Decision Analysis for Management Judgment Third Edition. Paul Goodwin The Management School, University of Bath. George Wright Durham Business School . Download as PDF, TXT or read online from Scribd. Flag for . Decision analysis for management judgment / Paul Goodwin, George Wright. – 3rd ed. p. cm. Decision Analysis for Management. Judgment. Fifth Edition. Paul Goodwin. The Management School, University of Bath. George Wright. Strathclyde Business.

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Decision Analysis for Management Judgment (2nd Edn). Article (PDF Available) in Journal of the Operational Research Society 44(2) · February with 3, . Decision Analysis for Management Judgment Third Edition Paul Goodwin The Management School, University of Bath George Wright Durham Business School . Decision Analysis for Management Judgment is unique in its breadth of coverage of decision analysis methods. It covers both the psychological problems that.

Cam- bridge. Moore and Thomas1 discuss an experiment at a business school where a large num- ber of executives were asked to rank 10 words or phrases in decreasing order of uncertainty. Permissions Request permission to reuse content from this site. Despite this. As we saw above. New York.

We will consider, in later chapters, techniques which are designed to be used where risk and uncertainty are central concerns of the decision maker.

Nevertheless, there are exceptions to this rule, and we will show later how the method can be adapted to problems involving risk and uncertainty. It should be emphasized that the main role of our analysis is to enable the decision maker to gain an increased understanding of his or her decision problem.

Often the insights gained may suggest other approaches to the problem or lead to a greater common understanding among a heterogeneous group of decision makers. They may lead to a complete reappraisal of the nature of the problem or enable a manager to reduce a large number of alternatives to a few, which can then be put forward to higher management with arguments for and against.

Indeed, it is likely that this will happen as a deeper understanding of the nature of the problem is gained through the analysis. Basic terminology Objectives and attributes Before proceeding, we need to clarify some of the basic terms we will be using.

Typical objectives might be to minimize costs or maximize market share. An attribute is used to measure performance in relation to an objective. Sometimes we may have to use an attribute which is not directly related to the objective. Such an attribute is referred to as a proxy attribute.

For each course of action facing the decision maker we will be deriving a numerical score to measure its attractiveness to him.

If the decision involves no element of risk and uncertainty we will refer to this score as the value of the course of action. Alternatively, where the decision involves risk and uncertainty, we will refer to this score as the utility of the course of action. Utility will be introduced in Chapter 5. While the owner would like to keep his costs as low as possible, he would also like to take other factors into account. It is also an old, dark building which will not be. The owner is unsure how to set about making his choice, given the number of factors involved.

Because of the simplicity of both the responses required of the decision maker and the manner in which these responses are analyzed, SMART has been widely applied. This, coupled with the relative speed by which the method can be applied, means that SMART has been found to be a useful vehicle for decision conferences see Chapter 12 , where groups of decision makers meet to consider a decision problem.

The cost of this simplicity is that the method may not capture all the detail and complexities of the real problem. Nevertheless, in practice, the approach has been found to be extremely robust see Watson and Buede5.

The main stages in the analysis are shown below:. Stage 1: Identify the decision maker or decision makers. In our problem we will assume that this is just the business owner, but in Chapter 13 we will look at the application of SMART to prob- lems involving groups of decision makers. Stage 2: Identify the alternative courses of action. Stage 3: Identify the attributes which are relevant to the decision problem.

In the next section we will show how a value tree can be of use when identifying relevant attributes. Stage 4: For each attribute, assign values to measure the performance of the alternatives on that attribute. For example, how well do the. Stage 5: Determine a weight for each attribute. Stage 6: For each alternative, take a weighted average of the values assigned to that alternative. Stage 7: Make a provisional decision. Stage 8: Constructing a value tree Stages 1 and 2 of our analysis have already been completed: The next step is to identify the attributes which the decision maker considers to be relevant to his problem.

You will recall that an attribute is used to measure the performance of courses of action in relation to the objectives of the decision maker. This means that we need to arrive at a set of attributes which can be assessed on a numeric scale. However, the initial attributes elicited from the decision maker may be vague e. A value tree can be useful here, and Figure 3. We start constructing the tree by addressing the attributes which represent the general concerns of the decision maker.

There is, of course, no restriction on the number of attributes which the decision maker can initially specify e. In some applications e. Having established the main attributes for our business owner, we need to decompose them to a level where they can be assessed. If the tree is complete, all the attributes which are of concern to the decision maker will have been included. This criterion requires that the performance of an option on one attribute can be judged independently of its performance on other attributes.

If two attributes duplicate each other because they actually represent the same thing then one of these attributes is clearly redundant. One way of identifying redundancy is to establish whether the decision would in any way be affected if a given attribute was eliminated from the tree.

If the deletion of the attribute would not make any difference to the choice of the best course of action then there is no point in including it. If the tree is too large any meaningful analysis may be impossible. To ensure that this does not happen, attributes should not be decomposed beyond the level where they can be evaluated.

Sometimes the size of the tree can be reduced by elim- inating attributes which do not distinguish between the options. For example, to make the tree operational it may be necessary to increase its size. Often several attempts at formulating a tree may be required before an acceptable structure is arrived at.

The tree went through a number of stages of development as new insights were gained into the nature of the problem. The owner already knows the annual rent and he is able to obtain estimates of cleaning and electricity costs from companies which supply these services.

Details of all these costs are given in Table 3. In measuring these attributes our task will be made easier if we can identify variables to represent the attributes. Table 3. Addison Square 30 3 2 35 Bilton Village 15 2 17 Carlisle Walk 5 1 6 Denver Street 12 1 1 14 Elton Street 30 2 2 34 Filton Village 15 1 2 18 Gorton Square 10 1 12 His rankings are:.

Addison Square, the best location for image, can now be given a value for image of and Carlisle Walk, the location with the least appealing image, can be given a value of 0. As we explain below, any two numbers could have been used here as long as the number allocated to the most- preferred location is higher than that allocated to the least preferred. Figure 3. This shows that the improvement in image between Carlisle Walk and Gorton Square is perceived by the owner to be twice as preferable as the improvement in image between Carlisle Walk and Bilton Village.

Similarly, the improvement in image between Carlisle Walk and Addison Square is seen to be ten times more preferable than the improvement between Carlisle Walk and Bilton Village. Note that it is the interval or improvement between the points in the scale which we compare.

This is because the allocation of a zero to represent the image of Carlisle Walk was arbitrary, and we therefore have what is known as an interval scale, which allows only intervals between points to be compared.

The Fahrenheit and Celsius temperature scales are the most well-known examples of interval scales. You can verify this by converting the temperatures to degrees Fahrenheit to obtain. Having established an initial set of values for image, these should be checked to see if they consistently represent the preferences of the decision maker.

We can achieve this by asking him, for example, if he is happy that the improvement in image between Elton Street and Addison Square is roughly as preferable as the improvement in image between Gorton Square and Denver Street. Similarly, is he happy that the improvement in image between Carlisle Walk and Denver Street is less preferable than that between Denver Street and Elton Street? The answers to these questions may lead to a revision of the values. In mathematical notation we can say that: Because of this.

Elton Street. This can be achieved as follows. He judges this to be ft2. He decides that this is ft2. There are several methods which can be used to elicit a value function.

The curve also suggests that a move from 0 to 2 miles from the town center is far more damaging to business than a move from 6 to 8 miles. The increases from smaller areas will.

SMART function. This implies that an increase in area from ft2 to ft2 is just as attractive as an increase from ft2 to ft2. He is then offered other candidates for the midpoint position for example.

Note that the greater the distance from the town center. Measuring how well the options perform on each attribute 39 75 Value 50 25 0 Office floor area ft2 Figure 3. X and Y. To see this. Then he is asked. If the options perform very similarly on a particular attribute.

Taking this to extremes. These are derived by asking the decision maker to compare a change or swing from the least-preferred to the most-preferred value on one attribute to a similar change in another attribute. The simplest approach is to proceed as follows. The problem with importance weights is that they may not take into account the range between the least. In this case. Determining the weights of the attributes 41 According to this.

After this change has been made. As shown below. After some thought. Normalization is achieved by simply dividing each weight by the sum of the weights and multiplying by The owner is asked to compare a swing from the least visible location to the most visible. The other weights are assessed as follows. Normalized weights to nearest Attribute Original weights whole number Closeness to customers 32 Visibility 80 26 Image 70 23 Size 30 10 Comfort 20 6 Car-parking facilities 10 3 The weights for the higher-level attributes in the value tree.

The procedure is repeated for all the other lower-level attributes and Figure 3. To do this. The additive model is by far the most widely used. The calculations which the additive model involves are shown below for Addison Square. As we show below. We will examine the limitations of the additive model later. Each value is multiplied by the weight attached to.

Any inconsistencies between the two sets of weights can then be discussed and. SMART that attribute. If our decision maker had not found this to be a problem then we could have treated cost as just another attribute. In Figure 3. We could therefore have allocated values to the various costs.

Thus the only locations which are worth considering are Addison Square A. At the very least. It would not therefore be worth considering Elton Street. This is to make this graph comparable with ones we will meet later in the book. Gorton Square would be an intermediate choice. He may be surprised that Bilton Village has fared so. This topic is covered in the next section.

If this is the case then the following procedure suggested by Edwards and Newman9 can be used. Now we need to determine how much each extra value point is worth to the owner. These normalize to 0. The owner is a little worried about the weight of turnover i. If the decision maker accepts these axioms. At the other extreme. These assumptions. Similar analysis could be carried out on the lower-level weights. As we saw above. They represent a set of postulates which may be regarded as reasonable.

Since the owner assigned a weight of 81 to turnover. Filton Village F. In many cases sensitivity analysis also shows that the data supplied do not need to be precise.

In technical terms. It was implicitly assumed that such a distance existed. Let us now consider the axioms: We assumed that the owner was able to decide which of two options he preferred. As we pointed out. Theoretical considerations 49 should also accept the preference rankings indicated by the method.

It may have been that the owner was very unsure about making this comparison or he may have refused to make it at all. He also preferred the image of Bilton Village to Carlisle Walk i. This implies that if the owner prefers A to B and B to C.

If transitivity applies then the owner must therefore also prefer the image of Addison Square to Carlisle Walk i. Here the owner was asked to identify a distance from the center of town which had a value halfway between the worst and best distances.

Assumptions made when aggregating values In our analysis we used the additive model to aggregate the values for the different attributes. The owner preferred the image of Addison Square to Bilton Village i. This assumption was necessary for the bisection method of obtaining a value function. To see what can happen when the additive model is applied to a problem where mutual preference independence does not exist. P and Q. When choosing a holiday destination.

If this is the case. Note that mutual preference independence does not automatically follow. These are both the same size ft2 but X is closer to customers. Longer discussions can be found in Bodily12 and von Winterfeldt and Edwards. The most well known of these is the multiplicative model. In the occasional problems where this is not possible. Consider again the case of the house purchase decision where the quality of the architecture and attractiveness of the garden complemented each other.

Because the multiplicative model is not widely used we will not consider it in detail. This view is supported by research which suggests that the correlation of preference rankings derived from holistic judgments with those derived from SMART-type analyses decreases as the number of attributes in the problem gets larger.

An analogy can be made with an attempt to answer a math- ematical problem by using mental arithmetic rather than a calculator. In other words. It is possible that the decision maker could argue the case for a different set of sensible axioms. As long as he behaved consistently with these axioms.

If exploration of the discrepancy between holistic judgment and model results shows the model to be at fault. We can. We will examine this idea in more detail in Chapter 12 in the context of group decision making. SMART different order from that obtained through our analysis.

This begs the question: Thus at the point where the decision maker knows what to do next a requisite model has been achieved. Phillips argues that: We identify the alternative courses of action stage 2 before we determine the relevant attributes stage 3. As a deeper understanding of the problem is obtained the model will be revised and the discrepancy between the analytical and intuitive judgments will be reduced.

These alternatives are then evaluated in the same way as for alternative-focused thinking. The model can be considered requisite only when no new intuitions emerge about the problem.

Only then do you create alternatives that might help you to achieve these objec- tives. Edwards and Barron recommend that preliminary checks should be made. We could then ask how much more desirable is the improvement at the bottom compared to the top. Underlying this strategy is the idea that. If it does not matter. As a rule of thumb. Never- theless. Ward Edwards and F. Consistent with this strategy. SMART situation? We know that. While a set of equations.

This suggests an approximate weight of Even when they did not agree. This means we should be very care- ful before we exclude dominated options from further consideration. These discrepancies become less important if we recall that the main purpose of a decision analysis model is not to tell us what to do in a mechanistic fashion. This is because the assessment of the worth of a value point to the decision maker is based on the normalized weights and differences between the SMART and ROC weights can lead to large discrepancies in this assessment.

The central idea was that. The method through which they are derived involves some sophisticated mathematics. Both of these problems can be mitigated to some extent by using an alternative weight approximation method.

Several methods exist. Summary 57 we hope to obtain. To be told that your implied weight for an attribute is We saw that the method required the decision maker to quantify his or her strengths of preferences. Summary In this chapter we have looked at a method for analyzing decision problems where each alternative had several attributes associated with it.

Belton and Stewart17 point out that the ratio of the ROC weights between the most and least important attributes is generally very high. This makes the relative importance of the lowest ranked attribute so low that. Suppose that 3 attributes have been ranked. While this may not have been an easy process. Rank-sum weights are easily calculated and hence are more transparent. For 4 attributes the weights will be 0. We stated in the Introduction that this method is normally applied where risk and uncertainty are not major concerns of the decision maker.

We will consider a similar approach to risk in the context of a group decision problem in Chapter A graph such as Figure 3. SMART The decision problem presented in this chapter was designed to be amenable to hand calculations. Wooler and Barclay6 describe such an application involving a strike-prone production facility.

This would. The analysis involved a group of managers in a decision conference. Exercises 1 Formulate a value tree to identify the attributes which are of concern to you when choosing a vacation. Because your employer will pay for the package you are not concerned about the cost.

What does this imply about your method of analysis in b? Four companies have submitted designs for the equipment which will be installed in the mill and a choice has to be made between them. Assuming that mutual preference independence exists between the two attributes. These values are shown below. He has assessed how well each design performs on each attribute by allocating values on a scale from 0 the worst design to the best.

Four methods of transport are being considered: Safety of Cargo. The values she assigned are shown below together with the estimated annual cost of using each form of transport.

She has also allocated weights of 30 to punctuality. The manager then rated the performance of each form of transport on the different attributes.

Assuming that the two weights sum to and that mutual preference independence exists between the attributes. Which design should the manager choose? Convenience and Costs. Note that 0 represents the worst and the best score on an attribute. Jones Wood. Red Beach and Treehome Valley. Exercises 61 b For each form of transport. In order to help them to choose between the sites the managers involved in the decision arranged for a decision analyst to attend one of their meetings.

The scores they eventually agreed are shown below. Red Beach is only two miles from the main town in the area and is close to a main highway while Inston Common is in a remote spot and its use would lead to a major increase in the volume of transport using the minor roads in the area. In the case of risk. Inston Common.

The main criteria which will be used to compare candidates are: What are the dangers of this approach? The managers agreed that they would move to a site offering the least risk of contamination. SMART The decision analyst then asked the managers to imagine a site which had the worst visual impact. Discuss whether the method the personnel manager used to assess these weights is appropriate.

The alternative banks are listed below. Her decision on which bank to choose will be based not only on the estimated annual bank charges which each bank will levy. State any assumptions you have made.

Her ranks are given as: Bank Aggregate score Central She is then asked to imagine that each attribute could be switched to its best possible value and asked to rank the attractiveness of these possible switches.

Cardi- gan. DRT and Ellton. These are i the cost of using the hotel. The organizers started with an initial list of 40 possible hotels. A value tree was used to identify the attributes relating to the decision. Exercises 67 ii ease of transport to the hotel. Explain what these criteria will mean in the context of the problem. Explain why the weights might still be valid. Show how the score for the Alton hotel was obtained. The results are given below.

Improvements from the worst to best performance in staff. Five universities have submitted bids for the course and the television company has to make a decision between them. Determine which hotel they should choose. It was agreed that improving the course structure to that equivalent to the best course structure would be the most desirable. Cambridge Uni- versity Press. Beverly Hills. Engineering Economist.

University Aggregate score Barchester Summer Sympo- sium Series. Larichev eds Strategic Decision Support Systems. Identify the university it should choose assuming that the original weights apply.

Vecsenyi and O. Prefer- ences and Value Tradeoffs. New York.

European Journal of Operational Research. Von Winterfeldt. Vrolijk H. Organizational Behavior and Human Decision Processes. Harvard University Press. SMART 7. An Integrated Approach. Defense Management Journal. Acta Psycho- logica. Hammond eds Judgment and Decision Making. Arkes and K. Kluwer Academic Publishers.

Numbers offer a much more precise way of measuring uncertainty. Moore and Thomas1 discuss an experiment at a business school where a large num- ber of executives were asked to rank 10 words or phrases in decreasing order of uncertainty.

There are a number of ways in which uncertainty can be measured and expressed. In the next chapter we will be looking at how to analyze decisions which involve uncertainty. Suppose that a company is thinking of simultaneously launching two new products.

The ways in which decision analysts elicit subjective probabilities from decision makers will be described and evaluated in Chapter Product A succeeds but B fails. An event consists of one or more possible outcomes. His list is shown below: Both products fail. In this chapter we will introduce the main ideas and rules which are used in probability calculations. Probabilities are measured on a scale which runs from 0 to 1.

At the opposite extreme. If the probability of an outcome occurring is zero then this implies that the outcome is impossible. It is worth pointing out that odds can be converted to probabilities. Product A fails but B succeeds. Each of the four possible things that can happen is called an outcome. Thus the chances of the event occurring are. Outcomes and events Before proceeding.

Both products succeed. You know that 80 of the components are damaged beyond repair. In most. The classical approach Consider the following problem. In 80 of these outcomes a component is selected which is damaged beyond repair so: In contrast. You work for a company which is a rather dubious supplier of electronic components and you have just sent a batch of components to a customer. What are the chances that the customer will select a component which is damaged beyond repair? The classical approach to probability involves the application of the following formula: Approaches to probability 73 Approaches to probability There are three different approaches to deriving probabilities: This raises the problem of specifying a suitable reference class.

The relative frequency approach In the relative frequency approach the probability of an event occurring is regarded as the proportion of times that the event occurs in the long run if stable conditions apply.

This probability can be estimated by repeating an experiment a large number of times or by gathering relevant data and determining the frequency with which the event of interest has occurred in the past. Judgment is therefore required to strike a balance between these two considerations.

If you decide to risk not insuring the contents of your house this year then you must have made some assessment of the chances of the contents remaining safe over the next 12 months. As we argued in the Introduction. Some people may be concerned that subjective probability estimates are likely to be of poor quality.

The company may have access to data relating to the success or otherwise of earlier products or machines. Many people are skeptical about subjective probabilities and yet we make similar sorts of judgments all the time. Thus a sales manager may say: Contact your Rep for all inquiries. View Student Companion Site. Request permission to reuse content from this site.

Undetected country. NO YES. Decision Analysis for Management Judgment, 5th Edition. Selected type: Any inconsistencies between the two sets of weights can then be discussed and. We will examine the limitations of the additive model later. The calculations which the additive model involves are shown below for Addison Square. As we show below. To do this. The additive model is by far the most widely used. Each value is multiplied by the weight attached to.

If our decision maker had not found this to be a problem then we could have treated cost as just another attribute. We could therefore have allocated values to the various costs.

In Figure 3. SMART that attribute. This is to make this graph comparable with ones we will meet later in the book. At the very least. It would not therefore be worth considering Elton Street. He may be surprised that Bilton Village has fared so. Gorton Square would be an intermediate choice. Thus the only locations which are worth considering are Addison Square A. Now we need to determine how much each extra value point is worth to the owner.

This topic is covered in the next section. If this is the case then the following procedure suggested by Edwards and Newman9 can be used. The owner is a little worried about the weight of turnover i. These normalize to 0. Filton Village F. Since the owner assigned a weight of 81 to turnover. Similar analysis could be carried out on the lower-level weights. If the decision maker accepts these axioms. At the other extreme. These assumptions. In many cases sensitivity analysis also shows that the data supplied do not need to be precise.

As we saw above. They represent a set of postulates which may be regarded as reasonable. This assumption was necessary for the bisection method of obtaining a value function. It may have been that the owner was very unsure about making this comparison or he may have refused to make it at all. If transitivity applies then the owner must therefore also prefer the image of Addison Square to Carlisle Walk i.

Assumptions made when aggregating values In our analysis we used the additive model to aggregate the values for the different attributes. Theoretical considerations 49 should also accept the preference rankings indicated by the method.

Let us now consider the axioms: We assumed that the owner was able to decide which of two options he preferred. As we pointed out. The owner preferred the image of Addison Square to Bilton Village i.

Here the owner was asked to identify a distance from the center of town which had a value halfway between the worst and best distances. He also preferred the image of Bilton Village to Carlisle Walk i. It was implicitly assumed that such a distance existed.

This implies that if the owner prefers A to B and B to C. In technical terms. P and Q. Note that mutual preference independence does not automatically follow. If this is the case. These are both the same size ft2 but X is closer to customers. To see what can happen when the additive model is applied to a problem where mutual preference independence does not exist. When choosing a holiday destination. In the occasional problems where this is not possible.

Because the multiplicative model is not widely used we will not consider it in detail. Longer discussions can be found in Bodily12 and von Winterfeldt and Edwards. The most well known of these is the multiplicative model. Consider again the case of the house purchase decision where the quality of the architecture and attractiveness of the garden complemented each other. An analogy can be made with an attempt to answer a math- ematical problem by using mental arithmetic rather than a calculator.

This view is supported by research which suggests that the correlation of preference rankings derived from holistic judgments with those derived from SMART-type analyses decreases as the number of attributes in the problem gets larger. It is possible that the decision maker could argue the case for a different set of sensible axioms.

As long as he behaved consistently with these axioms. Thus at the point where the decision maker knows what to do next a requisite model has been achieved. In other words. This begs the question: SMART different order from that obtained through our analysis.

Phillips argues that: We can. We will examine this idea in more detail in Chapter 12 in the context of group decision making. If exploration of the discrepancy between holistic judgment and model results shows the model to be at fault. The model can be considered requisite only when no new intuitions emerge about the problem. We identify the alternative courses of action stage 2 before we determine the relevant attributes stage 3.

Only then do you create alternatives that might help you to achieve these objec- tives. As a deeper understanding of the problem is obtained the model will be revised and the discrepancy between the analytical and intuitive judgments will be reduced. These alternatives are then evaluated in the same way as for alternative-focused thinking. As a rule of thumb.

Never- theless. SMART situation? Consistent with this strategy. We could then ask how much more desirable is the improvement at the bottom compared to the top. If it does not matter. Edwards and Barron recommend that preliminary checks should be made. Ward Edwards and F. Underlying this strategy is the idea that.

While a set of equations. This suggests an approximate weight of We know that. This is because the assessment of the worth of a value point to the decision maker is based on the normalized weights and differences between the SMART and ROC weights can lead to large discrepancies in this assessment.

Even when they did not agree. This means we should be very care- ful before we exclude dominated options from further consideration. These discrepancies become less important if we recall that the main purpose of a decision analysis model is not to tell us what to do in a mechanistic fashion. The central idea was that. Several methods exist. This makes the relative importance of the lowest ranked attribute so low that.

While this may not have been an easy process. Rank-sum weights are easily calculated and hence are more transparent.

The method through which they are derived involves some sophisticated mathematics. To be told that your implied weight for an attribute is Summary 57 we hope to obtain. Summary In this chapter we have looked at a method for analyzing decision problems where each alternative had several attributes associated with it. We saw that the method required the decision maker to quantify his or her strengths of preferences. For 4 attributes the weights will be 0. Both of these problems can be mitigated to some extent by using an alternative weight approximation method.

Belton and Stewart17 point out that the ratio of the ROC weights between the most and least important attributes is generally very high. Suppose that 3 attributes have been ranked. We stated in the Introduction that this method is normally applied where risk and uncertainty are not major concerns of the decision maker. Wooler and Barclay6 describe such an application involving a strike-prone production facility.

Because your employer will pay for the package you are not concerned about the cost. This would. SMART The decision problem presented in this chapter was designed to be amenable to hand calculations. Exercises 1 Formulate a value tree to identify the attributes which are of concern to you when choosing a vacation. We will consider a similar approach to risk in the context of a group decision problem in Chapter A graph such as Figure 3. The analysis involved a group of managers in a decision conference.

Four companies have submitted designs for the equipment which will be installed in the mill and a choice has to be made between them. These values are shown below. He has assessed how well each design performs on each attribute by allocating values on a scale from 0 the worst design to the best.

Assuming that mutual preference independence exists between the two attributes. What does this imply about your method of analysis in b? Four methods of transport are being considered: The values she assigned are shown below together with the estimated annual cost of using each form of transport.

The manager then rated the performance of each form of transport on the different attributes. Assuming that the two weights sum to and that mutual preference independence exists between the attributes.

Safety of Cargo. Which design should the manager choose? Convenience and Costs. She has also allocated weights of 30 to punctuality. Note that 0 represents the worst and the best score on an attribute.

Red Beach and Treehome Valley. In the case of risk. Jones Wood. Exercises 61 b For each form of transport.

In order to help them to choose between the sites the managers involved in the decision arranged for a decision analyst to attend one of their meetings. Red Beach is only two miles from the main town in the area and is close to a main highway while Inston Common is in a remote spot and its use would lead to a major increase in the volume of transport using the minor roads in the area.

The scores they eventually agreed are shown below. Inston Common. The managers agreed that they would move to a site offering the least risk of contamination.

The main criteria which will be used to compare candidates are: What are the dangers of this approach? SMART The decision analyst then asked the managers to imagine a site which had the worst visual impact. Discuss whether the method the personnel manager used to assess these weights is appropriate. State any assumptions you have made. Her decision on which bank to choose will be based not only on the estimated annual bank charges which each bank will levy.

The alternative banks are listed below. Cardi- gan. Her ranks are given as: Bank Aggregate score Central DRT and Ellton. She is then asked to imagine that each attribute could be switched to its best possible value and asked to rank the attractiveness of these possible switches. The organizers started with an initial list of 40 possible hotels.

These are i the cost of using the hotel. A value tree was used to identify the attributes relating to the decision. The results are given below. Show how the score for the Alton hotel was obtained. Exercises 67 ii ease of transport to the hotel. Explain why the weights might still be valid. Explain what these criteria will mean in the context of the problem. Improvements from the worst to best performance in staff.

Five universities have submitted bids for the course and the television company has to make a decision between them. Determine which hotel they should choose. It was agreed that improving the course structure to that equivalent to the best course structure would be the most desirable.

Prefer- ences and Value Tradeoffs. Identify the university it should choose assuming that the original weights apply. University Aggregate score Barchester New York.

Beverly Hills. Vecsenyi and O. Engineering Economist. Cambridge Uni- versity Press. Larichev eds Strategic Decision Support Systems.

Summer Sympo- sium Series. Vrolijk H. Organizational Behavior and Human Decision Processes. Kluwer Academic Publishers. Defense Management Journal. Hammond eds Judgment and Decision Making.

SMART 7.

An Integrated Approach. European Journal of Operational Research. Arkes and K.

Acta Psycho- logica. Harvard University Press. Von Winterfeldt. There are a number of ways in which uncertainty can be measured and expressed. Numbers offer a much more precise way of measuring uncertainty. Moore and Thomas1 discuss an experiment at a business school where a large num- ber of executives were asked to rank 10 words or phrases in decreasing order of uncertainty.

In the next chapter we will be looking at how to analyze decisions which involve uncertainty. Suppose that a company is thinking of simultaneously launching two new products.

An event consists of one or more possible outcomes. Outcomes and events Before proceeding. His list is shown below: Both products fail. In this chapter we will introduce the main ideas and rules which are used in probability calculations. At the opposite extreme. If the probability of an outcome occurring is zero then this implies that the outcome is impossible. Product A fails but B succeeds. Probabilities are measured on a scale which runs from 0 to 1. It is worth pointing out that odds can be converted to probabilities.

The ways in which decision analysts elicit subjective probabilities from decision makers will be described and evaluated in Chapter Each of the four possible things that can happen is called an outcome. Both products succeed. Thus the chances of the event occurring are. Product A succeeds but B fails. The classical approach Consider the following problem. Approaches to probability 73 Approaches to probability There are three different approaches to deriving probabilities: In most.

In 80 of these outcomes a component is selected which is damaged beyond repair so: In contrast. You know that 80 of the components are damaged beyond repair. You work for a company which is a rather dubious supplier of electronic components and you have just sent a batch of components to a customer. What are the chances that the customer will select a component which is damaged beyond repair?

The classical approach to probability involves the application of the following formula: This raises the problem of specifying a suitable reference class. Judgment is therefore required to strike a balance between these two considerations. This probability can be estimated by repeating an experiment a large number of times or by gathering relevant data and determining the frequency with which the event of interest has occurred in the past. The relative frequency approach In the relative frequency approach the probability of an event occurring is regarded as the proportion of times that the event occurs in the long run if stable conditions apply.

We will review this research in Chapter 9 while in Chapter The company may have access to data relating to the success or otherwise of earlier products or machines. Approaches to probability 75 The subjective approach Most of the decision problems which we will consider in this book will require us to estimate the probability of unique events occurring i.

In these circumstances the probability can be estimated by using the subjective approach. Thus a sales manager may say: As we argued in the Introduction. If you decide to risk not insuring the contents of your house this year then you must have made some assessment of the chances of the contents remaining safe over the next 12 months. In organizations. Some people may be concerned that subjective probability estimates are likely to be of poor quality.

The resulting statement can be precisely communicated to others and. Many people are skeptical about subjective probabilities and yet we make similar sorts of judgments all the time. The addition rule In some problems we need to calculate the probability that either one event or another event will occur if A and B are the two events.

Mutually exclusive and exhaustive events Two events are mutually exclusive or disjoint if the occurrence of one of the events precludes the simultaneous occurrence of the other. At this stage. Having looked at the three approaches to probability.

In these cases the addition rule can be used to calculate the required probability but. These calculations apply equally well to classical. If you make a list of the events which can occur when you adopt a particular course of action then this list is said to be exhaustive if your list includes every possible event. Time to launch product Probability 1 year 0.

The table gives details of rainfall during April for the past 20 years and also Table 4. The addition rule 77 If the events are mutually exclusive then the addition rule is: Let us now see what happens if the addition rule for mutually exclusive events is wrongly applied. Consider Table 4. This has meant that we have double-counted the nine years when both events did occur. We decide to use the relative frequency approach based on the records for the past 20 years and we then proceed as follows: Since it is certain that either the event or its complement must occur their probabilities always sum to one.

If the events are not mutually exclusive we should apply the addition rule as follows: This leads to the useful expression: Thus the correct answer to our problem is: The required probability is known as a conditional probability because the probability we are calculating is conditional on the fact that the worker has been exposed to the chemical.

Suppose that now we wish to calculate the probability of a worker suffering from cancer given that he or she was exposed to the chemical. The probability of event A occurring Table 4.

Marginal and conditional probabilities 79 For example. Suppose that we want to determine the probability that a worker in this industry will contract cancer irrespective of whether or not he or she was exposed to the chemical. Assuming that the survey is representative and using the relative frequency approach.

The answer is easily found: These two events are therefore said to be dependent. The multiplication rule We saw earlier that the probability of either event A or B occurring can be calculated by using the addition rule. In the previous section we saw that the probability of a worker contracting cancer was affected by whether or not he or she has been exposed to a chemical.

We will consider this sort of relationship between events next. Independent and dependent events Two events. If two events. In many circumstances. We only have records of workers who were exposed to the chemical and of these have contracted cancer. If they are. Before applying this rule we need to establish whether or not the two events are independent. C and D.

To see how the rule can be applied. The multiplication rule 81 example. It is estimated that the probability that the bridge construction will be completed on time is 0. If the test marketing is successful. Thus we have the probability of A occurring multiplied by the probability of B occurring. The probability of A and B occurring is known as a joint probability. A new product is to be test marketed in Florida and it is estimated that there is a probability of 0.

The teams involved with the two projects operate totally independently. Since it seems reasonable to assume that the two completion times are independent. What is the. The four routes through the tree represent the four joint events which can occur e. One device which can prove to be particularly useful when awkward problems need to be solved is the probability tree.

If the Socialist Party wins then it is estimated that there is a 0. Note that the tree shows the possible events in chronological order from left to right. It is estimated that there is a 0. The company wants to estimate the probability that their assets will be nationalized after the election. Applying the multiplication rule we have: The calculations shown on the tree are explained below. Socialists win and assets are not nationalized. The probability tree for this problem is shown in Figure 4.

Probability distributions 83 d 0. This is therefore an example of what is known as a discrete probability distribution. This complete statement of all the possible events and their probabilities is known as a probability distribution. If we plotted. Since the company is certain that the completion time will be between 10 and 22 weeks. There is no reason why the time should be restricted to a whole number of minutes. The probability that the completion time will be between two values is found by considering the area under the pdf between these two points.

Note that the vertical axis of the graph has been labeled probability density rather than probability because we are not using the graph to display the probability that exact values will occur. The curve shown is known as a probability density function pdf. Because half of the area under the curve falls between times of 14 and 18 weeks this implies that there is a 0. Figure 4. Because continuous uncertain quantities can.

Probability distributions 85 In contrast. The cdf for the above project is shown in Figure 4. It can be seen that there is a 0. Project completion time Probability 10 to under 14 weeks 0. A summary of the probability distribution is shown below. Sometimes it is useful to use continuous distributions as approxima- tions for discrete distributions and vice versa. Expected values 87 easier to treat it as a continuous variable.

This might also apply to the sales of a product. The num- ber of color sets she sells per week follows the probability distribution shown below: Project completion time Probability 12 weeks 0. In practice. This is easily done by multiplying each sales level by its probability of occurrence and summing the resulting products as shown below.

The result is known as an expected value. What is the expected saving of purchasing early? The calculations are shown below: As a result of the coup. It estimates that there is a 0.

Note that an expected value does not have to coincide with an actual value in the distribution. Summary As we shall see in the next chapter. Thus axioms 1 and 2 imply that the probability of an event occurring must be at least zero and no greater than 1. Axiom 1: Positiveness The probability of an event occurring must be non-negative. Summary 89 of the two monetary values taking into account their probabilities of occurrence.

Axiom 3: Unions If events A and B are mutually exclusive then: These axioms have been implied by the preceding discussion. Axiom 2: Certainty The probability of an event which is certain to occur is 1.

In the next few chapters we will use subjective probability assessments in our calculations without attempting to evaluate the quality of these judgmental inputs to our analyses.

In Chapters 9 and 10 we will consider the degree to which probability judgments comply with the axioms and have validity as predictions of future events. The axioms of probability theory If you use subjective probabilities to express your degree of belief that events will occur then your thinking must conform to the axioms of probability theory.

In most practical problems the probabilities used will be. In later chapters we will look at methods which are designed to help the decision maker to generate coherent assessments.

Often the receipt of new information. State the approach to probability which you used and any assumptions which you needed to make. Exercises 1 Determine the probability of each of the following events occur- ring. You are therefore urged to attempt the following exercises before reading further. Number of units sold 0 1 2 3 4 5 Probability 0. An analysis of sales records for the last weeks gives the following results: Level of sales no.