7. Decision Theory and Introduction to Quantitative Methods – Operations Research, 2nd Edition

7

Decision Theory and Introduction to Quantitative Methods

7.1 INTRODUCTION TO DECISION ANALYSIS

Everyday we make decisions and occasionally we make an important decision that can have immediate and/or long-term effects on our lives. Decisions like where to attend school, whether to rent or buy, whether your company should accept a merger proposal, whether to run a cool drink stall or a snacks stall and so on, are important decision—for which we would prefer to make the correct choice. A decision, in general, may be defined as the selection by the decision-maker of an act, considered to be best according to some predestinated standard, from among the several options.

7.1.1 Steps in Decision Theory Approach

Step 1: Determine the various alternative courses of action (or choices or strategies) from which the final decision is to be made.

Step 2: Identify the possible outcomes, called the states of nature or events for the decision problem. The events are beyond the control of the decision-maker.

Step 3: Construct a pay off table

The decision-maker now constructs a pay off table (table representing profit, benefit and so on) for each possible combination of alternative course of action and state of nature.

If there are m alternative courses of action A1, …, Am and n states of nature E1, …, En, then the pay off matrix will be represented as

TABLE 7.1

where aij — the pay off resulting from the combination of iih event and jth action.

Step 4: The decision-maker will choose the criterion which results in largest pay off. The criterion may be economic, quantitative or qualitative (for example, market share, profit, fragrance of a perfume and so on)

Example

A firm manufactures three types of products. The fixed and variable costs are given below:

 

The likely demand (units) of the products is given below:

  Poor demand : 3,000
  Moderate demand : 7,000
  High demand : 11,000

If the sale price of each type of product is 25 then prepare the pay off matrix.

Solution: Let D1, D2 and D3 denote poor, moderate and high demands. Then, pay off is given by Pay off = Sales revenue − Cost (fixed + variable).

The calculations for pay off (in thousand) for each pair of alternative demand (course of action) and the types of product (states of nature) are shown below:

 

D1 A = 3 × 25 − 25 − 3 × 12 = 14
D1 B = 3 × 25 − 35 − 3 × 9 = 13
D1 C = 3 × 25 − 53 − 3 × 7 = 1
D2 A = 7 × 25 − 25 − 7 × 12 = 66
D2 B = 7 × 25 − 35 − 7 × 9 = 77
D2 C = 7 × 25 − 53 − 7 × 7 = 73
D3 A = 11 × 25 − 25 − 11 × 12 = 118
D3 B = 11 × 25 − 35 − 11 × 9 = 141
D3 C = 11 × 25 − 53 − 11 × 7 = 145

The pay off table is

TABLE 7.2

7.1.2 Decision-making Environments

Decision analysis is used to determine optimum strategies where a decision-maker is faced with several decision alternatives. There are four types of decision-making environment:

  1. Decisions under certainty

    Whenever there exists only one outcome for a decision, we are dealing with this category. For example, the decision to purchase either a N.S.C. (National Saving Certificate), Indira Vikas Patra, or deposit in N.S.S. (National Saving Scheme) is one in which it is reasonable to assume the complete information about the further because there is no doubt that the Indian Government will pay the interest as it falls due and the principal at maturity.

    As other examples we can take linear programming transportation problem, assignment problem and sequencing and so on.

  2. Decisions under uncertainty

    These refer to situations where more than one outcome can result from any single decision. That is, here more-than one states of nature exist but the decision-maker lacks sufficient knowledge to allow him to assign probabilities to the various states of nature. For example, the probability that Mr X will be the Prime Minister of the country 15 years from now is not known.

  3. Decisions under risk

    These refer to decision situations wherein the decision-maker chooses from among several possible outcomes where the probability of occurrence can be stated objectively from the past data.

  4. Decisions 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’.

    As an example, we can consider the situation of conflict involving two or more competitors marketing the same product.

7.2 DECISION UNDER UNCERTAINTY

Under conditions of uncertainty, the decision-maker has a knowledge about the states of nature that happen but lacks the knowledge about the probabilities of their occurrence. Situations like launching a new product fall under this category. The insufficient data lead to a more complex decision model and perhaps, a less satisfactory solution. However, one uses scientific methods to exploit the available data to the fullest extent.

In decisions under uncertainty, situations exist in which two (or more) opponents with conflicting objectives try to make decisions with each trying to gain at the cost of the other(s). These situations are different since the decision-maker is working against an intelligent opponent. The theory governing these types of decision problems is called the theory of games and will be taken up later in this chapter.

Different persons have suggested several decision rules for making a decision under such situations.

7.2.1 Criterion of Pessimism (Minimax or Maximin)

The Maximin criterion is based upon the ‘Conservative approach’ to assume that the worst possible is going to happen. The decision-maker considers each strategy and locates the minimum pay off for each, and then select that alternative which maximises the minimum pay off.

Thus, this criterion involves two steps.

Step 1: Find the minimum assured pay off for each alternative (course of action).

Step 2: Choose that alternative which corresponds to the maximum of the above minimum pay off.

When dealing with costs, the maximum cost associated with each alternative is considered and the alternative that minimises this maximum cost is chosen. This is known as Minimax criterion and it involves two steps.

Step 1: Determine the maximum possible cost for each alternative.

Step 2: Choose that alternative which corresponds to the minimum of the above costs.

7.2.2 Criterion of Optimism (Maximax or Minimin)

This criterion is based upon extreme optimism.

In this criterion the decision-maker ensures that be should not miss the opportunity to achieve the greatest possible pay off or lowest possible cost. In maximax criterion the decision-maker selects that particular strategy which corresponds to the maximum pay off for each strategy.

Thus, the maximax criterion consists of the following two steps.

Step 1: Determine the maximum possible pay off for each alternative

Step 2: Select that alternative which corresponds to the maximum of the above maximum pay offs.

In decision problems dealing with costs, the minimum for each alternative is considered and then the alternative which minimises the above minimum cost is selected. This is termed as minimin principle.

7.2.3 Laplace Criterion or Equally Likely Decision Criterion

This criterion is based upon what is known as the principal of insufficient reason.

Since the probabilities associated with the occurrence of various events are unknown, there is not enough information to conclude that these probabilities will be different. Hence, it is assumed that all states of nature will occur with equal probability. That is, each state of nature is assigned an equal probability. As states of nature are mutually exclusive and collectively exhaustive, so the probability of each of these much be 1/(number of states of nature).

The working procedure is as follows.

Step 1: Assign equal probabilities 1/(number of states of nature) to each pay off of a strategy.

Step 2: Determine the expected pay off value for each alternative.

Step 3: Select that alternative which corresponds to the maximum (and minimum for cost) of the above expected pay offs.

7.2.4 Criterion of Realism (Hurwicz Criterion)

This criterion suggests that a rational decision-maker should be neither be completely optimistic nor pessimistic and therefore, must display a mixture of both. Hurwicz, who suggestes 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 completely pessimistic attitude about the future and 1 a completely optimistic attitude about the future. Thus, if α is the coefficient of optimism, then (1 − α) will represents the coefficient of pessimism.

The working procedure is summarised as follows:

  1. Decide the coefficient of optimism a and then the coefficient of pessimism 1 − α.
  2. Determine the maximum as well as minimum pay off for each alternative and obtain the quantities h = α × maximum for each alternative + (1 − α) × minimum for each alternative.
  3. Select an alternative with value of h as maximum.

7.2.5 Criterion of Regret (Savage Criterion) or Minimax Regret 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 action (or alternative) resulting in an opportunity loss of pay off.

The working procedure is as follows.

Step 1: From the given pay off matrix, develop an opportunity-loss (or regret) matrix.

  1. Find the best pay off corresponding to each state of nature (maximum for profit and minimum for cost)

Step 2: Determine the maximum regret amount for each alternative.

Step 3: Choose that alternative which corresponds to the minimum regrets.

Remark: The reader may observe that while the other decision rules do not take into account the cost of opportunity loses by making the wrong decision, the minimax regret criterion does so.

Example 1

A farmer wants to decide which of the three crops he should plant on his 100-acre farm. The profit from each is dependent on the rainfall during the growing season. The farmer has categorised the amount of rainfall as high, medium and low. His estimated profit for each is shown in the Table below.

 

TABLE 7.3

If the farmer wishes to plant only one crop, decide which should be his ‘best crop’ using

  1. Maximax Criterion
  2. Maximin Criterion
  3. Hurwicz criterion (farmer’s degree of optimism being 0.6)
  4. Laplace criterion
  5. Minimax regret criterion.

Solution:

 

TABLE 7.4

  1. Maximax Criterion: From Table 7.4 we observe that the maximum pay off for each alternative crop A, crop B and crop C are 8000, 5000 and 5000, respectively. Maximum among these is 8000 corresponding to crop A. Hence, best strategy under this criterion is crop A.
  2. Maximin Criterion: The minimum off for each alternative crop A, crop B and crop C are 2000, 3500 and 4000, respectively. The maximum among these is 4000 which corresponds to crop C. Hence, the best alternative according to this criterion is C.
  3. Hurwicz Criterion: Given that framers degree of optimism α = 0.6. Hence, 1 − α = 0.4 The quantity h is calculated in the following Table.

     

    TABLE 7.5

    The maximimin value of h is 5600 corresponding to crop A. Hence, the best choice is crop A.

  4. Laplace Criterion: Assign equal probabilities (1/3) [∵ there are three outcomes or states of nature namely high, medium and low] to each pay off of a strategy. The expected pay offs are calculated for each alternative as follows.

    E(Crop A) = 1/3 (8000) + 1/3 (2000) ≃
    E(Crop B) = 1/3 (2000 + 3500 + 4000) ≃ 4333
    E(Crop C) = 1/3 (5000 + 5000 + 4000) ≃ 4666

    The maximum expected value is 4833 for Crop A. Hence, the best alternative under this criterion is A.

  5. Minimax Regret Criterion

TABLE 7.6

Since the pay offs are profits the regret value is called as follows.

ith regret = (maximum pay off − ith pay off)

Since the minimum regret is 3000 corresponding to Crop A and Crop C, the best alternative under this category is either Crop A or Crop C.

Example 2

A manufacturer makes a product, of which the principle ingredient is a chemical X. At the moment, the manufacturer spends, 1000 per year on the supply of X, but there is a possibility that the price may soon increases to four times its present figure because of a worldwide shortage of the chemical. There is another chemical Y, which the manufacturer could use in conjunction with a third chemical Z, in order to give the same effect as chemical X. Chemicals Y and Z would together cost the manufacturer 3000 per year; but their prices are unlikely to rise. What action should the manufacturer take? Apply the maximin and minimax regret criteria for decision-making and give two sets of solutions.

If the coefficient of optimism is 0.4, find the course of action that minimises the cost.

Solution: The data of the problem is summarised in the following Table (negative figures in the Table represents profit).

 

TABLE 7.7

  1. Maximin Criterion

    Since the maximum of the minimum pay off is −3000 which corresponds to S1. Hence the best alternative is S1.

  2. Minimax Regret Criterion

    The opportunity loss Table is given below,

     

    TABLE 7.8

    Table 7.8 shows that the best criteria is S2.

  3. Hurwicz Criterion

    Given that α = 0.4. Hence, 1 − α = 0.6. The quantity h can be calculated as follows. h = α × maximum pay off + (1 − α) × minimum pay off.

 

TABLE 7.9

Since the pay offs or profits, the maximum h is − 2800, corresponding to S2. Hence, the best alternative is S2.

7.3 DECISION UNDER CERTAINTY

Since under this environment, only one state of nature exists, the decision-maker simply picks up the best pay off in the one column and chooses the associated alternative. Under conditions of certainty, the particular state of nature is associated with probability 1. Though the state of nature is only one, possible alternatives could be numerous.

Linear programming, transportation and assignment techniques, input out analysis, activity analysis and economic order quantity models are used for such situations. Few complex managerial decision making problems, however, ever enjoy the luxury of having complete information about the future and thus decision-making under certainty is of little consequential interests.

7.4 DECISION-MAKING UNDER RISK

Most business decisions may have to be made under conditions of risk. Here, more than one states of nature exist and the decision-maker has sufficient information to assign probabilities to each of these states. These probabilities could be obtained form the past records or from simply the subjective judgment of the decision-maker. Under conditions of risk, a number of decision criteria are available which could be of help to the decision-maker. The most popular criterion for evaluating the alternatives is the expected monetary value/expected opportunity loss of the expected pay off.

7.4.1 Expected Monetary Value (EMV) Criterion

The Expected Monetary Value (EMV) for a given course of action is the expected value of the conditional pay off for that action. The conditional pay off are obtained for each action by considering various act-event combinations. The Expected Monetary Value (EMV) for a given course of action is the weighted average pay off, which is the sum of the pay offs for each course of action multiplied by the probabilities associated with each state of nature. Mathematically, EMV is stated as follows:

 

EMV(Aj) =
n = number of states of nature
Pi = probability of occurrence of state of nature i.
pij = pay off for the ith state of nature and jth courses of action.

Procedure:

Step 1: List conditional profit for each act-event combinations, along with the corresponding event probabilities.

Step 2: For each act, determine the expected conditional profits.

Step 3: Determine EMV for each act.

Step 4: Choose the act which corresponds to the optimal EMV.

Example 1

A person has the choice of running a hot snack stall or an ice-cream and cool drink stall at Ooty. If the weather is cool and rainy, he can expect to make a profit of 15000 and if it is warm he can expect to make a profit of 3000 by running a hot snack stall. On the other hand, if his choice is to run an ice-cream and cool drink stall, he can expect to make a profit of 18000 if the weather is warm and 3000 if the weather is cool any rainy. There is 40% chance of weather being warm in the coming season. Should he opt for running the hot snack stall or an ice-cream stall?

Solution: The following pay off table shows the conditional profits resulting from the given act-event combinations.

 

TABLE 7.10

The Expected Monetary Value (EMV) for the hot snack stall is = 0.6 × 15000 + 0.4 × 3000 = 10200

The Expected Monetary Value (EMV) for cool drink stall is = 0.6 × 3000 + 0.4 × 18000 = 9000.

Since the EMV for hot snacks is maximum, person should opt for running a hot-snack stall.

Example 2

A newspaper boy has the following probabilities of selling a magazine.

No. of copies sold Probability
10 0.10
11 0.15
12 0.20
13 0.25
14 0.30

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

Solution: The demand varies from 10 to 14. Hence, there is no reason for the newsboy to buy less than 10 or more than 14 copies. The following Table shows the profit resulting from any possible combination of supply and demand. Stocking 10 copies each day will always result in a profit of 200 paise irrespective of the demand. For instance, even if the demand on some day is 13 copies, he can sell only 10 and hence his conditional profit is 200 paise.

When he stocks 11 copies, his profit will be 220 paise on days when buyers request 11, 12, 13, or 14 copies. But on days he has 11 copies on stock and buyers buy only 10 copies, his profit decreases to 170 paise. The profit of 200 paise on the 10 copies sold must be reduced by 30 paise, the cost of one copy left unsold. The same will be true when he stocks 12, 13 or 14 copies. Thus, the conditional profit in paise is given by

 

Pay off = 20 × copies sold − 30 × copies unsold.

TABLE 7.11 Conditional Profit Table

Next, the expected value of each decision alternative is obtained by multiplying its conditional profit by the associated probability and adding the resulting values.

 

TABLE 7.12 Expected Profit Table

The newsboy must, therefore, order 12 copies to earn the highest possible average daily profit of 222.5 paise. This stock will maximise the total profits over a period of time. Of course, there is no guarantee that he will make a profit of 222.5 paise the next day. However, if he stocks 12 copies each day under the conditions given, he will have a average profit of 222.5 paise per day. This is the best he can do because the choice of any one of the other four possible stock actions will result in a lower daily profit.

7.4.2 Expected Opportunity Loss (EOL) Criterion

An alternative approach to maximising Expected Monetary Value (EMV) is to minimise the Expected Opportunity Loss (EOL), also called expected value of regret.

The EOL is defined as the difference between the highest profit (or pay off) for a state of nature (event) and the actual profit obtained for the particular course of action taken. In other words, EOL is the amount of pay off that is lost by not selecting the course of action that has the greatest pay off for the state of nature that actually occurred. The course of action for 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 in 7.4.1. Thus, only one of the two methods should be applied to reach a decision. Mathematically, it is stated as follows:

where,

 

lij = opportunity loss due to state of nature Ni and course of action Sj,
Pi = probability of occurrence of state of nature Ni.

Procedure for calculating EOL

Step 1: Construct the conditional profit table for each act-event combination, along with the corresponding event probabilities.

Step 2: For each state of nature calculate the Conditional Opportunity Loss (COL) values by subtracting each pay off from the maximum pay off for that event.

Step 3: For each act, determine the expected COL values and add these values to get the Expected Opportunity Loss (EOL) for that act.

Step 4: Choose that act which corresponds to the minimum EOL value.

Example 1

Consider the newspaper boy problem given in example 2 of Section 7.4.1. Obtain the decision alternative using EOL criterion.

Solution: The conditional profit table is given in Table 7.11. The Conditional Opportunity Loss and EOL are calculated in Table 7.13.

 

TABLE 7.13

Since the minimum EOL is 27.5 the best choice is to have a stock of 12 copies.

7.4.3 Expected Value of Perfect Information (EVPI)

In decision-making under risk each state of nature is associated with a probability of its occurrence. However, if the decision-maker can acquire perfect information about the occurrence of various states of nature, then he will be able to select a course of action that yields the desired pay off for whatever state of nature that actually occurs.

Expected profit with perfect information (EPPI) represents the expected profit, in the long run, if we have perfect information before a decision is made.

Expected value of perfect information (EVPI) represents the maximum amount of money the decision-maker has to pay to get this additional information about the occurrence of various states of nature before a decision has to be made

 

EVPI = EPPI − Expected profit without information
= EPPI − EMV*

where,

 


pij − best pay off for state of nature, Ni
pi − probability of states of nature, Ni
EMV* − maximum expected monetary value.

Example 1

A dairy firm wants to determine the quantity of butter it should produce to meet the demand past records have shown the following demand patterns:

The stock levels are restricted to the range 15 to 50 kg due to inadequate storing facilities. Butter costs 40 per kg. and is sold at 50 per kg.

  1. Construct a conditional profit table.
  2. Determine the action alternative associated with maximisation of expected profit.
  3. Determine EVPI.

Solution: The range is from 15 kg to 50 kg.

The probabilities are calculated as follows:

The conditional profit will be calculated as follows:

 

Conditional profit = (50 − 40) S = 10 S when DS
= 50 D − 41 S, when DS

where,

 

D − Demand  and  S − stock.

 

TABLE 7.14 Conditional Profit Table

TABLE 7.15 Expected Profit Table

Since the maximum EMV is 217.50, the dairy firm may produce 25 kg or 30 kg of butter.

If perfect information is known then the stock level is kept equal to the demand level. Then, the EPPI is calculated as follows:

 

TABLE 7.16

EVPI = EPPI − EMV*
= 318.50 − 217.50
= 101

Example 2 EMV for Items that have a Salvage (Resale) Value

An ice-cream retailer buys ice-cream at a cost of 5 per cup and sells it for 8 per cup; any ice-cream remaining unsold at the end of the day can be disposed of at a salvage price of 2 per cup. Past sales have ranged between 15 and 18 cups per day; there is no reason to believe that sales volume will take on any other magnitude in future. Find the EVPI, if the sales history has the following probabilities.

Solution: The market size ranges from 15 to 18 cups. Since the resale value unsold ice-cream is 2 per cup, the conditional profit is calculated as follows:

  Conditional profit = (8 − 5) S = 3S when DS
    = D − 5S + 2 (SD), when D < S

where, D − Demand and S − Stock

The conditional profit and the expected conditional profit is calculated in Table 7.17.

 

TABLE 7.17

Since the maximum EMV is 48.60, the best strategy is to have a stock of 17 cups. If the perfect information is known as which event would occur the retailer can keep the stock level equal to the demand level. Then, the EPPI is calculated as follows:

 

TABLE 7.18

EVPI = EPPI − EMV*
= 50.52 − 48.60
= 1.92.

Example 3

A television dealer finds that the cost of a TV in stock for a week is 30 and the cost of a unit shortage is 70. For one particular model of TV the probability distribution of weekly sales is as follows:

How many units per week should the order? Also find EVPI.

Solution: Here, the cost of stocking = 30 week and shortage cost = 70. The conditional cost are calculated as follows:

Conditional cost = 30 × S when D < S
  = 30 S + 70 (DS) when DS.

TABLE 7.19 Conditional Cost (in )

The expected conditional cost is calculated by multiplying the conditional cost with the corresponding probabilities.

TABLE 7.20 Expected Cost ()

Since the minimal expected cost is 132, the TV dealer should order 3 unit/week.

If the perfect information is known the TV dealer makes the stock level is equal to the demand level. Then, the expected cost is calculated as follows:

 

TABLE 7.21

Now,

 

EVPI = Minimum expected cost − ECPI
= 132 − 87
= 45.

Thus, the TV dealer may give a maximum of 45 to get the perfect information.

7.5 DECISION TREES

Decision-making problems discussed so far referred to as single stage decision problems. Because the pay offs, 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 decision-maker needs to revise his previous decisions on getting new information and make a sequence of other decisions. Thus, the problem becomes a multi-stage decision problem because the consequence of one decision affects future decisions. For example, in the process of marketing a new product, the first decision is often test marketing and the alternative courses of action might be either intensive testing or gradual testing. Given various possible consequences—favourable fair, or poor, the decision-maker may be required to decide between redesigning the product, and aggressive advertising campaign or complete withdrawal of product and so on. Given that decision, there will be an outcome which leads to another decision and so on.

Decision Tree is a graphical representation of various decision alternatives and the sequence of events as if they were branches of a tree.

A decision node is usually represented by a square and indicates places where a decision-maker must make a decision. Each branch leading away from a decision node represents one of the several possible courses of action available to the decision-maker. The chance node is represented by a circle and indicates a point at which the decision-maker will discover the response to his decision, that is, different possible outcomes which can result from a chosen course of action. The respective pay offs and the probabilities associated with alternative courses and the chance events are shown alongside these branches. At the terminal of the chance branch are shown the expected values of the outcome are shown.

The general approach used in decision tree analysis is to work back ward through the tree from right to left, computing the expected value of each chance node. We then choose the particular branch leaving a decision node which leads to the chance node with the highest expected value. This is known as roll back or fold back process.

Steps in Decision Tree Analysis

Step 1: Identify the decision points and the alternative courses of action at each decision point systematically.

Step 2: At each decision point determine the probability and the pay off associated with each course of action.

Step 3: Commencing from the extreme right end, compute the expected pay offs (EMV) for each course of action.

Step 4: Choose the course of action that yields the best pay off for each decisions.

Step 5: Proceed backwards to the next stage of decision points.

Step 6: Repeat above steps till the first decision point is reached.

Step 7: Finally, identify the course of action to be adopted from the beginning to the end under different possible outcomes for the situation as a whole.

Advantages of Decision Tree Approach

  1. It structures the decision process and helps decision-making in an orderly, systematic and sequential manner.
  2. It requires the decision-maker to examine all possible outcomes, whether desirable or undesirable.
  3. It communicates the decision-making process to other in an easy and clear manner, illustrating each assumption about the future.
  4. It displays the logical relationship between the parts of a complex decision and identifies the time sequence in which various actions and subsequent events would occur.
  5. It is especially useful in situations wherein the initial decision and its outcome affects the subsequent decisions. It can be applied in various fields such as introduction of a new product, marketing, make or buy decisions, investment decisions and so on.

Limitations of Decision Tree Approach

  1. Decision tree diagrams become more complicated as the number of decision alternative increases and more variable are introduced.
  2. It becomes highly complicated when inter-dependent alternatives and dependent variables are present in the problem.
  3. It assumes that utility of money is linear with money.
  4. It analyses the problem in terms of expected values and thus yields an ‘average’ valued solution.
  5. There is often inconsistency in assigning probabilities for different events.

Example 1

An electronics company makes a profit of 10000 per day at present. The company has the option to go in for licence from another company and make a profit of 20000 per day gross but has to pay a royalty of 6000 per day. It has another option to go in for research and development at a cost of 10000. The company may make a profit of 25000 with a probability of 70% success and 30% failure. Draw a decision tree and choose the appropriate action.

Solution:

FIGURE 7.1 Decision tree representation

EMV (A) = [0.7 × 25000 + 0.3 × 0] − 10000
= 7500
EMV (node 1) = Maximum [10000, 20000 − 6000, 7500]
= 14,000 (corresponding to loyalty)

Hence, the best choice is to go for licence by paying 6000 as loyalty.

Example 2

Mr. Anish had to decide whether or not to drill on his farm. In his town, only 40% of the wells drilled were successful at 200 feet of depth. Some of the farmers who did not get water at 200 feet, drilled further upto 250 feet but only 20% struck water at 250 feet. Cost of drilling is 50 per foot. Mr. Anish estimated that he would pay 18000 during 4 – year period in the present value terms, if he continues to buy water from the neighbour rather than go for the well which would have a life of 4 years. Mr. Ansih has three decision to make (a) should he drill upto 200 feet and (b) if no water is found at 200 feet, should he drill upto 250 feet? (c) should he continue to buy water from his neighbour?

Solution:

FIGURE 7.2 Decision tree representation

EMV of node B = [0.2 × 12500 + 0.8 × 30500]
= (2500 + 24400)
= 26900

 

EMV of node 2 = 26900 (Lesser of the two values of 26900 and 28000).

 

EMV of node A = [0.4 × 10000 + 0.6 × 26900]
= [4000 + 16140]
= 20140
EMV of node 1 = 18000 [Lesser of the two values of 20140 and 18000].

Hence, the optimal choice of Mr. Anish is not to drill the well any pay 18,000 for water to his neighbour for five years.

7.6 INTRODUCTION TO QUANTITATIVE METHODS

The quantitative methods are powerful tools for achieving the following:

  1. Augmentation of production
  2. Maximisation of profits
  3. Minimisation of costs
  4. Accomplishment of pre-determined objectives

Many of the problems that arise in a business or industrial enterprise can be solved by using quantitative methods.

7.6.1 Definition and Classification

Quantitative methods are those statistical and operations research or programming techniques that help in the decision-making process, especially concerning business and industry. They involve the use of elements of quantities, namely numbers, symbols and other mathematical expressions. Quantitative methods can also be defined as those techniques that provide the decision-maker with a systematic and powerful means of analysis and help, based on quantitative data, in exploring policies for achieving predetermined goals.

Quantitative methods are classified under two major categories, namely (a) statistical techniques and (b) programming techniques.

The following chart enlists the names of the important quantitative techniques:

Programming techniques

These techniques occupy the major part of this book.

Statistical techniques

Those techniques are used in conducting the statistical inquiry concerning a certain phenomenon.

The methods of collection of statistical data, the technique of classification and tabulation of the collected data, the calculation of various statistical measures such as mean, standard deviation, coefficient of correlation, and so on, the techniques of analysis and interpretation, and finally the task of deriving inferences and judging their reliability are some of the important statistical methods.

The following are some of the important techniques often used in business and industry:

  1. Probability theory and sampling analysis
  2. Correlation and regression analysis
  3. Index numbers
  4. Time series analysis
  5. Interpolation and extrapolation
  6. Ratio analysis
  7. Statistical quality control

7.6.2 Role of Quantitative Methods in Business and Industry

Quantitative methods, especially operations research techniques, have gained increasing importance since World War II in the technology of business administration. These techniques greatly help in tackling the intricate and complex problems of modern business and industry. Quantitative techniques for decision-making are, in fact, examples of the use of scientific method of management.

The various roles of the quantitative methods are as follows:

  1. They provide a tool for scientific analysis.
  2. They provide solution for various business problems.
  3. They enable proper deployment of resources.
  4. They help in minimising waiting and servicing costs.
  5. They enable the management to decide when to buy and how much to buy.
  6. They assist in choosing an optimum strategy.
  7. They render great help in optimum resource allocation.
  8. They facilitate the process of decision-making.
  9. Through various quantitative techniques, management can know the reactions of the integrated business systems.
  10. Statistical techniques are also of great help to businessmen in more than one way.

7.6.3 Quantitative Techniques and Business Management

Quantitative techniques are useful to the production management in the following:

  1. Selecting the building site for a plant, scheduling and controlling its development, and designing its layout
  2. Locating within the plant and controlling the movements of required production materials and finished goods inventories
  3. Scheduling and sequencing production by adequate preventive maintenance with optimum number of operations by proper allocation of machines
  4. Calculating the optimum product-mix

Quantitative methods are useful to the personnel management for the following:

  1. Determining optimum manpower planning
  2. Identifying the number of persons to be maintained on the permanent or full-time roll
  3. Identifying the number of persons to be kept in a work pool intended for meeting the absenteeism
  4. Determining the optimum number of sequencing and routing of personnel to a variety of jobs
  5. Studying personnel recruiting procedures, accident rates and labour turnover

Quantitative techniques equally help the marketing management to determine the following:

  1. Where distribution points and warehousing should be located and their size, quantity to be stocked and the choice of customers
  2. The optimum allocation of sales budget to direct selling and promotional expenses
  3. The choice of different media of advertising and bidding strategies
  4. The consumer preferences relating to size, colour, packaging and so on for various products as well as to outbid and outwit competitors

Quantitative techniques are also very useful in financial management for the following:

  1. Finding long-range capital requirements as well as methods to generate these requirements
  2. Determining optimum replacement policies
  3. Working out a profit plan for the firm
  4. Developing capital investment plans
  5. Estimating credit and investment risks

7.6.4 Limitations of Quantitative Techniques

The following are the limitations of quantitative techniques:

  1. There are inherent limitations concerning mathematical expressions.
  2. High costs are involved in the use of quantitative techniques.
  3. Quantitative techniques do not take into consideration the intangible factors (i.e. non-measurable human factors).
  4. Quantitative techniques are just the tools of analysis and not the complete decision-making process.
EXERCISES

Section 7.1

1. Discuss the difference between decision-making under certainty and risk.

2. Discuss the difference between decision-making under uncertainty and risk.

3. What are the steps in decision-making analysis?

Section 7.2

4. Explain various quantitative methods which are useful for decision-making under uncertainty.

5. What techniques are used to solve decision-making problems under uncertainty? Which technique results in an optimistic decision? Which technique results in a pessimistic decision?

6. Give an example of a good decision you made that resulted in a bad outcome. Also, give an example of a good decision you made that had a good outcome. Why was each decision good or bad?

7. A super Bazar 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 for levels of supplies are thus suggested with level j being ideal (from the view point 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 satisfied. The Table below provides these costs in thousands of rupees.

What is the best alternative under

  1. Laplace criterion
  2. Minimax criterion
  3. Maximum criterion
  4. Savage criterion
  5. Hurwicz criterion.

[Answer: (i) A2 or A3 (ii) A3 (iii) A4 (iv) A2 (v) A3.]

8. A food product company is contemplating the introduction of a revolutionary new product with new packaging to 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 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 (N2). The marketing department of the company worked out the pay offs in terms of yearly net profits for each of the strategies of three events (expected sales). This is represented in the following Table.

Which strategy should the concerned executive choose on the basis of

  1. Maximin criterion
  2. Maximax criterion
  3. Minimax regret criterion
  4. Laplace criterion.

[Answer: (i) S3 (ii) S1 (iii) S1 (iv) S1.]

9. Dr. Vivekanandan ha been thinking about starting his own independent nursing home. The problem is to decide how large the nursing home should be. The annual returns will depend on both size of nursing and a number of marketing factors. After a careful analysis, Dr. Vivekanandan developed the following Table:

  1. What is the maximax decision?
  2. What is the maximin decision?
  3. What is equally likely decision?
  4. What is the criterion of realism decision? Use α = 0.8
  5. Develop an opportunity loss table and determine the minimax decision.

[Answer: (a) VL (b) S (c) VL (d) VL (e) L or VL.]

10. In the Anish Toy manufacturing company, suppose the product acceptance probabilities are not known, but the following data is known.

Determine the optimal decision under each of the following criteria and show how you arrived at it.

  1. Maximax
  2. Maximin
  3. Equally likelihood
  4. Minimax regret

[Answer: (a) Full (b) Minimal (c) Full or partial (d) Partial.]

11. Mr Nishok has 10,000 to invest in one of three options. A, B or C. The return on his investment depends on whether the economy experiences inflation recession, or no change at all. His possible returns under each economic condition are given below:

What should he decide using the pessimistic criterion, optimistic criterion, equally likely criterion and regret criterion?

[Answer: Pessimistic A 120, optimistic: B 300, equally likely: C 176.6, regret: C 50.]

Section 7.4

12. A physician purchases a particular vaccine on Monday of each week. The vaccine must be used within the following week, otherwise it becomes worthless. The vaccine costs 2 per dose and the physician charges 4 per dose. In the past 50 weeks, the physician had administered the vaccine in the following quantities.

Determine how many doses the physician should buy every week.

[Answer: EMV = 54; purchase 40 doses.]

13. Mr. Navin quite often files from town A to town B. He can use the airport bus which costs 25 but if he takes it, there is a 0.08 change that he will miss the flight. A hotel limousine costs 270 with a 0.96 chance of being on time for the flight. For 350 he can us a taxi which will make 99 of 100 flights. If Mr. Navin catches the plane on time, he will conclude a business transaction which will produce a profit of 10000, otherwise he will lose it. Which mode of transport should Mr. Navin use? Answer on the basis of the EMV criterion.

[Answer: EMV: 9500 Best choice is: Taxi.]

14. The Blossom flower shop promises its customers delivery within four hours on all flower orders. All flowers are purchased on the previous day and delivered to Blossom by 8.00 AM the next morning. Parker’s daily demand for roses is as follows:

Blossom purchases roses for 10.00 per dozen and sells them for 30.00. All unsold roses are donated to a local hospital. How many dozens of roses should Blossom order each evening to maximise its profits? What is the optimum expected profit?

[Answer: EMV = 168. Best Course of action is 9.]

15. The probability of the demand for lorries for hiring on any day in a given district is as follows:

Lorries have a fixed cost of 90 each day to keep the daily hire charges (net of variable costs of running) 200. If the lorry-hire company owns four 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?

[Answer: EMV = 140. Best course of action: 2.]

16. You are given the following pay offs of three acts A1, A2 and A3 and the events E1, E2, E3.

The probabilities of the states of nature are respectively 0.1, 0.7 and0.2. Calculate and tabulate EMV and conclude which of the acts can be choosen as the best.

[Answer: EMV (A1) = 412.5; EMV (A2) = 455, EMV (A3) = 417.5 Best choice is A2.]

17. The marketing staff of a certain industrial organisation has submitted the following pay off table, giving profit in million rupees, concerning a certain proposal depending upon the rate of technology advance.

The probabilities are 0.2, 0.5 and 0.3 for much, little and none technological advance, respectively. What decision should be taken?

[Answer: EMV (accept) = 2.6, EMV (reject) = 2.8, Best choice is reject.]

18. A company buys certain items which sells for 16 and costs 12. A tabulation of recent demand for the product appears as follows:

Future demands for this product during the next 30 days should be comparable to past demand.

  1. What is the expected profit and loss from stocking 83 units?
  2. What is the expected profit under uncertainty?
  3. What would be the maximum expected profits certainly if this were possible?

[Answer: (a) Profit = 328.72, loss = 11.28]

(b) EMV = 328.72

(c) EPPI = 32.72.]

19. A motor-parts dealer finds that the cost of a particular item in stock for a week is 20 and the cost of a unit shortage is 50. The probability distribution of weekly sales (in thousand items) is as follows:

How many units per week should the dealer order? Also, find the EVPI.

[Answer: 3000 or 4000 units; EVPI = 33.50.]

20. A bicycle repairman has an opportunity to purchase a stock of discontinued bicycles. They were originally supposed to be sold for 400 each. The repairman is offered all five bicycles for 500, which makes his cost for each bicycle 100. If he sells them he believes he can get 250 for each bicycle, thereby making a profit of 150. He has two options: either to buy all the discontinued bicycles or not to buy at all. There are six states of nature; these being the demand for 0, 1, 2, 3, 4 and 5 bicycles.

  1. prepare the pay off as well as regret tables for the problem.
  2. If the repairman has the option of buying any number of bicycles (0 to 5), find the average expected pay off and average expected regret for each stock action.

[Answer: EMV = 200 or EOL = 175 and the optimum stick level is 300 discontinued bicycles.]

21. A certain piece of equipment can be purchased for a construction project at a remote location. This equipment contains an expensive part which is subject to random failure. Spare of this part can be purchased at the same time the equipment is purchased. Their unit cost is 1,500 and they have no scarp 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 9000 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:

  1. Determine optimal EMV and optimal number of spares to purchase initially.
  2. Based on opportunity loss, determine the optimal course of action and optimal value of EOL.
  3. Determine expected profit with perfect information and expected value of perfect information.

[Answer: (a) EMV (one spare purchased) = 1950. Hence, the best choice is to purchase one spare part.

(b) Minimum EOL is (one spare purchased) = 1575. Hence, the best choice is one spare purchased.

(c) EPPI = − 375; EVPI = 1575.]

22. A producer of boats has estimated the following distribution of demand for a particular kind of boat.

Each boat costs him 7000 and he sells them for 10000 each. Boats left unsold at the end of the season must be disposed off for 6000 each. How many boats should be in stock so as to maximise his expected profit?

[Hint: Conditional profit

Maximum EMV = 4080, stock 3 boats.]

23. A small industry finds from the past data, and the cost of making an item is 25, the selling price of the item is 30, if it is sold within a week; and it could be disposed off at 20 per item at the end of the week.

Find the optimum number of items per week the industry should produce.

[Answer: Conditional profit

Maximum EMV = 48.60, produce 6 items.]

24. A firm makes pastries which it sells at 8 per dozen in special boxes containing one dozen each. The direct cost of pastries for the firm is 4.50 per dozen. At the end of the week the stale pastries are sold off for a lower price of 2.50 per dozen. The overhead expenses attributable to pastry production are 1.25 per dozen. Fresh pastries are sold in special boxes which cost 50 paise each and the stale pastries are sold wrapped in ordinary paper. The probability distribution of demand per week is as under:

Find the optimal production level of pastries per week.

[Hint: Conditional profit
Maximum EMV = 4.28, produce 4 dozen pastries.]

25. The probability distribution of monthly sales of an item as follows:

The cost of carrying inventory (unsold during the month) is 30 per unit per month and cost of unit shortage is 70. Determine optimum stock to minimise expected cost.

[Answer: Cost function

where,

 

D–demand and S–stock

Since expected cost, 46 is minimum for course of action 4, optimum stock to minimise the cost is 4 units per month.]

26. A modern home appliances dealer finds that the cost of holding a mini cooking range in stock for a month is 200 (insurance, minor deterioration, interest on borrowed capital and others).

Customers who cannot obtain a cooking range immediate tend to go to other dealers and the estimates that for every customer who cannot get immediate delivery, he loses an average of 500. The probabilities of a demand of 0, 1, 2, 3, 4, 5 mini cooking ranges in a month are 0.05, 0.10, 0.20, 0.30, 0.20 and 0.15, respectively. Determine the optimal stock of mini cooking ranges. Also, find EVPI.

[Answer: Cost function

Since expected cost, 315 is minimum for course of action 4, optimum stock to minimise the cost is four mini cooking ranges.]

Section 7.5

27. Nishok Company is currently working with a process which after paying for materials, labour and so on, brings a profit of 12000. The following alternatives are made available to the company;

  1. The company can conduct research (R1) which is expected to cost 10000 having 90% chances of success. If it proves a success, the company gets a gross income of 25000.
  2. The company can conduct research (R2) which is expected to cost 8000 having a probability of 60% success, the gross income will be 25000.
  3. The company can pay 6000 as royalty for a new process which will bring a gross income of 20000.
  4. The company continues the current process. Due to limited resources it is assumed that only one of the two types research can be carried out at a time. Use decision tree analysis, locate the optimal strategy for the company.

[Answer: Optimal decision is to procure new process on royalty basis.]

28. A manager has a choice between (i) A risky contract promising 7 lakhs with probability 0.6 and 4 lakhs with probability 0.4, and (ii) A diversified portfolio consisting of two contracts with independent outcomes each promising 3.5 lakhs with probability 0.6 and 2 lakhs with probability 0.4.

Construct a decision tree for using EMV Criteria. Can you arrive at the decision using EMV criteria?

[Answer: EMV = 5.8lakhs.]

29. A client asks an estate agent to sell three properties A, B and C for him and agrees to pay him 5% commission on each sale. He specifies certain conditions. The estate agent must sell A first, and this he must do within 50 days. The agent receives his 5% commission on that sale only after A is sold. He can then either back out at this stage or nominate and try to sell one of the two remaining properties within 60 days. If he does not succeed in selling the nominated property in that period, he is not given the opportunity to sell the other. If he does sell it in the period, he is given the opportunity to sell the third property on the same conditions, The following Table summarizes the prices, selling costs and the estate agent’s estimated probability of making a sale.

  1. Draw up an appropriate decision tree for the estate agent.
  2. What is the estate agent’s best strategy under EMV approach?

[Answer: Accept the offer and try to sell A, then B and then C.]

30. An industry has three alternatives namely:

  1. Start production commercially.
  2. Build a pilot plant and then decide about the commercial production.
  3. Stop production.

    It is estimated that the pilot plant has a 0.08 change of high yield and 0.2 chance of low yield. If the yield is high, the commercial production will have a 0.85 change of high profit and 0.15 chance of low profit. If the yield is low the commercial production will have 0.1 chance of high profits and 0.9 chance of low profits. However, direct commercial production is expected to have 0.7 chance of high profit and 0.3 chance of low profit.

    Draw a decision tree to illustrate the situation for decision-making. Excepted values of high profits and low profits are 12 lakh and 15 lakh. The cost of the pilot plant is 3.5 lakh. What should be your decision?

31. A company is currently working with a process, which, after paying for materials, labour and so on, brings a profit of 12000. The company has the following alternatives.

  1. The company can conduct research R1 which is excepted to cost 10000 and having 90% probability of success. If successful, the gross income will be 26000.
  2. The company can conduct research R2, which is expected to cost 6000 and having a probability of 60% success. If successful, the gross income will be 24000.
  3. The company can pay 5000 as royalty of a new process which will bring a gross income of 20000.
  4. The company may continue the current process.

[Answer: The company should pay 5000 as royalty of the new process to earn maximum expected profit of 15000.]

32. Raman Industries Ltd, has a new product which they expect has great potential. At the moment they have two courses of action open to them. To test market (S1) and to drop product (S2).

If they test it, it will cost 50000 and the response could be positive or negative with probabilities 0.70 and 0.30, respectively. If it is positive, they could either market it with full scale or drop the product. If they market with full scale, them the result might be low, medium or high demand and the respective net pay offs would be 100000, 100000 or 500000. These outcomes have probabilities of 0.25, 0.55 and 0.20, respectively. If the result of the test marketing is negative they have decided to drop the product. If, at any point, they drop the product there is a net gain of 25000 from the sale of scrap. All financial values have been discounted to the present. Draw a decision tree for the problem and indicate the most preferred decision. [Answer: The company should test market rather than drop the product. If test market result is positive, company should market the product otherwise drop the product.]

33. XYZ company manufactures guaranteed tennis balls. At present, approximately 10% of the tennis balls are defective. A defective ball leaving the factory costs the company 0.50 to honour its guarantee. Assume that all defective balls are returned. At a cost of 0.10 per ball, the company can conduct a test, which always correctly identifies both good and bad tennis balls.

  1. Draw a decision tree and determine the optimal course of action and its expected cost.
  2. At what test cost the company should be indifferent to testing?

[Answer: (a) Do not test, 0.05 (b) 0.05.]