15

12 0.20

13 0.25

14 0.30

03 0

00

01524

8 41018

512 6

9 1213.594.5

67.5

63

12

E.O.L. 50 35 27.5 30 45

EXPECTED VALUE FOR PERFECT INFORMATION:Perfect Information means complete and

accurate information about the future demand and that remove all the uncertainty for future.

EVPI represents the maximum amount of money the decision maker has to pay to get this additional information about the occurrence of various state of nature before a decision has to be made. The procedure to calculate expected value of perfect information is as follows: Construct conditional profit table with perfect information.Construct expected profit table with perfect information.Determine EVPI from relation;

EVPI = EPPI – max EMV

SOLUTION

Cost Price = 30 paisa.Selling Price = 50 paisa.Profit = Selling price – Cost price = 20 paisa. STEP I : CONSTRUCT CONDITIONAL PROFIT TABLEPossible Demand(No. of Copies)

Probability


Copies11

Copies12

Copies13

Copies14

Copies

10 0.10

11 0.15

12 0.20

13 0.25

14 0.30

200

220

240

260

280

STEP II: CONSTRUCT EXPECTED PROFIT TABLE WITH PERFECT INFORMATION:

DEMAND (No. Of Copies)

Probability Conditional Profit Under Certainty

Expected Profit With Perfect Information

10 0.10

11 0.15

12 0.20

13 0.25

14 0.30

200220

240

260

280

2033486584EPPI = 250

STEP III: The expected value of perfect information is

given byEVPI = EPPI – max EMV

= 250 – 222.5= 27.5 Paise

Thus this is the maximum amount which the newsboy

willing to pay, per day, for a perfect information.

= min E.O.L.

ILLUSTRATIONUnder an employment promotion program, it is proposed

to allow sale of newspapers on the buses during off peak hours. The vendor can purchase the newspaper at a special concessional rate of 25 paise per copy against the selling price of 40 paise. Any unsold copies are, however a dead loss. A vendor has estimated the following probability distribution for the no. of copies demanded:

a) How many copies should he ordered so that his expected profit will be maximum?

b) Compute EPPIc) The vendor is thinking of spending on a small market

survey to obtain additional information regarding the demand levels. How much should he be willing to spend on such a survey?

No. of copies

15 16 17 18 19 20

Probability 0.04 0.19 0.33 0.26 0.11 0.07

SOLUTION(a) CALCULATION OF EXPECTED PROFIT:Cost Price = 25 paisa.Selling Price = 40 paisa.Profit = Selling price – Cost price = 40 – 25 = 15

paisa.STEP I : CONSTRUCT CONDITIONAL PROFIT TABLE:Demand(No. of Copies) Prob


Copies16

Copies17

Copies18

Copies19

Copies20

Copies

15 0.04

16 0.19

17 0.33

18 0.26

19 0.11

20 0.07

225

225 240

255

270

285

225

225

225 240240

240

255

255 270

150 125165190215

230 205

245

175200

260

100140

180

220

285225 240 255 270 300

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