G Decision Theory

(G) DECISION THEORY INTRODUCTION Every day we, are humans, make many decisions; and occasionally we make an important decision that can have immediate and/or long-term effects on our lives. Such decisions as where to attend school, whether to rent or buy, whether your company should accept a merger proposal, and so on, are important decisions for which we would prefer to make correct choice. The success or failure that an individual or organization experiences, depends to a large extent on the ability of making appropriate decisions. Making of a decision requires an enumeration of feasible and viable alternatives (courses of action or strategies), the projection of consequences associated with different alternatives, and the measure of effectiveness (or an objective) to identify best alternative to be used. Everyone engages in the process of making decisions on a daily basis. Some of these decisions are quite easy to make and almost automatic. Other decisions can be very difficult to make and almost debilitating. Likewise, the information needed to make a good decision varies greatly. Some decisions require a great deal of information whereas others much less. Sometimes there is not much if any information available and hence the decision becomes intuitive, if not just a guess. Many, if not most, people make decisions without ever truly analyzing the situation and the alternatives that exist. There is a subjective and intrinsic aspect to all decision making, but there are also systematic ways to think about problems to help make decisions easier. The purpose of decision analysis is to develop techniques to aid the process of decision making, not replace the decision maker. Earlier, the decisions were taken subjectively based on the skill, experience and intuition of the decision maker. But in today’s world of dynamism, the decision making has become very complex, particularly in business, marketing and management because they involve a number of interactive variables (factors) whose values and relationships cannot be determined accurately. In such situations, mere intuition and expertise of the decision maker are inadequate and we require well considered judgement and analysis based on the use of several quantitative techniques and even computers in solving problems. It is in this context that we need a full-fledged decision theory which provides a sound and scientific basis for improved decision making. Decision making is the essence of management. In general, the process of making decisions calls for (i) identifying the alternatives, (ii) gathering all the relevant information about them, and (iii) selecting the best alternative on the basis of some criterion. The decision theory, also called the decision analysis, is used to determine optimal strategies where a decision-maker is faced with several decision alternatives and an uncertain, or risky, pattern of future events. To recapitulate, all decision-making situations are characterized by the fact that two or more alternative courses of action are available to the decision-maker to choose from. Further, a decision may be defined as the selection by the decision-maker of an act, considered to be best according to some pre-designated standard, from among the available options. When analyzing the decision making process, the context or environment of the decision to be made allows for a categorization of the decisions based on the nature of the problem or the 1

nature of the data or both. There are two broad categories of decision problems: decision making under certainty and decision making under uncertainty. ELEMENTS OF DECISION MAKING Decision Maker: The entity responsible for making the decision. This may be a single person, a committee, company, and the like. It is viewed here as a single entity, not a group. Alternatives: A finite number of possible decision alternatives or courses of action available to the decision maker. The decision maker generally has control over the specification and description of the alternatives. These alternatives are also called courses of action (actions, acts or strategies) and are known to the decision-maker. States of Nature: The scenarios or states of the environment that may occur but are not under control of the decision maker. These are the circumstances under which a decision is made. The states of nature are mutually exclusive events and exhaustive. This means that one and only one state of nature is assumed to occur and that all possible states are considered. Payoff or Outcome: Outcomes are the measures of net benefit, or payoff, received by the decision maker. This payoff is the result of the decision and the state of nature. Hence, there is a payoff for each alternative and outcome pair. The measures of payoff should be indicative of the decisions maker’s values or preferences. The payoffs are generally given in a payoff matrix in which a positive value represents net revenue, income, or profit and a negative value represents net loss, expenses, or costs. This matrix yields all alternative and outcome combinations and their respective payoff and is used to represent the decision problem. General form of payoff matrix Courses of Action (Alternatives) States of Nature Pr obability S1 S2 L Sn N1 N2 M Nm

p1 p2 M pm

p11 p21 M pm1

p12 p22 M pm2

L L L L

p1n p2n M pmn

STEPS OF DECISION MAKING PROCESS The decision making process involves the following steps: 1. Identify and define the problem. 2. Listing of all possible future events, called states of nature, which can occur in the context of the decision problem. Such events are not under the control of decision-maker because these are erratic in nature. 3. Identification of all the courses of action (alternatives or decision choices) which are available to the decision-maker. The decision-maker has control over these courses of action. 2

4. Expressing the payoffs (Pij) resulting from each pair of course of action and state of

nature. These payoffs are normally expressed in a monetary value. 5. Apply an appropriate mathematical decision theory model to select best course of action

from the given list on the basis of some criterion (measure of effectiveness) that results in the optimal (desired) payoff. TYPES OF DECISION-MAKING ENVIRONMENTS To arrive at a good decision it is required to consider all available data, an exhaustive list of alternatives, knowledge of decision environment, and use of appropriate quantitative approach for decision-making. In this section four types of decision-making environments: Certainty, uncertainty, risk and conflict have been described. The knowledge of these environments helps in choosing appropriate quantitative approach for decision-making. Type 1 Decision-Making under Certainty The process of choosing an act or strategy when the state of nature is completely known, is called decision making under certainty. The decision-maker has the complete knowledge (perfect information) of consequence of every decision choice (course of action or alternative) with certainty. Obviously, he will select an alternative that yields the largest return (payoff) for the known future (state of nature). In such situation, each act will only result in one event and the outcome of the act can be predetermined with certainty. Hence, such situations are also termed as deterministic situations. For example, the decision to purchase either National Saving Certificate (NSC); Indira Vikas Patra, or deposit in National Saving Scheme (NSS) is one in which it is reasonable to assume complete information about the future because there is no doubt that the Indian government will pay the interest when it is due and the principal at maturity. In this decision-model, only one possible state of nature (future) exists. Type 2 Decision-Making under Risk In this case the decision-maker has less than complete knowledge with certainty of the consequence of every decision choice (course of action) because it is not definitely known which outcome will occur. This means there is more than one state of nature (future) and for which he makes an assumption of the probability with which each state of nature will occur. For example, probability of getting head in the toss of a coin is 0.5. Decision-making under risk is a probabilistic decision situation, in which more than one state of nature exists and the decisionmaker has sufficient information to assign probability values to the likely occurrence of each of these states. The probabilities of various outcomes may be determined objectively from the past data. Knowing the probability distribution of the states of nature, the best decision is to select that course of action which has the largest expected payoff value. The expected (average) payoff of an alternative is the sum of all possible payoffs of that alternative weighted by the probabilities of those payoffs occurring. However, past records may not be available to arrive at the objective probabilities. In many cases the decision-maker may, on the basis of his experience and judgement, be able to assign subjective probabilities to the various outcomes. The problem can then be solved as decision problem under risk. Under conditions of risk, the most popular decision criterions for evaluating the alternative is the expected monetary value/expected opportunity loss of the expected payoff. (i) Expected monetary value (EMV) 3

More generally, the decision-making in situations of risk is on the basis of the expectation principle, with the event probabilities assigned, objectively or subjectively as the case may be, the expected pay-off for each strategy is calculated by multiplying the pay-off values with their respective probabilities and then adding up these products. The strategy with the highest expected pay-off represents the optimal choice. It goes without saying that in problems involving pay-off matrix in terms of costs, optimal strategy is that corresponding to which the expected value is the least. Symbolically, for a decision problem involving n events and m strategies, the expected pay-offs, EP, can be expressed as under: Wherein aij represents the pay-off resulting from the combination of i th event and jth act, where pi represents the probability of ith event. (ii) Expected Opportunity Loss (EOL) An alternative approach to maximizing expected monetary value (EMV) is to minimize the expected opportunity loss (EOL), also called expected value of regret. The EOL is defined as the difference between the highest profit (or payoff) for a state of nature and the actual profit obtained for the particular course of action taken. In other words, EOL is the amount of payoff that is lost by not selecting the course of action that has the greatest payoff for the state of nature that actually occur. The course of action due to which EOL is minimum, is recommended. Since EOL is an alternative decision criterion for decision-making under risk, therefore, the results will always be the same as those obtained by EMV criterion discussed earlier. The steps for calculating EOL are summarized as follows: (a) Prepare a profit (cost) table for each course of action and state of nature combination along with the associated probabilities. (b) For each state of nature calculate the opportunity loss (OL) values by subtracting each payoff from the maximum payoff for that outcome. (For each state of nature calculate the opportunity loss (OL) values by subtracting the minimum payoff for that outcome from each payoff.) (c) Calculate EOL for each course of action by multiplying the probability of each state of nature with the OL value and then adding the values. (d) Select a course of action for which the EOL value is minimum. Expected value of perfect information (EVPI) The expected profit with perfect information is the expected return, in the long run, if we have perfect information before a decision is made. The Expected Value of Perfect Information (EVPI) may be defined as the maximum amount one would be willing to pay, to acquire perfect information as to which event would occur. EPPI represents the maximum obtainable EMV with perfect information as to which event will actually occur (as calculated before information is received). If EMV* represents the maximum obtainable EMV without perfect information, perfect information would increase expected profit from EMV* up to the value of EPPI, so the amount of that increase would be equal to EVPI. Thus, we have EVPI = EPPI – EMV* Type 3 Decision-Making under Uncertainty In this case the decision-maker is unable to specify the probabilities with which the various states of nature (futures) will occur. However, this is not the case of decision-making under ignorance, 4

because the possible states of nature are known. Thus, decisions under uncertainty are taken even with less information than decisions under risk. For example, the probability that Mr. X will be the prime minister of the country 15 years from now is not known. The decision situations where there is no way in which the decision-maker can assess the probabilities of the various states of nature are called decisions under uncertainty. In such situations, the decision-maker has no idea at all as to which of the possible states of nature would occur nor has he a reason to believe why a given state is more, or less, likely to occur as another. With probabilities of the various outcomes unknown, the actual decisions are based on specific criteria. The several principles which may be employed for taking decisions in such conditions include (i) Laplace Criterion, (ii) Maximin or Minimax Criterion, (iii) Maximax or Minimin Criterion, (iv) Savage Criterion, (v) Hurwicz Criterion (or Criterion of Realism). Such situations are frequent in business and management. Will the new plant or industrial unit be successful? Will the new product be able to compete with others in the market? How much to produce and stock to get maximum returns? (i) Optimism (Maximax (Profit) or Minimin (Cost)) Criterion In this criterion the decision-maker ensures that he should not miss the opportunity to achieve the largest possible profit (maximax) or lowest possible cost (minimin). Thus, he selects the alternative (decision choice or course of action) that represents the maximum of the maxima (or minimum of the minima) payoffs (consequences or outcomes). The working method is summarized as follows: (a) Locate the maximum (or minimum) payoff values corresponding to each alternative (or course of action), then (b) Select an alternative with best anticipated payoff value (maximum for profit and minimum for cost). Since in this criterion the decision-maker selects an alternative with largest (or lowest) possible payoff value, it is also called an optimistic decision criterion. (ii) Pessimism (Maximin (Profit) or Minimax (Cost)) Criterion This principle is adopted by pessimistic decision-makers who are conservative in their approach. Using this approach, the minimum pay-offs resulting from adoption of various strategies are considered and among these values the maximum one is selected. It involves, therefore, choosing the best (the maximum) profit from the set of worst (the minimum) profits. When dealing with the costs, the maximum cost associated with each alternative is considered and the alternative which minimizes this maximum cost is chosen. In this context, therefore, the principle is used minimax-the best (the minimum cost) of the worst (the maximum cost). The working method is summarized as follows: (a) Locate the minimum (or maximum in case of profit) payoff value in case of loss (or cost) data corresponding to each alternative, then (b) Select an alternative with best anticipated payoff value (maximum for profit and minimum for loss or cost). Since in this criterion the decision-maker is conservative about the future and always anticipates worst possible outcome (maximum for profit and minimum for loss or cost), it is called a pessimistic decision criterion. This criterion is also known as Wald’s criterion. (iii) Equal probabilities (Laplace) Criterion 5

Since the probabilities of states of nature are not known, it is assumed that all states of nature will occur with equal probability, i.e. each state of nature is assigned an equal probability. As states of nature are mutually exclusive and collectively exhaustive, so the 1 probabilities of each of these must be . The working method number of states of nature is summarized as follows: (a) Assign equal probability value to each state of nature by using the formula: 1 . number of states of nature (b) Compute the expected (or average) payoff for each alternative (course of action) by adding all the payoffs and dividing by the number of possible states of nature or by applying the formula: (Probability of state of nature j) × (Payoff value for the combination of alternative, i and state of nature j) (c) Select the best expected payoff value (maximum for profit and minimum for cost). This criterion is also known as the criterion of insufficient reason because, except in a few cases, some information of the likelihood of occurrence of states of nature is available. (iv) Coefficient of optimism (Hurwicz) Criterion This criterion suggests that a rational decision-maker should be neither completely optimistic nor pessimistic and, therefore must display a mixture of both. Hurwicz, who suggested this criterion, introduced the idea of a coefficient of optimism (denoted by α ) to measure the decision-maker’s degree of optimism. This coefficient lies between 0 and 1, where 0 represents a complete pessimistic attitude about the future and 1 a complete optimistic attitude about the future. Thus, if α is the coefficient of optimism, then (1 − α ) will represent the coefficient of pessimism. In case of profits, the Hurwicz approach suggests that the decision-maker must select an alternative that maximizes H (Criterion of realism) = α (Maximum in column) + (1 − α ) (Minimum in column) The working method is summarized as follows: (a) Decide the coefficient of optimism α (alpha) and then coefficient of pessimism (1 − α ). (b) For each alternative select the largest and lowest payoff value and multiply these with α and (1 − α ) values, respectively. Then calculate the weighted average, H by using above formula. (c) Select an alternative with best anticipated weighted average payoff value. In the case of costs, the principle works like this. The minimum of the costs for each course of action is multiplied by α (the indicator of the degree of optimism of the decision maker), and the maximum of the costs for each alternative is multiplied by(1 − α ). Then the sum of the products for each action strategy is obtained the alternative for which the sum is the least is selected. (v) Regret savage criterion This criterion is also known as opportunity loss decision criterion or minimax regret decision criterion because decision-maker feels regret after adopting a wrong course of 6

action (or alternative) resulting in an opportunity loss of payoff. Thus, he always intends to minimize this regret. The working method is summarized as follows: (a) From the given payoff matrix, develop an opportunity loss (or regret) matrix as follows: (i) Find the best payoff corresponding to each state of nature, and (ii)Subtract all other entries (payoff values) corresponding to each state of nature from this value. (b) For each course of action (strategy or alternative) identify the worst or maximum regret value. Record this number in a new row. (c) Select the course of action (alternative) with the smallest anticipated opportunity loss value. In the case of costs, the principle works like this. (a) From the given payoff matrix, develop an opportunity loss (or regret) matrix as follows: (i) Find the worst payoff corresponding to each state of nature, and (ii)Subtract all other entries (payoff values) corresponding to each state of nature from this value. (b) For each course of action (strategy or alternative) identify the best or minimum regret value. Record this number in a new row. (c) Select the course of action (alternative) with the greatest anticipated opportunity loss value. Type 4 Decision-Making under Conflict In many situations, neither states-of-nature are completely known nor are they completely uncertain. Partial knowledge is available and therefore it may be termed as decision-making under ‘partial uncertainty’. An example of this is the situation of conflict involving two or more competitors marketing the same product. Question 1: A food product company is contemplating the introduction of a revolutionary new product with new packaging or replace the existing product at much higher price (S1) or a moderate change in the composition of the existing product with a new packaging at a small increase in price (S2) or a small change in the composition of the existing product except the word ‘New’ with a negligible increase in price (S3). The three possible states of nature or events are: (i) high increase in sales (N1), (ii) no change in sales (N2) and (iii) decrease in sales (N3). The marketing department of the company worked out the payoffs in terms of yearly net profits for each of the strategies of three events (expected sales). This is represented in the following table: States of Strategies Nature S1 S2 S3 N1 7,00,00 5,00,00 3,00,000 0 0 N2 3,00,00 4,50,00 3,00,000 0 0 N3 1,50,00 0 3,00,000 0 Which strategy should the concerned executive choose on the basis of the following? 7

(a) Maximin criterion (b) Maximax criterion (c) Minimax regret criterion (d) Laplace criterion Solution The payoff matrix is rewritten as follows: (a) Maximin Criterion States of Nature Strategies S1 S2 S3 N1 7,00,00 5,00,00 3,00,000 0 0 N2 3,00,00 4,50,00 3,00,000 0 0 N3 1,50,00 0 3,00,000 0 Column (minimum) 1,50,00 0 3,00,000 ← Maximin 0 The maximum of column minima is 3,00,000. Hence, the company should adopt strategy S3. (b) Maximax Criterion States of Nature Strategies S1 S2 S3 N1 7,00,000 5,00,00 3,00,000 0 N2 3,00,000 4,50,00 3,00,000 0 N3 1,50,000 0 3,00,000 Column (maximum) 7,00,000 5,00,00 3,00,000 Maximax ↑ 0 The maximum of column maxima is 7,00,000. Hence, the company should adopt strategy S1. (c) Minimax Regret Criterion (opportunity loss in case of profits) States of Strategies Nature S1 S2 S3 − − N1 7,00,000 7,00,000 = 0 7,00,000 5,00,000 = 7,00,000 − 3,00,000 = 2,00,000 4,00,000 N2 4,50,000 − 3,00,000 = 4,50,000 − 4,50,000 = 0 4,50,000 − 3,00,000 = 1,50,000 1,50,000 N3 3,00,000 − 1,50,000 = 3,00,000 − 0 = 3,00,000 3,00,000 − 3,00,000 = 0 1,50,000 Column 1,50,000 3,00,000 4,00,000 (maximum) Minimax regret ↑ Hence, the company should adopt minimum opportunity loss strategy, S1. (d)Laplace Criterion Since, we do not know the probabilities of states of nature, assume that they are equal. For this example, we would assume that each state of nature has a probability 1/3 of occurrence. Thus, Strategy Expected Return (Rs) S1 (7,00,000 + 3,00,000 + 1,50,000)/3 = 3,83,333.33 S2 (5,00,000 + 4,50,000 + 0)/3 = 3,16,666.66 S3 (3,00,000 + 3,00,000 + 3,00,000)/3 = 3,00,000 8

Since, the largest expected return is from strategy S1; the executive must select strategy S1. Question 2: A Super Bazaar must decide on the level of supplies it must stock to meet the needs of its customers during Diwali days. The exact number of customers is not known, but it is expected to be in one of the four categories; 300, 350, 400 or 450 customers. Four levels of supplies are thus suggested with level j being ideal (from the viewpoint of incurred costs) if the number of customers falls in category j. Deviations from the ideal levels results in additional costs either because extra supplies are stocked needlessly or because demand cannot be specified. The table below provides these costs in thousands of rupees. Supplies level Customer category A1 A2 A3 A4 E1 7 12 20 27 E2 10 9 10 25 E3 23 20 14 23 E4 32 24 21 17 (a) Which level of inventory is chosen on the basis of (i) Laplace criterion (ii) minimax criterion (iii) minimin criterion? (b) Now consider payoff matrix as profit matrix then which level of inventory is chosen on the basis of Hurwicz criterion Solution (a) (i) Laplace Criterion Since, we do not know the probabilities of states of nature, assume that they are equal. For this question, we would assume that each state of nature has a probability 1/4 of occurrence. Thus, Strategy Expected Return (Rs) A1 (7 + 10 + 23 + 32)/4 = 18 A2 (12 + 9 + 20 + 24)/4 = 16.25 A3 (20 + 10 + 14 + 21)/4 = 16.25 A4 (27 + 25 + 23 + 17)/4 = 23 Since, the lowest expected return is from strategy A 2 and A3; the executive must select strategy A2 or A3. (ii)Minimax Criterion States of Nature Strategies A1 A2 A3 A4 E1 7 12 20 27 E2 10 9 10 25 E3 23 20 14 23 E4 32 24 21 17 Column 32 24 21 ← Minimax 27 (maximum) The minimum of column maxima is 21. Hence, the company should adopt strategy A3. (iii) Minimin Criterion States of Nature Strategies A1 A2 A3 A4 E1 7 12 20 27 E2 10 9 10 25 E3 23 20 14 23 9

E4 32 24 21 17 Column (minimum) 7 9 10 17 Minimin ↑ The minimum of column minima is 7. Hence, the company should adopt strategy A1. (b)In the context of profit data, Hurwicz Criterion, HC = α (Max Value) + (1 – α ) (Min Value). Its value for various strategies is as follows:

State of Profit from optimal Course of Action(thousand Nature Rs) (1 (2 (3 (4 (5) (6) ) ) ) ) A1 A2 A3 A4 Profit (Max in 0.5 x columns (1, 2, 3 (5) & 4)) E1 7 12 20 27 32 16 E2 10 9 10 25 24 12 E3 23 20 14 23 21 10.5 E4 32 24 21 17 27 13.5 Since, maximum is 22, so, it is optimal to adopt strategy A4.

(7)

(8)

Profit (Min in columns (1, 2, 3 & 4)) 7 9 10 17

0.5 x (6) + (7) (8) 3.5 4.5 5 8.5

19.5 16.5 15.5 22

Question 3: A manufacturer makes a product, of which the principle ingredient is a chemical X. at the moment, the manufacturer spends Rs 1,000 per year on supply of X, but there is a possibility that the price may soon increase to 4 times its present figure because of a worldwide shortage of the chemical, Z in order to give the same effect as chemical X. chemicals Y and Z would together cost the manufacturer Rs 3,000 per year, but their prices are unlikely to rise. What action should the manufacturer take? Apply the maximin criteria for decision-making and give the solution. Solution The data of the problem is summarized in the following table. States of Nature Courses of Action S1(use Y and Z) S2 (use X) N1 (Price of X increases) 3,000 4,000 N2 (Price of X does not increase) 3,000 1,000 Maximin Criterion (opportunity loss in case of cost) States of Nature Courses of Action S1(use Y and Z) S2 (use X) − N1 (Price of X increases) 3,000 3,000 = 0 3,000 − 4,000 = − 1000 N2 (Price of X does not increase) 1,000 − 3,000 = − 2,000 1,000 − 1,000 = 0 − 2,000 − 1000 ← Maximin Minimum Opportunity Hence, manufacturer should adopt the course of action S2 that minimizes the cost. Question 4: Technico Ltd has installed a machine costing Rs 4 lacs and is in the process of deciding on an appropriate number of a certain spare parts required for repairs. The spare parts cost Rs 4000 each but are available only if they are ordered now. In case the machine fails and no spares are available, the cost to the company of mending the plant would be Rs 10

18000. The plant has an estimated life of 8 years and the probability distribution of failures during the time, based on experience with similar machines, is as follows: No. of failures during 8-yearly period 0 1 2 3 4 5 Probability 0. 0. 0. 0. 0. 0.1 1 2 3 2 1 Ignoring any discounting for time value of money, determine the following: (a) The expected number of failures in the 8-year period. (b) The optimal number of units of the spare part on the basis of Hurwicz principle (taking α =0.7). (c) EVPI. Solution Since the availability of number of spares at the time of the failure of any machine is under the control of decision-maker, no. of spares per year is considered as ‘course of action’ (decision choice) and the no. of failures of machines is uncertain and only known with probability, therefore, it is considered as a ‘state of nature’ (event). Let S be the quantity (number of spares to be available). And F is the no. of failures within one year. It is given that cost of storing a spare is Rs. 4000. Cost of not storing the spare is Rs. 18000. Cost function = 4,000S, S ≥ F 4,000S + 18000 (F – S), S < F (a) The expected number of failures in the 8 year period, is given by 6

E(F) = ∑ pi Fi = 0.1 × 0 + 0.2 × 1 + 0.3 × 2 + 0.2 × 3 + 0.1 × 4 + 0.1 × 5 = 2.3 i =1

State of Probability Cost (thousand Rs) Due to Expected Cost (thousand Rs) Due to Nature Course of Action (purchase) Course of Action (F) (1) (2) (3) (4) (5) (6) (7) (1) x (1) x (1) x (1) x (1) x (1) (2) (3) (4) (5) (6) x (7) 0 1 2 3 4 5 0 1 2 3 4 5 0 0.10 0 4 8 12 16 20 0 0.4 0.8 1.2 1.6 2 1 0.20 18 4 8 12 16 20 3.6 0.8 1.6 2.4 3.2 4 2 0.30 36 22 8 12 16 20 10.8 6.6 2.4 3.6 4.8 6 3 0.20 54 40 26 12 16 20 10.8 8 5.2 2.4 3.2 4 4 0.10 72 58 44 30 16 20 7.2 5.8 4.4 3 1.6 2 5 0.10 90 76 62 48 34 20 9 7.6 6.2 4.8 3.4 2 Expected Cost (EC) 41.4 29.2 20.6 17.4 17.8 20 (b) In the context of cost data, Hurwicz Criterion, HC = α (Min Value) + (1 – α ) (Max Value). Its value for various strategies is as follows:

State of Nature

Probability Cost (thousand Rs) Due to Cost from optimal Course of Course of Action Action(thousand Rs) (1)

(2) (3) (4) (5) (6) (7) (8) 0 1 2 3 4 5 Cost (Min in columns (2, 3, 5, 6 11

(9) 0.7 x (8)

(10) Cost (Max in columns (2, 3, 5, 6

(11) 0.3 x (10)

(9) + (11)

& 7)) 0 0.05 0 4 8 12 16 20 0 1 0.10 18 4 8 12 16 20 4 2 0.20 36 22 8 12 16 20 8 3 0.30 54 40 26 12 16 20 12 4 0.20 72 58 44 30 16 20 16 5 0.15 90 76 62 48 34 20 20 Since, minimum is 20, so, it is optimal to keep 5 spare parts. (c) State of Probability Cost (thousand Rs) Due to Course Nature of Action (1) (2) (3) (4) (5) (6) (7) 0 1 2 3 4 5 0 0.05 0 4 8 12 1 0.10 18 4 8 12 2 0.20 36 22 8 12 3 0.30 54 40 26 12 4 0.20 72 58 44 30 5 0.15 90 76 62 48 Expected Cost with Perfect Information (ECPI) Now, EVPI = EC* – ECPI = 17.4 – 9.2 = 8.2 thousand rupees

16 16 16 16 16 34

20 20 20 20 20 20

& 7)) 0 90 2.8 76 5.6 62 8.4 48 11.2 34 14 20

27 22.8 18.6 14.4 10.2 6

27 25.6 24.2 22.8 21.4 20

Cost from optimal Course of Action(thousand Rs) (8) (1) x (8) Cost (Min in (2, Weighted 3, 5, 6 & 7)) Cost 0 0 4 0.8 8 2.4 12 2.4 16 1.6 20 2 9.2

Question 5: An investor is given the following investment alternatives and percentage rates of return. Investment alternatives State of Nature (Market Conditions) Low Medium High Regular Shares 7% 10% 15% Risky Shares -10% 12% 25% Property -12% 18% 30% Over the past 300 days, 150 days have been medium market conditions and 60 days have had high market increases. On the basis of these data, state the optimum investment strategy for the investment. Solution According to the given information, the probabilities of low, medium and high market conditions would be 0.30 (300 – (150 + 60) = 90/300), 0.50 (150/300) and 0.20 (60/300) respectively. The expected pay-offs for each of the alternatives are calculated and shown in the table below: Market Probability Profit (Rs) Due to Course of Expected Payoff (Rs) Due to Conditions Action Course of Action (1)

(2) Regular

(3) Risky 12

(4) (1) x (2) Property Regular

(1) x (3) Risky

(1) x (4) Property

shares Low 0.30 0.07 Medium 0.50 –0.10 High 0.20 –0.12 Expected monetary value (EMV) Since the expected return of 23% is the alternative.

shares 0.10 0.12 0.18

shares 0.15 0.021 0.25 –0.05 0.30 –0.024 –0.053 highest for property, the investor

shares 0.03 0.045 0.06 0.125 0.036 0.06 0.126 0.230 should invest in this

Question 6: A company manufactures goods for a market in which the technology of the product is changing rapidly. The research and development department has produced a new product which appears to have potential for commercial exploitation. A further Rs 60,000 is required for development testing. The company has 100 customers and each customer might purchase at the most one unit of the product. Market research suggests that a selling price of Rs 6000 for each unit with total variable costs of manufacturing and selling estimate are Rs 2,000 for each unit. From previous experience, it has been possible to derive a probability distribution relating to the proportion of customers who will buy the product as follows: Proportion of 0.0 0.0 0.1 0.1 0.20 customers 4 8 2 6 Probability 0.1 0.1 0.2 0.4 0.20 0 0 0 0 Determine the expected opportunity losses, given no other information than that stated above, and state whether or not the company should develop the product. Solution If p is the proportion of customers who purchase the new product, the profit is: (6,000 – 2,000) x 100p – 60,000 = Rs (4,00,000p – 60,000). Let Ni (I = 1, 2, …, 5) be the possible states of nature, i.e. proportion of the customers who will buy the new product and S1 (develop the product) and S2 (do not develop the product) be the two courses of action. The profit values (payoffs) for each pair of Ni’s and Sj’s are shown in the following table: State of Prob Profit = Rs Opportunity Loss (Rs) (1) x (1) x Nature Ni abilit (4,00,000p – (2) (3) (Proportion y 60,000)Cour of se of Action (2) (1) (3) Customers, p) S1 S2 S1 S2 S1 S2 0.04 0.1 – 0 0 – (–44,000) = 0 – 0 = 0 4,40 0 44,000 44,000 0 0.08 0.1 – 0 0 – (–28,000) = 0 – 0 = 0 2,80 0 28,000 28,000 0 0.12 0.2 – 0 0 – (–12,000) = 0 – 0 = 0 2,40 0 12,000 12,000 0 0.16 0.4 4,000 0 4,000 – 4,000 = 0 4,000 – 0 = 4,000 0 1,60 0 0.20 0.2 20,000 0 20,000 – 20,000 = 0 20,000 – 0 = 0 4,00 20,000 0 13

Expected Opportunity Loss (EOL)

9,60 5,60 0 0 (Note: All the entries of column S2 would be 0. Since, we are not developing anything then no profit will be earned) Since, the company seeks to minimize the expected opportunity loss, the company should select course of action S2 (do not develop the product) with minimum EOL. Question 7: A TV dealer finds that the cost of holding a TV in stock for a week is Rs. 50. Customers who cannot obtain new TVs immediately tend to go to other dealers and he estimates that for every customer who cannot get immediate delivery he loses an average of Rs. 200. For one particular model of TV, the probabilities of demand of 0, 1, 2, 3, 4 and 5 TVs in a week are 0.05, 0.10, 0.20, 0.30, 0.20 and 0.15 respectively. (i) How many televisions per week should the dealer order? Assume that there is no time lag between ordering and delivery. (ii) Compute EVPI. (iii) The dealer is thinking of spending on a small market survey to obtain additional information regarding the demand levels. How much should be willing to spend on such a survey? Solution Since number of TV sets ordered (purchased) is under the control of decision-maker, purchase per week is considered as ‘course of action’ (decision choice) and the weekly demand of the TV sets is uncertain and only known with probability, therefore, it is considered as a ‘state of nature’ (event). Let P be the quantity (number of TV sets to be purchased). And D is the demand within a week. It is given that cost of storing a TV set is Rs. 50. Cost of not satisfying the customer demand is Rs. 200. Cost function = 50P, P ≥ D 50P + 200 (D – P), P=P (80-50)D – (50-20) (P-D), D < P The resulting profit values and corresponding expected payoffs are computed in the following table: 15

States of Probability Nature D (Demand per week)

Profit (Rs) Due to Courses Expected Payoff (Rs) Due to Courses of Action P (Purchase per of Action (Purchase per Day) day)

15 (2) = 450

16 (3) 420

17 (4) 390

18 (5) 360

(1) 15 12/120 0.1 16 24/120 = 450 480 450 420 0.2 17 48/120 = 450 480 510 480 0.4 18 36/120 = 450 480 510 540 0.3 Expected monetary value (EMV) Since the highest EMV of Rs 486 is corresponding to purchase 17 cases of cherries every morning.

15 (1)x(2) 45

16 (1)x(3) 42

17 (1)x(4) 39

18 (1)x(5) 36

90

96

90

84

180

192

204

192

135

144

153

162

450 474 486 474 course of action 17, the retailer must

Question 9: The probability of the demand for Lorries for hiring on any day in a given district is as follows: No. of lorries demanded 0 1 2 3 4 Probability 0. 0. 0. 0. 0.2 1 2 3 2 Lorries have a fixed cost of Rs 90 each day to keep the daily hire charges (net of variable costs of running) Rs 200. If the lorry-hire company owns 4 Lorries, what is its daily expectation? If the company is about to go into business and currently has no Lorries, how many Lorries should it buy? Solution It is given that Lorry Hire Company owns 4 Lorries then its daily expectation would be: No. of Lorries demanded Probability Payoff (With 4 Expected Value Lorries) (1) (2) (1) x (2) 0 0.1 200 x 0-90 x 4 = -360 -36 1 0.2 200 x 1-90 x 4 = -160 -32 2 0.3 200 x 2-90 x 4 = 40 12 3 0.2 200 x 3-90 x 4 = 240 48 4 0.2 200 x 4-90 x 4 = 440 88 Daily Expectation 80 Since number of Lorries purchased is under the control of decision-maker, purchase per day is considered as ‘course of action’ (decision choice) and the daily demand of the Lorries is uncertain and only known with probability, therefore, it is considered as a ‘state of nature’ (event). Let P be the quantity (number of Lorries to be purchased). And D is the demand within a day. Profit = (200–90) P, D ≥ P (200–90)D – (90) (P–D), D < P The resulting profit values and corresponding expected payoffs are computed in the following table: 16

States of Probability Profit (Rs) Due to Courses Expected Payoff (Rs) Due to Courses of Nature D of Action P (Purchase per Action (Purchase per Day) (Deman day) d per week) 0 1 2 3 4 0 1 2 3 4 (1) (2 (3) (4) (5) (6) (1)x(2 (1)x(3 (1)x(4 (1)x(5 (1)x(6) ) ) ) ) ) 0 0.1 0 -90 0 -9 -18 -27 -36 18 27 36 0 0 0 1 0.2 0 11 20 -70 0 22 4 -14 -32 0 16 0 2 0.3 0 11 22 13 40 0 33 66 39 12 0 0 0 3 0.2 0 11 22 33 24 0 22 44 66 48 0 0 0 0 4 0.2 0 11 22 33 44 0 22 44 66 88 0 0 0 0 Expected monetary value (EMV) 0 90 140 130 80 Since the highest EMV of Rs 140 is corresponding to course of action 2, the retailer must purchase 2 Lorries. Question 10: A company needs to increase its production beyond its existing capacity. It has narrowed the alternatives to two approaches to increase the production capacity: (a) expansion, at a cost of Rs 8 million, or (b) modernization at a cost of Rs 5 million. Both approaches would require the same amount of time for implementation. Management believes that over the required payback period, demand will either be high or moderate. Since high demand considered being somewhat less likely than moderate demand, the probability of high demand has been set at o.35. If the demand is high, expansion would gross estimated additional Rs 12 million but modernization only additional Rs 6 million, due to lower maximum product capability. On the other hand, if the demand is moderate, the comparable figures would be Rs 7 million for expansion and Rs 5 million for modernization. (a) Calculate the profit in relation to various action and outcome combinations and states of nature. (b) If the company wishes to maximize its EMV, should it modernize or expand? (c) Calculate the EVPI. (d) Construct the conditional opportunity loss table and also calculate EOL. Solution Defining the states of nature: high demand and moderate demand (over which the company has no control) and courses of action (company’s possible decisions): Expand and Modernize. Since the probability that the demand is high estimated at 0.35, the probability of moderate demand must be (1 – 0.35) = 0.65. The resulting profit values, corresponding expected payoffs and Expected Opportunity Loss (EOL) values are computed in the following table: 17

State of Nature (Dema nd)

Prob Profit (million abilit Rs) Due to y Course of Action (1)

(2)

(3)

Expan d (S1)

Mode rnize( S2)

Expected Payoff (million Rs) Due to Course of Action (1) x (1) x (2) (3) Expa Mod nd erniz (S1) e(S2)

Profit from optimal Course of Action(million Rs) (4)

Opportunity (1) Loss (million x Rs) Due to (5) Course of Action

(1) x (5) (4) Profit Weig S1 (Max hted in (2 Profit & 3)) 4 1.40 4–4= 0

High 0.35 12 – 8 6 – 5 1.4 0.35 Dema =4 =1 nd (N1) Moder 0.65 7 – 8 = 5 – 5 –0.65 0 0 ate –1 =0 Dema nd (N2) Expected monetary value 0.75 0.35 (EMV) Expected Profit with Perfect Information (EPPI) Expected Opportunity Loss (EOL)

0

(1) x (6)

(6) S2

S1

4–1 0 =3

0 – (– 0 – 0 0.6 1) = 1 = 0 5

S2

1.05

0

1.40

0.6 1.05 5 (b) Since the highest EMV of Rs 0.75 million is corresponding to course of action Expand, the company must expand it. (c) EVPI = EPPI – EMV =1.40 – 0.75 = Rs. 0.65 Million (d)Since, the company seeks to minimize the expected opportunity loss (EOL), the company should select course of action S1 (Expand). Question 11: A certain piece of equipment has to be purchased for a construction project at a remote location. This equipment contains an expensive part which is subjected to random failure. Spares of this part can be purchased at the same time the equipment is purchased. Their unit cost is Rs 1,500 and they have no scrap value. If the part fails on the job and no spare is available, the part will have to be manufactured on a special order basis. If this is required, the total cost including down time of the equipment, is estimated as Rs 9,000 for each such occurrence. Based on previous experience with similar parts, the following probability estimates of the number of failures expected over the duration of the project are provided as given below: Failure 0 1 2 Probability 0.8 0.1 0.05 0 5 18

(a) Determine optimal EMV* and optimal number of spares to purchase initially.

(b) Based on opportunity losses, determine the optimal course of action and optimal value of EOL. (c) Determine expected profit with perfect information and EVPI. Solution Let N1 (no failure), N2 (1 failure) and N3 (2 failures) be the possible states of nature (i.e. number of parts failures or number of spares required). Similarly, let S1 (no spare purchased), S2 (one spare purchased) and S3 (two spares purchased) are possible courses of action or strategies. The costs for each pair of course of action and state of nature combination are shown in the following table: State of Nature Course of Action Purchase Emergency Total Conditional (spare required) (number of spare Cost (Rs) Cost (Rs) Cost (Rs) purchased) 0 0 0 0 0 0 1 1,500 0 1,500 0 2 3,000 0 3,000 1 0 0 9,000 9,000 1 1 1,500 0 1,500 1 2 3,000 0 3,000 2 0 0 18,000 18,000 2 1 1,500 9,000 10,500 2 2 3,000 0 3,000 Using the conditional costs as given in the above table and the probabilities of states of nature, the expected monetary value EMV can be calculated for each of three states of nature as shown in the following table: State Nature

of Probability Cost (Rs) Due to Course of Expected Cost (Rs) Due to Course Action of Action (1)

(2) (3) (4) (1) x (2) (1) x (3) (1) x (4) S1 S2 S3 S1 S2 S3 N1 0.80 0 1,500 3,000 0 1,200 2,400 N2 0.15 9,000 1,500 3,000 1,350 225 450 N3 0.05 18,000 10,500 3,000 900 525 150 Expected monetary value (EMV) 2,250 1,950 3,000 Since the lowest EMV of Rs 1,950 is corresponding to course of action S2, the company should purchase only 1 spare initially. If the EMV is expressed in terms of profit, then EMV* = – Rs 1950 State Probability Cost (Rs) Due to Opportunity Loss (Rs) Expected of Course of Action Due to Course of Action Opportunity Loss Nature (Rs) Due to Course of Action (1) (2) (3) (4) (5) (6) (7) (1) x (1) x (1) x (5) (6) (7) S1 S2 S3 S1 S2 S3 S1 S2 S3 N1 0.80 0 1,500 3,000 0 – 0 = 1500 – 3000 0 1200 2400 19

0

0 = 1500 3,000 9,000 1,500 – 1500 – 1500 = 7500 = 0

– 0 = 3000 N2 0.15 9,000 1,500 3,000 1125 0 225 – 1500 = 1500 N3 0.05 18,000 10,500 3,000 18,000 10,500 3,000 750 375 0 – 3000 – 3000 – = = 7500 3000 15000 =0 Expected Opportunity Loss (EOL) 1875 1575 2625 Since, the company seeks to minimize the expected opportunity loss (EOL), the company should select course of action S2 (Purchase one spare). State of Probability Cost (Rs) Due to Course of Cost from optimal Course of Nature Action Action(million Rs) (1) (2) (3) (4) (4) (1) x (4) S1 S2 S3 Cost (Min in (2, 3 Weighted & 4)) Cost N1 0.80 0 1,500 3,000 0 0 N2 0.15 9,000 1,500 3,000 1500 225 N3 0.05 18,000 10,500 3,000 3000 150 Expected Cost with Perfect Information (ECPI) 375 And, EPPI = –375 Now, EVPI = EPPI – EMV* = –375 – (–1950) = Rs 1575 Here, it can be observed that, EVPI = EOL* = Rs 1575. Question 12: A toy manufacturer is considering a project of manufacturing a dancing doll with three different movement designs. The doll will be sold at an average of Rs 10. The first movement design using ‘gears and levels’ will provide the lowest tooling and set up cost of Rs 1,00,000 and Rs 5 per unit of variable cost. A second design with spring action will have a fixed cost of Rs. 1, 60,000 and variable cost of Rs 4 per unit. Yet another design with weights and pulleys will have a fixed cost of Rs. 3, 00,000 and variable cost of Rs 3 per unit. One of the following demand events can occur for the doll with the probabilities: Demand (units) Probability Light demand 25,000 0.10 Moderate demand 1,00,000 0.70 Heavy demand 1,50,000 0.20 (a) Construct a payoff table for the above project. (b) Which is the optimum design? (c) How much can be decision-maker afford to pay to obtain perfect information about the demand? Solution Payoff (Profit) = Revenue – Cost = (Selling Price x no. of units demanded) – (fixed cost + variable cost) = (Selling Price x no. of units demanded) – (fixed cost + (no. of units demanded x per unit cost)) 20

State of Probability Profit (Rs) Due to Course of Expected Payoff (Rs) Due to Nature Action Course of Action (Demand) (1) (2) (3) (4) (1) x (2) (1) x (3) (1) x (4) Gears & Spring Weights Gears & Spring Weights Levels Action & Pulleys Levels Action & Pulleys Light 0.10 25,000 –10,000 –1,25,000 2,500 –1,000 –12,500 Moderate 0.70 4,00,000 4,40,000 4,00,000 2,80,000 3,08,000 2,80,000 Heavy 0.20 6,50,000 7,40,000 7,50,000 1,30,000 1,48,000 1,50,000 Expected monetary value (EMV) 4,12,500 4,55,000 4,17,500 Since, EMV is largest for spring action, it must be selected. State of Probability Profit (Rs) Due to Course of Action Profit from optimal Course Nature of Action(Rs) (Demand) (1) (2) (3) (4) (4) (1) x (4) Gears & Spring Weights & Profit (Max Weighted Levels Action Pulleys in (2, 3 & 4)) Profit Light 0.10 25,000 –10,000 –1,25,000 25,000 2,500 Moderate 0.70 4,00,000 4,40,000 4,00,000 4,40,000 3,08,000 Heavy 0.20 6,50,000 7,40,000 7,50,000 7,50,000 1,50,000 Expected Profit with Perfect Information (EPPI) 4,60,500 The maximum amount of money that the decision-maker would be willing to pay to obtain perfect information regarding demand for the doll will be EVPI = EPPI – EMV =4,60,000 – 4,55,000 = Rs 5,500 DECISION TREE ANALYSIS Decision-making problems discussed so far have been limited to a single decision over one period of time, because the payoffs, states of nature, courses of action and probabilities associated with the occurrence of states of nature are not subject to change. However, situations may arise when a decision-maker needs to revise his previous decisions on getting new information and make a sequence of several interrelated decisions over several future periods. Thus he should consider the whole series of decisions simultaneously. Such a situation is called a sequential or multi period decision process. Decision tree is a network which exhibits graphically the logical relationship between the different parts of the complex decision process. It is a graphic model of each combination of various acts and states of nature {Si, Aj}; (I = 1, 2, …, m; j = 1, 2, …, n) along with their payoffs, the probability distribution of the various states of nature and the EMV or EOL for each act. Decision tree is a very effective device in making decisions in various diversified problems relating to personnel, investment, portfolios, project management, new project strategies, etc. Each combination (Si, Aj) is depicted by a distinct path through the decision tree. An essential feature of the decision tree is that the flow should be from left to right in a chronological order. Standard symbols are used in drawing a decision tree. (i) A square ( ) is used to represent a decision point or decision node at which the decision maker has to decide about one of the various acts or alternatives available to him. (ii) Each act or alternative is shown as a line, representing a branch of the tree emanating from the square. 21

(iii)

A circle ( ) is used to represent a chance event or chance node at which various events or states of nature are represented by lines, which depict the sub-branches of the tree emanating from the circle. (iv)Each branch of the tree (corresponding to each act or strategy) has as many sub-branches as the number of events or states of nature. (v) Along the branches/sub-branches are also shown the probabilities of various states of nature and the payoffs for each combination (Si, Aj); I = 1, 2, …, m; j = 1, 2, …, n along with the EMV or EOL for each act. (vi)As a branch can sub-branch again, we obtain a tree like structure, which represents the various steps in a decision problem. Roll Back Technique of Analyzing a Decision Tree A decision tree is extremely useful in multistage situations which involve a number of decisions, each depending on the preceding one. At any stage, to decide about any strategy or act, the decision maker has to take into consideration all future outcomes that may result from choosing the said act. Consequently to analyze a decision tree, we start from the end of the tree (extremely RHS) i.e., we start from the last decision/event node, say D l and work backwards. This technique of analyzing the decision tree, called the roll-back technique is explained in the following steps. 1. (a) for each branch of the event node (of Dl) we compute the conditional expected payoffs. (b) Adding these expected payoffs for each event-nodal branch, we obtain the EMV for the given path (act or strategy) emanating from the square (decision node Dl). (c) The optimal act or strategy at Dl is the one which corresponds to the highest EMV. 2. Next we move to the last but one decision node (D l-1), make the EMV analysis as in steps 1 (a), (b) and (c) and then move back to the preceding decision node (Dl-2) and so on. 3. This roll-back process is continued till we reach the first decision node (Dl). Question 13: A manufacturing company has to select one of the two products X or Y for manufacturing. Product X requires investment of Rs. 30,000 and product Y, Rs. 50,000. Market result survey shows high, medium and low demands with corresponding probabilities and return from sales, (in thousand rupees), for the two products, as given in the following table. Demand Probability Return from Sales (‘ooo Rs.) Product Product Product X Product Y X Y High 0.4 0.3 75 100 Medium 0.4 0.4 55 80 Low 0.2 0.3 35 70 Construct the appropriate decision tree. What decision the company should take? Solution

22

( 0.4 and m e D High 00 750

)

Medium Demand (0.4)

L ow

tX d uc o r P - 300

55000

De m a nd

35 000

00

( 0.2 )

a nd De m

P rod

) (0 .3

Hig h 00000 1

uct Y - 500 00

Medium Demand (0.4) 80000

Low D em an d (0.3) 700

Net Payoff (Rs.) 75000-30000=45000 55000-30000=25000 35000-30000=5000 Total 100000-50000=50000 80000-50000=30000 70000-50000=20000 Total

Expected Payoff (Rs.) 45000 × 0.4=18000 25000 × 0.4=10000 5000 × 0.2=1000 29000 (EMV) 50000 × 0.3=15000 30000 × 0.4=12000 20000 × 0.3=6000 33000 (EMV)

00

Question 14: A businessman has two independent investments A and B available to him but he lacks the capital to undertake both of them simultaneously. He can choose to take A first and then stop, or if A is successful then take B, or vice versa. The probability of success for A is 0.7 while for B it is 0.4. Both investments require an initial capital outlay of Rs. 2000; and both return nothing if the venture is unsuccessful. Successful completion of A will return Rs. 3000 (over cost), and successful completion of B will return Rs. 5,000 (over cost). Draw and evaluate the decision tree by the roll back technique and determine the best strategy. Solution

23

St

3000 (0.7)

s es c c Su

op

D2 (C o Ac st 20 cep 00) tB

EMV = 2060

(C pt A os t2 00 0)

Fai l

EMV = 800 Fai lur e

-2000 (0.6)

e

-2000 (0.3)

Do Nothing

pt ce Ac

(C os t2

5000 (0.4)

ure

Ac c D1

ss ce c Su

(0) op St

B

00 0) S

EMV = 1400

Decision Node D3

(i) AcceptA

D2

(ii) Stop (i) Accept B

s es c uc

Fa il

ur e

Event

5000 (0.4)

cep 00) tA

-2000 (0.6)

Probability (p) Success 0.7 Failure 0.3 Success 0.4

D3 (C Ac ost 20

ss ce c Su

EMV = 1500 Fai lu re

3000 (0.7)

-2000 (0.3)

Conditional Payoff (in Expected Payoff (Rs.) Rs.) P p×P 3000 2100 -2000 -600 EMV = 1500 0 5000 2000 24

Failure D1

(ii) Stop (i) Accept A (ii) Accept B

0.6

-2000

Success 0.7

3000 + 800 = 3800

Failure

-2000

0.3

Success 0.4

5000 + 1500 = 6500

Failure

-2000

0.6

-1200 EMV = 800 0 2660 -600 EMV = 2060 2600

-1200 EMV = 1400 (iii)DoNothing 0 From the above table we conclude that the best strategy is to accept investment A first and if it is successful, then accept the investment B.

25