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Chapter 07 – Decision Model
Learning Objectives After completing this chapter, you will be able to: • Know the steps of the decision-making process. • Describe the types of decision-making environments. • Make decisions under uncertainty. • Use probability values to make decisions under risk. • Revise probabilities using Bayesian analysis. • Develop accurate and useful decision trees. • Use POM-QM to solve basic decision-making problems.
Decision Analysis – The Concept  A decision is a choice between alternatives based on estimates of the values of those alternatives. Good Decision Bad Decision • based on reasoning • Not based on reasoning • consider all available data and possible alternatives • Do not consider all available data and possible alternatives • employ a quantitative approach • Do not employ a quantitative approach • occasionally result in an unexpected outcome; it is still a good decision if made properly • occasionally result in a good outcome if you are lucky; it is still a bad decision Decision Theory - an analytic and systematic approach to decision making.
Decision Analysis – The Concept  Decision analysis is a method used to develop an optimal strategy when a decision maker is faced with several decision alternatives with an uncertain or risk-filled pattern of future events.  A decision problem is characterized by: – Decision alternatives: a course of action or strategy that may be chosen by the decision maker , – States of nature: an outcome over which the decision maker has little or no control , and – Resulting payoffs (cost or revenues): A reward for all possible combinations of alternatives and states of nature ` The goal of decision analysis is to make a choice among alternatives that optimizes the resulting payoff in terms of a decision criterion.
Decision Analysis – The Concept The six steps in Decision Analysis: 1. Clearly define the problem (to maximize revenue or minimize cost?) 2. List the possible alternatives (actions/decisions) 3. Identify the possible outcomes (state on nature) 4. List the payoff (profit/reward or cost) 5. Select one of the decision theory models (on the basis of the operating environment and degree of uncertainty). 6. Apply the model and make your decision
Decision Analysis – Payoff Table Analysis  Decision analysis can be made by:  Payoff Table  Decision Tree
Decision Analysis – Payoff Table Analysis  Payoff table analysis can be applied when:  There is a finite set of discrete decision alternatives.  The outcome of a decision is a function of a single future event.  In a Payoff table -  The rows correspond to the possible decision alternatives.  The columns correspond to the possible future events.  Events (states of nature) are mutually exclusive and collectively exhaustive.  The table entries are the payoffs.
Decision Analysis – Payoff Table Analysis  Example: An investor would like to invest on four potential projects. – Project A – Project B – Project C – Project D  The return on each investment depends on the (uncertain) market behavior during the year  The investor builds a payoff table to help make the investment decision.
Decision Analysis – Payoff Table Analysis Payoff Table  The payoff table shows potential “payoff” depending upon likely economic conditions. Decision Decision Alternatives Alternatives Economic Condition/State of Nature Economic Condition/State of Nature Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Project D Project D 1500 1500 6000 6000 9500 9500  Note: Since the payoff in project C is higher than the payoff for D for every economic condition, project C is dominant. We can eliminate project D from consideration.
Decision Making Environments  Decision-making environments are classified into: 1. Decision making under certainty.  The future state-of-nature is assumed known. 2. Decision making under uncertainty.  There is no knowledge about the probability of the states of nature occurring.  Various outcomes are possible 3. Decision making under risk.  There is some knowledge of the probability of the states of nature occurring.
1. Decision Making under certainty  Here, the state of nature is certain (one state).  Only one outcome for each decision alternative.  Select decision that yields the highest return  Examples:  Product Mix problem  Blending / Diet problem  Distribution problem/ Transport Problem  Scheduling Problem All the topics we have covered so far! Optimization Problems
2. Decision Making under Uncertainty  Here, state of nature is uncertain (several possible states).  Various outcomes are possible for each decision alternative;  Decision maker cannot assign probabilities to the States of Nature  Many business problems contain uncertain elements that are impossible to ignore. Examples:  Developing a New Product (E.g. Concrete mixer – Market?)  Construction problem (E.g. Which model condominium house to construct? See exercise 5 later)
2. …under Uncertainty  The decision criteria are based on the decision maker’s attitude towards life (Optimistic, pessimistic, neither both). The criteria include:  Maximax Criterion - seeks the largest of maximum payoffs.  Maximin Criterion - seeks the largest of the minimum payoffs among the actions.  Minimax Regret Criterion - seeks the smallest of the maximum regrets among the actions.  The Criterion of Realism – seeks a weighted average where maximum and minimum rewards are weighted by some coefficient.  Principle of Insufficient Reasoning – seeks the largest payoffs among the sum obtained on each alternative across all events. Note: But, the outcome is still uncertain. Uncertain parameters become known only after a decision is made.
 This criterion is based on the best possible scenario.  It fits both an optimistic (Risk Taking) and an aggressive decision maker.  An optimistic decision maker believes that the best possible outcome will always take place regardless of the decision made.  An aggressive decision maker looks for the decision with the highest payoff (when payoff is profit).  To find an optimal decision.  Find the maximum payoff for each decision alternative.  Select the decision alternative that has the maximum of the “maximum” payoff. i) The Maximax Criterion
Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 If you are an optimist, you will decide on the basis of Maximax. Step 1: Step 1: Pick the max value for each alternative. 6100 6100 12080 12080 10375 10375 Step 2: Step 2:Then pick the alternative with max payoff. i) The Maximax Criterion …
 This criterion is based on the worst-case scenario.  It fits both a pessimistic (Risk Averse) and a conservative decision maker’s styles. o A pessimistic decision - the worst possible result will always occur. o A conservative decision - ensure a guaranteed minimum possible payoff. ii) Maxmin criterion
 To find an optimal decision  Record the minimum payoff across all states of nature for each decision.  Identify the decision with the maximum “minimum payoff.” ii) Maxmin criterion… Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 1: Pick the min value 1: Pick the min value for each alternative. for each alternative. 4075 4075 0 0 2500 2500 2: Then pick the alternative 2: Then pick the alternative with max payoff. with max payoff.
iii) The Minimax Regret Criterion  The Minimax Regret Criterion again fits both a pessimistic and a conservative decision maker approach.  The payoff table is based on “lost opportunity,” or “regret.”  The decision maker incurs regret by failing to choose the “best” decision. If I knew the future, how much I’d regret for my decision…
 To find an optimal decision, for each state of nature:  Determine the best payoff over all decisions.  Calculate the regret as the difference between its payoff value and this best payoff value.  For each decision find the maximum regret over all states of nature.  Select the decision alternative that has the minimum of these “maximum regrets.” iii) The Minimax Regret Criterion …
iii) The Minimax Regret Criterion … Alternatives Alternatives Regret Table Regret Table (opportunity loss Table) (opportunity loss Table) Recession Recession Normal Normal Boom Boom Project A Project A 0 0 2000 2000 5980 5980 Project B Project B 4075 4075 1750 1750 0 0 Project C Project C 1575 1575 0 0 1705 1705 Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Step 1: Step 1: Calculate Calculate the maximum for the maximum for each outcome. each outcome. 4075| 7000| 4075| 7000| 12080 12080 Stet 2: Stet 2: Prepare “Regret Table” by subtracting each outcome cell value from its maximum.
iii) The Minimax Regret Criterion … Alternatives Alternatives Regret Table Regret Table Recession Recession Normal Normal Boom Boom Project A Project A 0 0 2000 2000 5980 5980 Project B Project B 4075 4075 1750 1750 0 0 Project C Project C 1575 1575 0 0 1705 1705 Step 3: Pick the max value for each alternative. 5980 5980 4075 4075 1705 1705 Step 4: Pick the alternative with minimum regret.
 Also known as the weighted average or Hurwicz criterion.  A compromise between an optimistic and pessimistic decision (Risk Tradeoff) .  A coefficient of realism,  (0 <  <1), is selected by the decision maker to indicate optimism or pessimism about the future – When  is close to 1, the decision maker is optimistic. – When  is close to 0, the decision maker is pessimistic. Criterion of realism = (row maximum) + (1- )(row minimum)  A weighted average where maximum and minimum rewards are weighted by  and (1 - ), respectively. v) The Criterion of Realism
Assume coefficient of realism of =0.80. Then, weighted Averages: – Project A = (0.8)(6100) + (0.2)(4075) = 5695 – Project B = (0.8)(12080) + (0.2)(0) =9664 – Project C = (0.8)(10375) + (0.2)(2500) = 8800 Decision: Select the alternative with the highest weighted value, i.e. Project B Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Step 1: Step 1: (row maximum) + (1- )(row minimum) 5695 5695 9664 9664 8800 8800 Step 2: Step 2:Then pick the alternative with the highest weighted value. v) The Criterion of Realism
 This criterion might appeal to a decision maker who is neither pessimistic nor optimistic.  It assumes all the states of nature are equally likely to occur. i.e. no information about their likelihood.  The procedure to find an optimal decision.  For each decision add all the payoffs.  Select the decision with the largest sum (for profits). vi) The Principle of Insufficient Reason(Equal likelihood) Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Step 1: Step 1: Sum the payoffs for each alternative. 15175 15175 17330 17330 19875 19875 Step 2: Step 2:Then pick the alternative with max payoff.
1. A company would like to expand its production by setting up new plants. The course of actions to be decided and the state of nature are shown in the following pay off table. Sample Problem Make a decision based on different decision criteria. Let us do it with excel Alternatives Alternatives Economic Condition Economic Condition High Demand High Demand Moderate Demand Moderate Demand Low Demand Low Demand Large Plant Large Plant 200,000 200,000 100,000 100,000 -120,000 -120,000 Small Plant Small Plant 90,000 90,000 50,000 50,000 -20,000 -20,000 No Plant No Plant 0 0 0 0 0 0
3. Decision Making under Risk – i) The expected value (EV) criterion  The above four approaches we used involved Decision Making without Probabilities.  The probability estimate for the occurrence of each state of nature (if available) can be incorporated in the search for the optimal decision.  For each decision, calculate the expected payoff as follows: (The summation is calculated across all the states of nature)  Select the decision with the best expected payoff.  It is a kind of weighted average method. Expected Payoff = (Probability)(Payoff)
EV = (0.2)(4075) + (0.55)(5000) + (0.25)(6100) = 5090 i) The Expected value (EV) Criterion… Decision Alternatives State of Nature Expected Value Recession Normal Boom Project A 4075 5000 6100 5090 Project B 0 5250 12080 5907.50 Project C 2500 7000 10375 6943.75 Prior Prob. Prior Prob. 0.20 0.55 0.25 EV = (0.2)(0) + (0.55)(5250) + (0.25)(12080) = 5907.50 EV = (0.2)(2500) + (0.55)(7000) + (0.25)(10375) = 6943.75 The optimal decision
 The expected value criterion is useful generally in two cases:  Long run planning is appropriate, and decision situations repeat themselves.  The decision maker is risk neutral. i) The Expected value (EV) Criterion…
 Frequently information is available which can improve the probability estimates for the states of nature.  May be interested to purchase it and thus make a better decision How valuable would this information be?  EVPI is the gain in expected return obtained from knowing with certainty the future state of nature.  The EVPI provides an upper bound on the expected value for additional information. i.e. The maximum amount you would pay to the source/someone to provide information. ii) The Expected Value of Perfect Information (EVPI) EREV ERPI EVPI   criterion EV by return ected EREV ormation perfect with return ected ERPI where exp inf exp ,  
 If it was known with certainty that there will be a “Boom” in the economic condition, … ... the optimal decision would be to invest in …. B Decision Decision Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Prior Prob. Prior Prob. 0.20 0.55 0.25 Similarly,… ii) The EVPI…
Expected Return with Perfect information (ERPI) = Σ(best payoff in state of nature i) (probability of i) = 0.2(4075)+0.55(7000)+0.25(12080) = 7685 Expected Return without additional information= EV criterion = 6943.75 EVPI = ERPI - EREV = 7685 – 6943.75 = 741.25  “Perfect” information is extremely rare.  You would be willing to pay some amount less than $741 depending on how reliable the information is perceived to be. Decision Alternatives State of Nature Expected Value Recession Normal Boom Project A 4075 5000 6100 5090 Project B 0 5250 12080 5907.50 Project C 2500 7000 10375 6943.75 Prior Prob. Prior Prob. 0.20 0.55 0.25 ii) The EVPI…
iii) Decision Making with Imperfect Information or sample information (EVSI) (Bayesian Analysis )  Sometimes, additional information may assist in refining original probability estimates, and help improve decision making. E.g, conducting marketing research survey.  Such a market survey will not provide with perfect information but it may help quite a bit nevertheless.  Bayesian Statistics play a role in assessing additional information obtained from various sources.  The forecast predicts “negative” or “positive” economic condition.  Say, the statistics regarding the forecast are: The forecast Predicted When the economic condition shows… When the economic condition shows… Recession Recession Normal Normal Boom Boom Positive Positive 30% 30% 50% 50% 80% 80% Negative Negative 70% 70% 50% 50% 20% 20%
The EVSI is the expected gain from making decisions based on sample information. EVSI = ERSI – EREV To find Expected payoff with forecast or ERSI the investor should determine the revised probability when the forecast is Positive and Negative. i.e. posterior probabilities.  P(Recession| The forecast predicted “Positive”)  P(Normal | The forecast predicted “Positive”)  P(Boom | The forecast predicted “Positive ”)  P(Recession | The forecast predicted “Negative”)  P(Normal | The forecast predicted “Negative”)  P(Boom | The forecast predicted “Negative”) Positive condition Negative Condition iii) The EVSI…
 Bayes’ Theorem provides a procedure to calculate these probabilities. Joint probability Sum of joint probability P(Ai) = Posterior Probabilities Probabilities determined after the additional info becomes available. Joint Probability Prior Probability * probability after the additional info becomes available. iii) The EVSI…
 The tabular approach to calculating posterior probabilities for “Positive” economical forecast. 0.06 0.535 0.535 Probability(Forecast = positive) = 0.535 State of Nature Prior Prob. Prob. (state/+ve Joint Prob. Posterior Prob. Recession 0.20 0.30 0.06 0.112 Normal 0.55 0.50 0.275 0.514 Boom 0.25 0.80 0.20 0.374 X = The posterior/revised probability that the economy shows “Recession” given that the forecast is “positive” The joint probability that the forecast is “Positive” and the economy shows “Recession”. iii) The EVSI…
 The tabular approach to calculating posterior probabilities for “Negative” economical forecast. Probability(Forecast = Negative) = 0.465 0.14 0.465 0.465 State of Nature Prior Prob. Prob. (state/-ve) Joint Prob. Posterior Prob. Recession 0.20 0.70 0.14 0.301 Normal 0.55 0.50 0.275 0.591 Boom 0.25 0.20 0.05 0.108 X = iii) The EVSI…
EV(Invest in……. |“Positive” forecast) = =0.112( )+0.514( )+0.374( ) = EV(Invest in ……. | “Negative” forecast) = =0.301( )+0.591( )+0.108( ) = 4075 5000 6100 5307.48 A 4075 5000 6100 A 4839.78 Decision Decision Alternatives Alternatives Expected value of Sample Information Expected value of Sample Information Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Prob. (state/+ve) Prob. (state/+ve) 0.112 0.514 0.374 Prob. (state/-ve) Prob. (state/-ve) 0.301 0.591 0.108  Then, revise the expected return for each decision using the posterior probabilities as follows: iii) The EVSI…
 The revised expected values for each decision: Positive forecast Negative forecast EV(A|Positive) = 5307.48 EV(A|Negative) = 4839.78 EV(B|Positive) = 7214.49 EV(B|Negative) = 4403.76 EV(C|Positive) = 7757.01 EV(C|Negative) = 6008.06 If the forecast is “Positive” Invest in C. If the forecast is “Negative” Invest in C. iii) The EVSI…
 Since the forecast is unknown before it is purchased, the investor can only calculate the expected return from purchasing it.  Expected return when buying the forecast (ERSI) = P(Forecast is positive)x(EV(C|Forecast is positive))+ P(Forecast is negative”)·(EV(C|Forecast is negative)) = (.535)(7757.01) + (.465)(6008.06) = 6943.75  The expected gain from buying the forecast is: EVSI = ERSI – EREV = 6943.75 – 6943.75 = 0  Efficiency of Sample Information= (EVSI/EVPI)*100 =0/741.25 = 0 Thus, market survey is only 0% as efficient as perfect information! iii) The EVSI…
A company would like to expand its production by setting up new plants. The course of actions to be decided and the state of nature are shown in the following pay off table. Sample Problem • Make a decision based on EV criterion. • What is EVPI? Let us do it with excel Alternatives Alternatives Economic Condition Economic Condition High Demand High Demand Moderate Demand Moderate Demand Low Demand Low Demand Large Plant Large Plant 200,000 200,000 100,000 100,000 -120,000 -120,000 Small Plant Small Plant 90,000 90,000 50,000 50,000 -20,000 -20,000 No Plant No Plant 0 0 0 0 0 0 Probability Probability 0.30 0.30 0.50 0.50 0.20 0.20
Decision Trees  The Payoff Table approach is useful for a non-sequential or single stage.  However, many real-world decision problems consists of a sequence of dependent decisions.  Decision Trees are useful in analyzing multi-stage decision processes.  A Decision Tree is a chronological representation of the decision process.
Decision Trees - Structure  The Decision tree is composed of nodes and branches. A branch emanating from chance node corresponds to a particular state of nature, and includes the probability of this state of nature is Chance Branch (CB). A branch emanating from a decision node corresponds to a decision alternative is Decision Branch (DB). It includes a cost or benefit value. - A decision node is represented by a rectangle - A chance node is represented by a circle. - At the end of each branch, there is an end node represented by a triangle. Decision node Chance node DB 2 DB 3 P(S2) P(S1 ) P(S3 ) P(S2) P(S1 ) P(S3 ) D B 1 CB
 A decision tree is constructed from left to right.  The decision tree must show all the possible paths that the decision maker might follow through time. Including all possible decision alternatives. Decision Trees - Structure
Decision Trees - Analysis Six Steps of Decision Tree Analysis: 1. Define the problem 2. Structure/draw the decision tree 3. Assign probabilities and payoff to the states of nature (chance branch) 4. Compute the expected payoff at each end node. 5. At the chance node, calculate the average (i.e. expected) payoff, which is usually termed as Expected Monetary Value (EMV). If there is no chance event for a particular decision branch, it’s EMV is equal to the payoff. 6. Select the decision with the highest EMV for each chance node. Otherwise, if dealing with costs, select the decision with the lowest EMV.
 Suppose you bought 500 units of X at $10 each. A dealer has offered to buy these from you at $14 each (you can make $4/unit profit).  Alternatively, you can sell these yourself for $16 each ($6/unit profit) but the demand is uncertain. The demand distribution is shown in the table. Decision Trees - Example Demand: X 300 400 500 600 Pr(X) 0.30 0.45 0.20 0.05  Note: if demand exceeds 500, you will sell all 500. On the other hand, if demand is under 500, you will have leftover units. These leftover items can disposed off for $7 each ($3 loss, the dealer will no longer buy these leftover units from you). What’s your decision? What’s your decision?
 Start with the tree having 2 branches (DB) at the decision point.  There are no chance events in the dealer sale branch,  For the self sale, there are 4 mutually exclusive possibilities (state of nature). Decision Trees - Solution Dealer Dealer Sale Sale Self sale Self sale 500, 20% 500, 20% 600, 5% 600, 5% 400, 45% 400, 45% 300, 30% 300, 30%
Decision Trees - Solution Payoff = 300*6 – 200*3 = 1200 Payoff = 300*6 – 200*3 = 1200 Payoff = 400*6 – 100*3 = 2100 Payoff = 400*6 – 100*3 = 2100 Payoff = 500*6 = 3000 Payoff = 500*6 = 3000 Payoff = 500*6 = 3000 Payoff = 500*6 = 3000 Payoff = 500*4 = 2000 Payoff = 500*4 = 2000 EMV = 2000 EMV = 2000 EMV = 0.3*1200 + 0.45*2100 EMV = 0.3*1200 + 0.45*2100 + + 0.2* 3000 + 0.05*3000 0.2* 3000 + 0.05*3000 = 2055 = 2055 Your decision? Your decision? Self sale Self sale Dealer Dealer Sale Sale Self sale Self sale 500, 20% 500, 20% 600, 5% 600, 5% 400, 45% 400, 45% 300, 30% 300, 30%
1. A company would like to expand its production by setting up new plants. The course of actions to be decided and the state of nature are shown in the following pay off table. Sample Problem • Make a decision using a decision tree analysis. Let us do it with excel Alternatives Alternatives Economic Condition Economic Condition High Demand High Demand Moderate Demand Moderate Demand Low Demand Low Demand Large Plant Large Plant 200,000 200,000 100,000 100,000 -120,000 -120,000 Small Plant Small Plant 90,000 90,000 50,000 50,000 -20,000 -20,000 No Plant No Plant 0 0 0 0 0 0 Probability Probability 0.30 0.30 0.50 0.50 0.20 0.20
THANK YOU
1. What are the elements for rational decision making? 2. Why decisions are hard? 3. What is a good decision? A bad decision? 4. A broker with $1000 has five potential investments A, B, C, D and E. The return on each investment depends on the (uncertain) market behavior during the year. The broker built a payoff table to help make the investment decision. Find an optimal decision based on: a) Maximax Criterion; b) Maximin Criterion c) Minimax Regret Criterion; and d) principle of Insufficient Reasoning Assignment
Assignment Decision States of Nature Alternatives Large Rise Small Rise No Change Small Fall Large Fall A -100 100 200 300 0 B 250 200 150 -100 -150 C 500 250 100 -200 -600 D 60 60 60 60 60 E 200 150 150 -200 -150
5. A building construction company is planning to construct luxury condominiums on its site. The company has a capacity to construct up to 250 condominium blocks. The financial success of the project depends upon the number of condominium blocks to be constructed and the chance event concerning the demand for the condominiums. The decision problem is to select the number of condominium blocks to be constructed that will lead to the largest profit given the uncertainty concerning the demand for the condominiums. The cost and revenue of construction per block are 6000 and 14000 (x1000)birr, respectively. Assignment
Assignment Low (50 units) Medium (100 units) High (150 units) Build 50 400,000 400,000 400,000 Build 100 100,000 800,000 800,000 Build 150 (200,000) 500,000 1,200,000 State of Nature Demand Alternative Actions Hint: Payoff Table Find an optimal decision: a) Maxmin Criterion b) Minimax Regret Criterion c) Maximax Criterion d) Principle of Insufficient Reasoning

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Lecture notes about system analysis 7.ppt

  • 1.
    Chapter 07 –Decision Model
  • 2.
    Learning Objectives After completingthis chapter, you will be able to: • Know the steps of the decision-making process. • Describe the types of decision-making environments. • Make decisions under uncertainty. • Use probability values to make decisions under risk. • Revise probabilities using Bayesian analysis. • Develop accurate and useful decision trees. • Use POM-QM to solve basic decision-making problems.
  • 3.
    Decision Analysis –The Concept  A decision is a choice between alternatives based on estimates of the values of those alternatives. Good Decision Bad Decision • based on reasoning • Not based on reasoning • consider all available data and possible alternatives • Do not consider all available data and possible alternatives • employ a quantitative approach • Do not employ a quantitative approach • occasionally result in an unexpected outcome; it is still a good decision if made properly • occasionally result in a good outcome if you are lucky; it is still a bad decision Decision Theory - an analytic and systematic approach to decision making.
  • 4.
    Decision Analysis –The Concept  Decision analysis is a method used to develop an optimal strategy when a decision maker is faced with several decision alternatives with an uncertain or risk-filled pattern of future events.  A decision problem is characterized by: – Decision alternatives: a course of action or strategy that may be chosen by the decision maker , – States of nature: an outcome over which the decision maker has little or no control , and – Resulting payoffs (cost or revenues): A reward for all possible combinations of alternatives and states of nature ` The goal of decision analysis is to make a choice among alternatives that optimizes the resulting payoff in terms of a decision criterion.
  • 5.
    Decision Analysis –The Concept The six steps in Decision Analysis: 1. Clearly define the problem (to maximize revenue or minimize cost?) 2. List the possible alternatives (actions/decisions) 3. Identify the possible outcomes (state on nature) 4. List the payoff (profit/reward or cost) 5. Select one of the decision theory models (on the basis of the operating environment and degree of uncertainty). 6. Apply the model and make your decision
  • 6.
    Decision Analysis –Payoff Table Analysis  Decision analysis can be made by:  Payoff Table  Decision Tree
  • 7.
    Decision Analysis –Payoff Table Analysis  Payoff table analysis can be applied when:  There is a finite set of discrete decision alternatives.  The outcome of a decision is a function of a single future event.  In a Payoff table -  The rows correspond to the possible decision alternatives.  The columns correspond to the possible future events.  Events (states of nature) are mutually exclusive and collectively exhaustive.  The table entries are the payoffs.
  • 8.
    Decision Analysis –Payoff Table Analysis  Example: An investor would like to invest on four potential projects. – Project A – Project B – Project C – Project D  The return on each investment depends on the (uncertain) market behavior during the year  The investor builds a payoff table to help make the investment decision.
  • 9.
    Decision Analysis –Payoff Table Analysis Payoff Table  The payoff table shows potential “payoff” depending upon likely economic conditions. Decision Decision Alternatives Alternatives Economic Condition/State of Nature Economic Condition/State of Nature Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Project D Project D 1500 1500 6000 6000 9500 9500  Note: Since the payoff in project C is higher than the payoff for D for every economic condition, project C is dominant. We can eliminate project D from consideration.
  • 10.
    Decision Making Environments Decision-making environments are classified into: 1. Decision making under certainty.  The future state-of-nature is assumed known. 2. Decision making under uncertainty.  There is no knowledge about the probability of the states of nature occurring.  Various outcomes are possible 3. Decision making under risk.  There is some knowledge of the probability of the states of nature occurring.
  • 11.
    1. Decision Makingunder certainty  Here, the state of nature is certain (one state).  Only one outcome for each decision alternative.  Select decision that yields the highest return  Examples:  Product Mix problem  Blending / Diet problem  Distribution problem/ Transport Problem  Scheduling Problem All the topics we have covered so far! Optimization Problems
  • 12.
    2. Decision Makingunder Uncertainty  Here, state of nature is uncertain (several possible states).  Various outcomes are possible for each decision alternative;  Decision maker cannot assign probabilities to the States of Nature  Many business problems contain uncertain elements that are impossible to ignore. Examples:  Developing a New Product (E.g. Concrete mixer – Market?)  Construction problem (E.g. Which model condominium house to construct? See exercise 5 later)
  • 13.
    2. …under Uncertainty The decision criteria are based on the decision maker’s attitude towards life (Optimistic, pessimistic, neither both). The criteria include:  Maximax Criterion - seeks the largest of maximum payoffs.  Maximin Criterion - seeks the largest of the minimum payoffs among the actions.  Minimax Regret Criterion - seeks the smallest of the maximum regrets among the actions.  The Criterion of Realism – seeks a weighted average where maximum and minimum rewards are weighted by some coefficient.  Principle of Insufficient Reasoning – seeks the largest payoffs among the sum obtained on each alternative across all events. Note: But, the outcome is still uncertain. Uncertain parameters become known only after a decision is made.
  • 14.
     This criterionis based on the best possible scenario.  It fits both an optimistic (Risk Taking) and an aggressive decision maker.  An optimistic decision maker believes that the best possible outcome will always take place regardless of the decision made.  An aggressive decision maker looks for the decision with the highest payoff (when payoff is profit).  To find an optimal decision.  Find the maximum payoff for each decision alternative.  Select the decision alternative that has the maximum of the “maximum” payoff. i) The Maximax Criterion
  • 15.
    Alternatives Alternatives Economic Condition EconomicCondition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 If you are an optimist, you will decide on the basis of Maximax. Step 1: Step 1: Pick the max value for each alternative. 6100 6100 12080 12080 10375 10375 Step 2: Step 2:Then pick the alternative with max payoff. i) The Maximax Criterion …
  • 16.
     This criterionis based on the worst-case scenario.  It fits both a pessimistic (Risk Averse) and a conservative decision maker’s styles. o A pessimistic decision - the worst possible result will always occur. o A conservative decision - ensure a guaranteed minimum possible payoff. ii) Maxmin criterion
  • 17.
     To findan optimal decision  Record the minimum payoff across all states of nature for each decision.  Identify the decision with the maximum “minimum payoff.” ii) Maxmin criterion… Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 1: Pick the min value 1: Pick the min value for each alternative. for each alternative. 4075 4075 0 0 2500 2500 2: Then pick the alternative 2: Then pick the alternative with max payoff. with max payoff.
  • 18.
    iii) The MinimaxRegret Criterion  The Minimax Regret Criterion again fits both a pessimistic and a conservative decision maker approach.  The payoff table is based on “lost opportunity,” or “regret.”  The decision maker incurs regret by failing to choose the “best” decision. If I knew the future, how much I’d regret for my decision…
  • 19.
     To findan optimal decision, for each state of nature:  Determine the best payoff over all decisions.  Calculate the regret as the difference between its payoff value and this best payoff value.  For each decision find the maximum regret over all states of nature.  Select the decision alternative that has the minimum of these “maximum regrets.” iii) The Minimax Regret Criterion …
  • 20.
    iii) The MinimaxRegret Criterion … Alternatives Alternatives Regret Table Regret Table (opportunity loss Table) (opportunity loss Table) Recession Recession Normal Normal Boom Boom Project A Project A 0 0 2000 2000 5980 5980 Project B Project B 4075 4075 1750 1750 0 0 Project C Project C 1575 1575 0 0 1705 1705 Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Step 1: Step 1: Calculate Calculate the maximum for the maximum for each outcome. each outcome. 4075| 7000| 4075| 7000| 12080 12080 Stet 2: Stet 2: Prepare “Regret Table” by subtracting each outcome cell value from its maximum.
  • 21.
    iii) The MinimaxRegret Criterion … Alternatives Alternatives Regret Table Regret Table Recession Recession Normal Normal Boom Boom Project A Project A 0 0 2000 2000 5980 5980 Project B Project B 4075 4075 1750 1750 0 0 Project C Project C 1575 1575 0 0 1705 1705 Step 3: Pick the max value for each alternative. 5980 5980 4075 4075 1705 1705 Step 4: Pick the alternative with minimum regret.
  • 22.
     Also knownas the weighted average or Hurwicz criterion.  A compromise between an optimistic and pessimistic decision (Risk Tradeoff) .  A coefficient of realism,  (0 <  <1), is selected by the decision maker to indicate optimism or pessimism about the future – When  is close to 1, the decision maker is optimistic. – When  is close to 0, the decision maker is pessimistic. Criterion of realism = (row maximum) + (1- )(row minimum)  A weighted average where maximum and minimum rewards are weighted by  and (1 - ), respectively. v) The Criterion of Realism
  • 23.
    Assume coefficient ofrealism of =0.80. Then, weighted Averages: – Project A = (0.8)(6100) + (0.2)(4075) = 5695 – Project B = (0.8)(12080) + (0.2)(0) =9664 – Project C = (0.8)(10375) + (0.2)(2500) = 8800 Decision: Select the alternative with the highest weighted value, i.e. Project B Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Step 1: Step 1: (row maximum) + (1- )(row minimum) 5695 5695 9664 9664 8800 8800 Step 2: Step 2:Then pick the alternative with the highest weighted value. v) The Criterion of Realism
  • 24.
     This criterionmight appeal to a decision maker who is neither pessimistic nor optimistic.  It assumes all the states of nature are equally likely to occur. i.e. no information about their likelihood.  The procedure to find an optimal decision.  For each decision add all the payoffs.  Select the decision with the largest sum (for profits). vi) The Principle of Insufficient Reason(Equal likelihood) Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Step 1: Step 1: Sum the payoffs for each alternative. 15175 15175 17330 17330 19875 19875 Step 2: Step 2:Then pick the alternative with max payoff.
  • 25.
    1. A companywould like to expand its production by setting up new plants. The course of actions to be decided and the state of nature are shown in the following pay off table. Sample Problem Make a decision based on different decision criteria. Let us do it with excel Alternatives Alternatives Economic Condition Economic Condition High Demand High Demand Moderate Demand Moderate Demand Low Demand Low Demand Large Plant Large Plant 200,000 200,000 100,000 100,000 -120,000 -120,000 Small Plant Small Plant 90,000 90,000 50,000 50,000 -20,000 -20,000 No Plant No Plant 0 0 0 0 0 0
  • 26.
    3. Decision Makingunder Risk – i) The expected value (EV) criterion  The above four approaches we used involved Decision Making without Probabilities.  The probability estimate for the occurrence of each state of nature (if available) can be incorporated in the search for the optimal decision.  For each decision, calculate the expected payoff as follows: (The summation is calculated across all the states of nature)  Select the decision with the best expected payoff.  It is a kind of weighted average method. Expected Payoff = (Probability)(Payoff)
  • 27.
    EV = (0.2)(4075)+ (0.55)(5000) + (0.25)(6100) = 5090 i) The Expected value (EV) Criterion… Decision Alternatives State of Nature Expected Value Recession Normal Boom Project A 4075 5000 6100 5090 Project B 0 5250 12080 5907.50 Project C 2500 7000 10375 6943.75 Prior Prob. Prior Prob. 0.20 0.55 0.25 EV = (0.2)(0) + (0.55)(5250) + (0.25)(12080) = 5907.50 EV = (0.2)(2500) + (0.55)(7000) + (0.25)(10375) = 6943.75 The optimal decision
  • 28.
     The expectedvalue criterion is useful generally in two cases:  Long run planning is appropriate, and decision situations repeat themselves.  The decision maker is risk neutral. i) The Expected value (EV) Criterion…
  • 29.
     Frequently informationis available which can improve the probability estimates for the states of nature.  May be interested to purchase it and thus make a better decision How valuable would this information be?  EVPI is the gain in expected return obtained from knowing with certainty the future state of nature.  The EVPI provides an upper bound on the expected value for additional information. i.e. The maximum amount you would pay to the source/someone to provide information. ii) The Expected Value of Perfect Information (EVPI) EREV ERPI EVPI   criterion EV by return ected EREV ormation perfect with return ected ERPI where exp inf exp ,  
  • 30.
     If itwas known with certainty that there will be a “Boom” in the economic condition, … ... the optimal decision would be to invest in …. B Decision Decision Alternatives Alternatives Economic Condition Economic Condition Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Prior Prob. Prior Prob. 0.20 0.55 0.25 Similarly,… ii) The EVPI…
  • 31.
    Expected Return withPerfect information (ERPI) = Σ(best payoff in state of nature i) (probability of i) = 0.2(4075)+0.55(7000)+0.25(12080) = 7685 Expected Return without additional information= EV criterion = 6943.75 EVPI = ERPI - EREV = 7685 – 6943.75 = 741.25  “Perfect” information is extremely rare.  You would be willing to pay some amount less than $741 depending on how reliable the information is perceived to be. Decision Alternatives State of Nature Expected Value Recession Normal Boom Project A 4075 5000 6100 5090 Project B 0 5250 12080 5907.50 Project C 2500 7000 10375 6943.75 Prior Prob. Prior Prob. 0.20 0.55 0.25 ii) The EVPI…
  • 32.
    iii) Decision Makingwith Imperfect Information or sample information (EVSI) (Bayesian Analysis )  Sometimes, additional information may assist in refining original probability estimates, and help improve decision making. E.g, conducting marketing research survey.  Such a market survey will not provide with perfect information but it may help quite a bit nevertheless.  Bayesian Statistics play a role in assessing additional information obtained from various sources.  The forecast predicts “negative” or “positive” economic condition.  Say, the statistics regarding the forecast are: The forecast Predicted When the economic condition shows… When the economic condition shows… Recession Recession Normal Normal Boom Boom Positive Positive 30% 30% 50% 50% 80% 80% Negative Negative 70% 70% 50% 50% 20% 20%
  • 33.
    The EVSI isthe expected gain from making decisions based on sample information. EVSI = ERSI – EREV To find Expected payoff with forecast or ERSI the investor should determine the revised probability when the forecast is Positive and Negative. i.e. posterior probabilities.  P(Recession| The forecast predicted “Positive”)  P(Normal | The forecast predicted “Positive”)  P(Boom | The forecast predicted “Positive ”)  P(Recession | The forecast predicted “Negative”)  P(Normal | The forecast predicted “Negative”)  P(Boom | The forecast predicted “Negative”) Positive condition Negative Condition iii) The EVSI…
  • 34.
     Bayes’ Theoremprovides a procedure to calculate these probabilities. Joint probability Sum of joint probability P(Ai) = Posterior Probabilities Probabilities determined after the additional info becomes available. Joint Probability Prior Probability * probability after the additional info becomes available. iii) The EVSI…
  • 35.
     The tabularapproach to calculating posterior probabilities for “Positive” economical forecast. 0.06 0.535 0.535 Probability(Forecast = positive) = 0.535 State of Nature Prior Prob. Prob. (state/+ve Joint Prob. Posterior Prob. Recession 0.20 0.30 0.06 0.112 Normal 0.55 0.50 0.275 0.514 Boom 0.25 0.80 0.20 0.374 X = The posterior/revised probability that the economy shows “Recession” given that the forecast is “positive” The joint probability that the forecast is “Positive” and the economy shows “Recession”. iii) The EVSI…
  • 36.
     The tabularapproach to calculating posterior probabilities for “Negative” economical forecast. Probability(Forecast = Negative) = 0.465 0.14 0.465 0.465 State of Nature Prior Prob. Prob. (state/-ve) Joint Prob. Posterior Prob. Recession 0.20 0.70 0.14 0.301 Normal 0.55 0.50 0.275 0.591 Boom 0.25 0.20 0.05 0.108 X = iii) The EVSI…
  • 37.
    EV(Invest in……. |“Positive”forecast) = =0.112( )+0.514( )+0.374( ) = EV(Invest in ……. | “Negative” forecast) = =0.301( )+0.591( )+0.108( ) = 4075 5000 6100 5307.48 A 4075 5000 6100 A 4839.78 Decision Decision Alternatives Alternatives Expected value of Sample Information Expected value of Sample Information Recession Recession Normal Normal Boom Boom Project A Project A 4075 4075 5000 5000 6100 6100 Project B Project B 0 0 5250 5250 12080 12080 Project C Project C 2500 2500 7000 7000 10375 10375 Prob. (state/+ve) Prob. (state/+ve) 0.112 0.514 0.374 Prob. (state/-ve) Prob. (state/-ve) 0.301 0.591 0.108  Then, revise the expected return for each decision using the posterior probabilities as follows: iii) The EVSI…
  • 38.
     The revisedexpected values for each decision: Positive forecast Negative forecast EV(A|Positive) = 5307.48 EV(A|Negative) = 4839.78 EV(B|Positive) = 7214.49 EV(B|Negative) = 4403.76 EV(C|Positive) = 7757.01 EV(C|Negative) = 6008.06 If the forecast is “Positive” Invest in C. If the forecast is “Negative” Invest in C. iii) The EVSI…
  • 39.
     Since theforecast is unknown before it is purchased, the investor can only calculate the expected return from purchasing it.  Expected return when buying the forecast (ERSI) = P(Forecast is positive)x(EV(C|Forecast is positive))+ P(Forecast is negative”)·(EV(C|Forecast is negative)) = (.535)(7757.01) + (.465)(6008.06) = 6943.75  The expected gain from buying the forecast is: EVSI = ERSI – EREV = 6943.75 – 6943.75 = 0  Efficiency of Sample Information= (EVSI/EVPI)*100 =0/741.25 = 0 Thus, market survey is only 0% as efficient as perfect information! iii) The EVSI…
  • 40.
    A company wouldlike to expand its production by setting up new plants. The course of actions to be decided and the state of nature are shown in the following pay off table. Sample Problem • Make a decision based on EV criterion. • What is EVPI? Let us do it with excel Alternatives Alternatives Economic Condition Economic Condition High Demand High Demand Moderate Demand Moderate Demand Low Demand Low Demand Large Plant Large Plant 200,000 200,000 100,000 100,000 -120,000 -120,000 Small Plant Small Plant 90,000 90,000 50,000 50,000 -20,000 -20,000 No Plant No Plant 0 0 0 0 0 0 Probability Probability 0.30 0.30 0.50 0.50 0.20 0.20
  • 41.
    Decision Trees  ThePayoff Table approach is useful for a non-sequential or single stage.  However, many real-world decision problems consists of a sequence of dependent decisions.  Decision Trees are useful in analyzing multi-stage decision processes.  A Decision Tree is a chronological representation of the decision process.
  • 42.
    Decision Trees -Structure  The Decision tree is composed of nodes and branches. A branch emanating from chance node corresponds to a particular state of nature, and includes the probability of this state of nature is Chance Branch (CB). A branch emanating from a decision node corresponds to a decision alternative is Decision Branch (DB). It includes a cost or benefit value. - A decision node is represented by a rectangle - A chance node is represented by a circle. - At the end of each branch, there is an end node represented by a triangle. Decision node Chance node DB 2 DB 3 P(S2) P(S1 ) P(S3 ) P(S2) P(S1 ) P(S3 ) D B 1 CB
  • 43.
     A decisiontree is constructed from left to right.  The decision tree must show all the possible paths that the decision maker might follow through time. Including all possible decision alternatives. Decision Trees - Structure
  • 44.
    Decision Trees -Analysis Six Steps of Decision Tree Analysis: 1. Define the problem 2. Structure/draw the decision tree 3. Assign probabilities and payoff to the states of nature (chance branch) 4. Compute the expected payoff at each end node. 5. At the chance node, calculate the average (i.e. expected) payoff, which is usually termed as Expected Monetary Value (EMV). If there is no chance event for a particular decision branch, it’s EMV is equal to the payoff. 6. Select the decision with the highest EMV for each chance node. Otherwise, if dealing with costs, select the decision with the lowest EMV.
  • 45.
     Suppose youbought 500 units of X at $10 each. A dealer has offered to buy these from you at $14 each (you can make $4/unit profit).  Alternatively, you can sell these yourself for $16 each ($6/unit profit) but the demand is uncertain. The demand distribution is shown in the table. Decision Trees - Example Demand: X 300 400 500 600 Pr(X) 0.30 0.45 0.20 0.05  Note: if demand exceeds 500, you will sell all 500. On the other hand, if demand is under 500, you will have leftover units. These leftover items can disposed off for $7 each ($3 loss, the dealer will no longer buy these leftover units from you). What’s your decision? What’s your decision?
  • 46.
     Start withthe tree having 2 branches (DB) at the decision point.  There are no chance events in the dealer sale branch,  For the self sale, there are 4 mutually exclusive possibilities (state of nature). Decision Trees - Solution Dealer Dealer Sale Sale Self sale Self sale 500, 20% 500, 20% 600, 5% 600, 5% 400, 45% 400, 45% 300, 30% 300, 30%
  • 47.
    Decision Trees -Solution Payoff = 300*6 – 200*3 = 1200 Payoff = 300*6 – 200*3 = 1200 Payoff = 400*6 – 100*3 = 2100 Payoff = 400*6 – 100*3 = 2100 Payoff = 500*6 = 3000 Payoff = 500*6 = 3000 Payoff = 500*6 = 3000 Payoff = 500*6 = 3000 Payoff = 500*4 = 2000 Payoff = 500*4 = 2000 EMV = 2000 EMV = 2000 EMV = 0.3*1200 + 0.45*2100 EMV = 0.3*1200 + 0.45*2100 + + 0.2* 3000 + 0.05*3000 0.2* 3000 + 0.05*3000 = 2055 = 2055 Your decision? Your decision? Self sale Self sale Dealer Dealer Sale Sale Self sale Self sale 500, 20% 500, 20% 600, 5% 600, 5% 400, 45% 400, 45% 300, 30% 300, 30%
  • 48.
    1. A companywould like to expand its production by setting up new plants. The course of actions to be decided and the state of nature are shown in the following pay off table. Sample Problem • Make a decision using a decision tree analysis. Let us do it with excel Alternatives Alternatives Economic Condition Economic Condition High Demand High Demand Moderate Demand Moderate Demand Low Demand Low Demand Large Plant Large Plant 200,000 200,000 100,000 100,000 -120,000 -120,000 Small Plant Small Plant 90,000 90,000 50,000 50,000 -20,000 -20,000 No Plant No Plant 0 0 0 0 0 0 Probability Probability 0.30 0.30 0.50 0.50 0.20 0.20
  • 49.
  • 50.
    1. What arethe elements for rational decision making? 2. Why decisions are hard? 3. What is a good decision? A bad decision? 4. A broker with $1000 has five potential investments A, B, C, D and E. The return on each investment depends on the (uncertain) market behavior during the year. The broker built a payoff table to help make the investment decision. Find an optimal decision based on: a) Maximax Criterion; b) Maximin Criterion c) Minimax Regret Criterion; and d) principle of Insufficient Reasoning Assignment
  • 51.
    Assignment Decision States ofNature Alternatives Large Rise Small Rise No Change Small Fall Large Fall A -100 100 200 300 0 B 250 200 150 -100 -150 C 500 250 100 -200 -600 D 60 60 60 60 60 E 200 150 150 -200 -150
  • 52.
    5. A buildingconstruction company is planning to construct luxury condominiums on its site. The company has a capacity to construct up to 250 condominium blocks. The financial success of the project depends upon the number of condominium blocks to be constructed and the chance event concerning the demand for the condominiums. The decision problem is to select the number of condominium blocks to be constructed that will lead to the largest profit given the uncertainty concerning the demand for the condominiums. The cost and revenue of construction per block are 6000 and 14000 (x1000)birr, respectively. Assignment
  • 53.
    Assignment Low (50 units)Medium (100 units) High (150 units) Build 50 400,000 400,000 400,000 Build 100 100,000 800,000 800,000 Build 150 (200,000) 500,000 1,200,000 State of Nature Demand Alternative Actions Hint: Payoff Table Find an optimal decision: a) Maxmin Criterion b) Minimax Regret Criterion c) Maximax Criterion d) Principle of Insufficient Reasoning