Quantitative Analysis for Management Chapter 3 Fundamentals of

Quantitative Analysis for Management Chapter 3 Fundamentals of Decision Theory Models 3 -1

Chapter Outline 3. 1 Introduction 3. 2 The Six Steps in Decision Theory 3. 3 Types of Decision-Making Environments 3. 4 Decision Making Under Risk 3. 5 Decision Making Under Uncertainty 3. 6 Marginal Analysis with a Large Number of Alternatives and States of Nature 3 -2

Learning Objectives Students will be able to: § List the steps of the decision-making process § Describe the types of decision-making environments § Use probability values to make decisions under risk § Make decisions under uncertainty where there is risk but probability values are not known § Use computers to solve basic decisionmaking problems 3 -3

Introduction ¨ Decision theory is an analytical and systematic way to tackle problems ¨ A good decision is based on logic. 3 -4

The Six Steps in Decision Theory ¨ Clearly define the problem at hand ¨ List the possible alternatives ¨ Identify the possible outcomes ¨ List the payoff or profit of each combination of alternatives and outcomes ¨ Select one of the mathematical decision theory models ¨ Apply the model and make your decision 3 -5

Decision Table for Thompson Lumber Favorable Unfavorable Market ($) Construct a large plant Construct a small plant Do nothing 200, 000 -180, 000 100, 000 -20, 000 0 0 3 -6

Types of Decision-Making Environments ¨ Type 1: Decision-making under certainty § decision-maker knows with certainty the consequences of every alternative or decision choice ¨ Type 2: Decision-making under risk § The decision-maker knows the probabilities of the various outcomes ¨ Decision-making under uncertainty § The decision-maker does not know the probabilities of the various outcomes 3 -7

Decision-Making Under Risk Expected Monetary Value: n EMV(Altern ative i) = å (Payoff * P(S j )) j=1 Sj where j = 1 to the number of states of nature, n 3 -8

Decision Table for Thompson Lumber Favorable Unfavorable Market ($) Construct a large plant Construct a small plant Do nothing EMV 200, 000 -180, 000 100, 000 -20, 000 40, 000 0 0 0. 50 3 -9 0

Expected Value of Perfect Information (EVPI ) ( ¨ EVPI places an upper bound on what one would pay for additional information ¨ EVPI is the expected value with perfect information minus the maximum EMV 3 -10

Expected Value With Perfect Information (EV|PI ) ( n EV | PI = å ( best outcome for state of nature j) * P(S j ) j =1 where j = 1 to the number of states of nature, n 3 -11

Expected Value of Perfect Information ¨ EVPI = EV|PI - maximum EMV 3 -12

Expected Value of Perfect Information Favorable Unfavorable EMV Market ($) Construct a large plant Construct a small plant 200, 000 40, 000 Do nothing 0 0. 50 3 -13

Expected Value of Perfect Information EVPI = expected value with perfect information - max(EMV) EMV = $200, 000*0. 50 + 0*0. 50 - $40, 000 = $60, 000 3 -14

Expected Opportunity Loss ¨ EOL is the cost of not picking the best solution ¨ EOL = Expected Regret We want to maximize EMV or minimize EOL 3 -15

Computing EOL - The Opportunity Loss Table 3 -16

The Opportunity Loss Table continued 3 -17

The Opportunity Loss Table continued 3 -18

Sensitivity Analysis EMV(Large Plant) = $200, 000 P - (1 -P)$180, 000 1 -P EMV(Small Plant) = $100, 000 P - $20, 000(1 -P) 1 -P EMV(Do Nothing) = $0 P + 0(1 -P) 1 -P 3 -19

Sensitivity Analysis - continued EMV t) n a l P l (Smal t) n la P e g r a L ( V EM 3 -20

Decision Making Under Uncertainty ¨ Maximax ¨ Maximin ¨ Equally likely (Laplace) ¨ Criterion of Realism ¨ Minimax 3 -21

Decision Making Under Uncertainty Maximax - Choose the alternative with the maximum output Construct a large plant Construct a small plant Do nothing Favorable Unfavorable Market ($) 200, 000 -180, 000 100, 000 -20, 000 0 0 3 -22

Decision Making Under Uncertainty Maximin - Choose the alternative with the maximum minimum output Construct a large plant Construct a small plant Do nothing Favorable Unfavorable Market ($) 200, 000 -180, 000 100, 000 -20, 000 0 0 3 -23

Decision Making Under Uncertainty Equally likely (Laplace) - Assume all states of nature to be equally likely, choose maximum EMV Construct a large plant Construct a small plant Do nothing Favorable Unfavorable Market ($) EMV 200, 000 -180, 000 100, 000 -20, 000 40, 000 0 0 0. 50 3 -24 0

Decision Making Under Uncertainty Criterion of Realism (Hurwicz): CR = *(row max) + (1 - )*(row min) Construct a large plant Construct a small plant Favorable Unfavorable Market ($) CR 200, 000 -180, 000 124, 000 100, 000 -20, 000 76, 000 0 Do nothing 0. 50 3 -25

Decision Making Under Uncertainty Minimax - choose the alternative with the minimum maximum Opportunity Loss Construct a large plant Construct a small plant Do nothing Favorable Unfavorable Market ($) Max in row 0 180, 000 100, 000 20, 000 100, 000 200, 000 0. 50 3 -26

Marginal Analysis ¨ P = probability that demand is greater than or equal to a given supply ¨ 1 -P = probability that demand will be less than supply ¨ MP = marginal profit ML = marginal loss ¨ Optimal decision rule is: P*MP (1 -P)*ML ¨ or 1 - <#>

Marginal Analysis Discrete Distributions ¨ Steps using Discrete Distributions: § Determine the value for P § Construct a probability table and add a cumulative probability column § Keep ordering inventory as long as the probability of selling at least one additional unit is greater than P 3 -28

Café du Donut Example 3 -29

Café du Donut Example continued ¨ Marginal profit = selling price - cost = $6 - $4 = $2 ¨ Marginal loss = cost ¨ Therefore: ML P ML + MP 4 4 = = = 0. 66 4 +2 6 3 -30

Café du Donut Example continued 3 -31

Marginal Analysis Normal Distribution ¨ = average or mean sales ¨ = standard deviation of sales ¨ MP = marginal profit ¨ ML = Marginal loss 3 -32

Marginal Analysis Discrete Distributions ¨ Steps using Normal Distributions: § Determine the value for P. ML P= ML + MP § Locate P on the normal distribution. For a given area under the curve, we find Z from the standard Normal table. - X § Using Z = we can now solve for X* * 3 -33

Joe’s Newsstand Example A ¨ ML = 4 ¨ MP = 6 ¨ = Average demand = 50 papers per day ¨ = Standard deviation of demand = 10 3 -34

Joe’s Newsstand Example A continued 4 ML = = 0. 40 ¨ Step 1: P = ML+ MP 4 + 6 ¨ Step 2: Look on the Normal table for P = 0. 6 (i. e. , 1 -. 04) Z = 0. 25, and 0. 25 = X - 50 * 10 or: X = 10 * 0. 25 + 50 = 52. 5 or 53 newspapers * 3 -35

Joe’s Newsstand Example A continued 3 -36

Joe’s Newsstand Example B ¨ ML = 8 ¨ MP = 2 ¨ = Average demand = 100 papers per day ¨ = Standard deviation of demand = 10 3 -37

Joe’s Newsstand Example B continued 8 ML = = 0. 80 ¨ Step 1: P = ML+ MP 8 + 2 ¨ Step 2: Z = -0. 84 for an area of 0. 80 and - 0. 84 = X - 1000 * 10 or: X = -8. 4 + 100 = 91. 6 or 92 newspapers * 3 -38

Joe’s Newsstand Example B continued 3 -39