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----chapter 1 0--- Decision Analysis a Priori. Prior probabilities or probabilities subjectively established before an actual event.  Decision criteria. A rule used to select from among several alternatives. Decision tree. A drawing used to represent the flow o r alternatives and states . of nature.  Dominant strategy. A superior strategy over all other strategies u nder all states of nature.  Expected value. Weighted average value. Payoff. The result of a specific alt ernative under an assumed state of nature. State of nature. An environmental condition which influences the outcome of an alternative and over which the decision maker does not have control. Previously, we have looked at techniques that fit specific decision situations such as resource allocation. We will now journey into the realm of decision analysis that is both a decision technique and a decision-making philosophy. Unlike most other methods, decision analysis as a philosophy can be formulated to fit most any decision situation, although the technique contains more specific attributes. Decision analysis may be the most intuitive of all the methods presented in this  book. Indeed, decision analysis can closely follow intuitive thought processes. This chapter evaluates decisions that may be similar to ones you have made recently. Determine the process that you went through to reach your decision. You will undoubtedly come to the startling conclusion that you have been using intuitive decision analysis for years. Chapter 10 formalizes this decision-making process.

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8/13/2019 Dunn and Ramsing

----chapter 1 0---

Decision Analysis

a Priori. Prior probabilities or probabilities subjectively established before an actualevent.

Decision criteria. A rule used to select from among several alternatives. Decisiontree. A drawing used to represent the flow or alternatives and states . of nature.

Dominant strategy. A superior strategy over all other strategies under allstates of nature.

Expected value. Weighted average value.Payoff. The result of a specific alternative under an assumed state of nature. State ofnature. An environmental condition which influences the outcome of

an alternative and over which the decision maker does not have control.

Previously, we have looked at techniques that fit specific decision situations such

as resource allocation. We will now journey into the realm of decision analysis thatis both a decision technique and a decision-making philosophy. Unlike most othermethods, decision analysis as a philosophy can be formulated to fit most anydecision situation, although the technique contains more specific attributes.

Decision analysis may be the most intuitive of all the methods presented in this book. Indeed, decision analysis can closely follow intuitive thought processes. Thischapter evaluates decisions that may be similar to ones you have made recently.Determine the process that you went through to reach your decision. You willundoubtedly come to the startling conclusion that you have been using intuitivedecision analysis for years. Chapter 10 formalizes this decision-making process.

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When you conclude this chapter you should begin to synthesize your philosophyof decision making, for decision analysis can become the core of formalized

decision making.In

fact, virtually every other technique can use decision analysis asits coordinating theme.Decision analysis requires you to evaluate probability distributions; thus it is a

good area in which to consolidate your concepts of probability, particularlysubjective probabilities. Identify the unique circumstance of this problem based onan actual business situation.

The Case: Selki rk WinterRecreation Complex ___________________________ _

The Recreational Lands Division of Dramun Forests had recently acquired someland in the mountains of the western part of the country. The land proved ideal forskiing and other winter activities and had some potential for summer activities aswell. It was located close to western metropolitan centers and was readilyaccessible to eastern markets. The land was not presently in commercial timberproduction. The firm decided to use the property for a recreational development.

Mr. Dramun had decreed that the Selkirk project. the name of the proposedrecreational complex. wou Id be completed in several phases; the first phasewould involve some form ot a residential housing. Ms. Williams explained. "Wehave three options open to us. We can build a condominium complex. a rentalcomplex. or a combination rental-condominium complex. Our task is to determinewhich approach would likely lead to the highest net present value of cash flowfor the project over the next ten years." Mr. Dramun had indicated that he wouldaccept only one of these options and that he wished to select that option that hadthe highest payoff while allowing a high likelihood of achieving that payoff

The first proposal. the condominium. was a complex of family units that wouldbe sold to customers. The second. the rental option. would provide similar units forrental rather than for sale. The third proposal. a combination unit. included somerental units and some ownership units .

.. We can expect to experience one of three levels of demand." continued Ms.Williams. The first level. favorable for condominium development. the secondfavorable to rental units. and the third favorable to a combination. The likelihood ofeach level of demand was expected to be .3 . .4. and .3. in that order. Thus the firmwas only 30 percent sure that demand would be favorable for condominium units.

Now. if the firm followed option 1 (and built only condominium units) and thedemand was favorable for condominium units. the firm could expect to sell\$22.273,670 of. condominium units. Other revenue projections. expressed in netvalue. are shown in Table 10-1. These figures were forecasted by the MarketResearch Department of Dramun Forests and were based on I ikely consumerreactions to each of the alternatives. Before a decision is to be made. the costs ofgenerating the projected

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Table 10-1. Revenue Options

Revenue If Demand Is Favorable for

Option Taken

CondominiumRentalCombination

Condominium

\$22,273,67012,862,54116,665,030

Rental

\$12,273,67020,097,72016,968,100

Combination

\$16,109,19116,748,10019,511,537

revenues must be considered. Although each option would generate a similarphysical complex. fixed and variable costs would differ. These costs are indicatedin Table 10-2. This table implies that the net cash flow of any option selecteddepends on the costs. the type of demand. and the influence that demand has onvariable costs and revenues. For instance. if you recommend thatthefirm build acondominium (option 1) and demand is favorable for a condominium, the firmwould encounter the following cash flow (in net present value):

Revenues \$22.273.670 (cash inflow)Less:

Fixed costs (13,747.900) (cash outflow)Variable costs

(.161 x \$22,273,670) (3,586,061 ) (cash outflow)

Net cash flow \$4.939.709

The fixed costs. which are encountered over the entire ten-year time horizon of theproject. include construction, financing, overhead, and maintenance costs.Variable costs. in this case, include sales commissions, direct expenses. andsome semivariable costs, such as advertising.

The company president, Mr. Dramun. has stated that he wishes to select thatalternative that will maximize net present value of cash flow while explicitlyevaluating chances of achieving that cash flow. Your task, as a managementscience analyst is to evaluate the Selkirk project and recommend an option to thepresident. Your recommendation must include your rationale and the argumentsfor and against your proposal. Information not already presented need not beconsidered at this time.

Table 10-2. Selkirk Project Costs'

Option

CondominiumRentalCombination

Fixed Costs

\$13,747,90012,508,20012,128,000

Variable Costst

16.1%16.915.7

* All costs are shown in net present value.t Expressed as a percentage of gross revenues.

The Case: Selkirk Winter Recreation Complex 225

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The Probl em Elements

As you examine this problem you should specifically determine:

1. What the president wishes to achieve (i.e., his objective).2. What the data mean.3. What assumptions are made.

Specifying the Problem Objective

Previously, we have observed that the first step in any problem-solving procedureis to determine the objective. In this situation, Mr. Dramun has stated quite clearlythat he is concerned with two specific things: the net present value of cash flow andthe likelihood of achieving that cash flow.

How can we formulate Mr. Dramun's- directions into a clear, precise statement

that can be used as a measure of success? We could simply state, "take that actionthat produces the highest net present value (NPV) of cash flow and that we are likelyto achieve." However, the first part of the statement can conflict with the second.Thus, we would expect our NPV cash flow to decrease as the likelihood of achievingthat cash flow increases. Hence, which of Mr. Dramun's two criteria predominate?

A specific statement of objective can be developed using expected value criterion:"Select that action that maximizes expected net cash flow." We are now very clear onexactly what criterion we will use to evaluate various actions. This statement is thedecision criterion.

Because this decision criterion considerers net present value, another con-sideration arises. Mr. Dramun has identified ten years as the time frame for thisdecision. Any cash flow beyond ten years is not considered relevant and is therefore

ignored. Of course, this greatly simplifies the decision procedure. The decisionmaker must use discretion in selecting the time horizon so that relevant informationis not ignored. In any event, we must recognize that this decision situation has a timehorizon over which the data will be evaluated.

Identifying Possible Actions

A third element of the decision situation involves the various actions that we maytake. Mr. Dramun has identified three possible alternative decisions and hasdisallowed any other alternative at the moment. Our problem is beginning to takeform. All we have to do is state each alternative and consider the expected payoff of

each alternative. The three alternatives are as follows: A1: Construct a condominium complex and sell each unit. A2:Construct a rental complex and lease each unit.

A 3: Construct a combination complex utilizing alternatives one and two.

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Note that these alternatives are mutually exclusive and collectively exhaustive. Inother words, none of the alternatives overlap, so that, if alternative A1 is selected, A2cannot concurrently be selected. Often a status quo, or "donothing," alternative can

be considered, an option that has been precluded in this case by Mr. Dramun.

States of Nature

The fourth and fifth elements of the problem are closely related. Several distinctstates of demand must be evaluated. The demand could be favorable forcondominiums, favorable for rentals, or favorable for combination units. Naturally,the occurrence of any specific state of demand will influence the decision we make.However, we do know, before the decision is made (a priori), which state of demandwill occur. Environmental variables over which we have no control can be classifiedinto mutually exclusive conditions called states of nature.Therefore,

N 1: Demand favors condominiums. N2: Demand favors rental units.N 3: Demand favors a combination of units.

The fifth element of the problem is the payoff. Each outcome of a decision will be

influenced by a state of nature. Every combination of a state of nature and analternative will have a payoff. This is the net figure we can expect to attain if wefollow a specific alternative and a particular state of nature occurs. The data from theSelkirk project identify three alternative courses of action and three states of naturefor each alternative. Thus, if we select alternative Ab the condominium approach,and demand is favorable for condominiums, N b we can expect a net cash flow of\$4,939,709,as explained in the previous section.

Probabilities of States of Nature

One final element remains. Each state of nature has an associated likelihood or probability of occurring. In other words, each state of nature can occur with aspecific likelihood in the form of a probability. The probability that demand isfavorable for condominium units is .3, that is, Pr(N 1) = .3. Thus, we are 30 percentsure that the market favors a condominium development over the other alternatives.This information can be estimated by knowledgeable managers, or it can begenerated by the Market Research Department.

We have now identified the components of the decision problem and the decisioncriterion. This problem reflects all the components necessary to evaluate thesituation using decision analysis.

The Problem Elements227

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Single-Stage Decision Analysis Model ____________ _

The Selkirk project is an example of a single-stage decision analysis problem. It issingle-stage because there is only one level of decision to be made. That is, we needonly to decide which one of the three alternatives (Al> A 2 , or A 3) to select. Ifsubsequent alternatives were generated and were dependent on the outcome of thealternatives given, we would have a multistage problem.

In summary, decision analysis models have the following conditions:

1. A choice must be made between mutually exclusive alternative courses ofaction, Ai'

2. States of nature, N j , exist that are outside the control of the decision maker butthat influence the outcome of any decision.

J. Payoffs, P ij, are associated with each combination of an alternative (A;) and astate of nature (NJ

4. The problem has either an expressed or implicit time horizon over which thedata and decision are relevant.

Decision analysis has several additional components:

5. Each state of nature (N j) has an associated probability of occurrence, Pr(N j).6. The sum of Pr(N j) over all j is equal to one.7. The decision criterion is to select that alternative that maximizes expected payoff.

The relationship between this technique and Bayes law will be discussed in Chapter11.

Solving a Single-State Model-

Payoff Table ApproachThis problem may be solved using either of two approaches-the payoff table

approach or the decision tree approach. Both approaches which generate the sameresults, attack the problem by identifying the expected payoff of each alternative andthen selecting that alternative with the highest expected payoff.

The payoff table approach focuses on the net payoffs or net cash flows for eachaction under each state of nature. Using the mathematical notation developed earlier,we can generate a general table for the Selkirk project.

The task now is to complete Table 10-3 with the problem data. The procedure forcompleting this analysis is shown in the flow chart in Figure 10-1. Refer to Figure10-1 as you read the following discussion.

1. Identify the decision criteria. Mr. Dramun's objectives clearly coincide with theexpected value criteria previously presented. Therefore, select the alternative thatresults in the highest expected net present value of cash flows.

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3. 4.

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2. Identify the allowable decision alternatives (A;). Mr. Dramun has clearlyspecified three allowable alternatives:

AI: Construct a condominium complex and sell each individual unit. A 2:Construct a rental complex and lease each unit.

A3: Construct a combination complex, selling half of the units and leasing half ofthe units.

At this point we must review the alternatives and ask if they are unique and complete. None of the alternatives overlap other alternatives, so we may conclude that they aremutually exclusive. Although we could develop other new alternatives, such as "donothing," Mr. Dramun has ruled out such possibilities. Therefore, it appears that this setof alternatives meets our evaluation criteria as set forth in Figure 10-1.

3. Determine the relevant states of nature (N j).The research staff has reviewed the

various environmental influences of the problem and has consolidated them into three possible states of nature:

N I: Demand favors condominium units. N 2:Demand favors rental units.

N 3: Demand favors combination units.

Notice that we could expand these states to include a multitude of possibilities.Another state, N 4, could read

N 4: 35 percent of the respondents favor a condominium. 20 percent ofthe respondents favor a rental complex. 45 percent of therespondents are neutral.

This information could be generated from a marketing research study. However, Ms.Williams feels that N 1> N 2, and N 3 are sufficient for the initial analysis. In essence,although we may add many different states of nature, we must narrow the number to onlythose few that have the greatest impact on our decision.

4. Determine the probabilities for each N]. The estimates of these probabilities are:

Pr(N 1 } = .3Pr(N 2 } = .4Pr(N 3 } = .3

We must realize that these are best estimates of the true probabilities of each state ofnature. We actually will not know the true underlying probabilities, although we could

purchase additional information about the unknown underlying probabilities. Currently,we will work only with these subjective probabilities.

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Examine, for a moment, the characteristics of these probabilities. Each event ismutually exclusive; thus:

Pr(N l n N 2) = 0Also,

3

L.Pr(N i) = 1i= 1

Therefore, these events represent all the possible states of nature. Finally, each ofthe figures represents simple or marginal probabilities rather than conditional probabilities. This is because we are dealing with a single-stage model. Our probabilities appear to meet the requirements set forth in Figure 10-1.

5. Compute each Pij. Pi} is the net present value of pursuing alternative i and thestate of nature j occurs. P u (which has already been computed) is equal to\$4,939,709. Some of the remaining payoffs are as follows:

Pij = Revenue., - fixed cost, - variable costs.,Pu = \$22,273,670 - \$13,747,900 - \$3,586,061 = \$4,939,709P12 = \$12,273,670 - \$13,747,900 - (\$12,273,670x 0.161) = -\$3,450,291P13 =\$16,109,191 - \$13,747,900 - (\$16,109,191 x 0.161) = -\$232,289

Each of the remaining Pi} is calculated in a like manner.Armed with this information, a payoff table can be constructed. Construct your

own payoff table and compare it with Table 10-4 (using the generalized model fromTable 10-3). If your figures are different, review the discussion in this section andreconstruct your table before continuing. If your figures are close, you are ready forstep 6.

6. Calculate the expected payoff for each alternative. The objective of this step isto complete our calculations so that we may recommend a strategy or action. Theanalysis can be shortened if a dominant strategy exists. In other words, does onestrategy have the highest payoff under all states of nature? If so, it would be a

dominant strategy and would always be preferred to the other alternatives.Evaluating the Selkirk project we see that, if N 1 occurs, Al

Table 10-4. Initial Payoff Table for Selkirk Project

State of Nature

Nt N2 N3Favor Condominium Favor Rental Neutral

Action Pr{Nt) = .3 Pr(N z } = .4 Pr{N 3 } = .3

At: Condominium \$4,939,709 - \$3,450,291 -\$232,289

A2 : Rental - \$1,819,428 \$4,193,005 \$1,409,471 A3 Combination \$1,920,620 \$2,176,108 \$4,320,226

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Table 10-5. Final Payoff Table for Selkirk Project

State of Nature NI N2 N3

Action Pr(Nt) =.3 Pr(N 2) = .4 Pr(N 3)= .3 EV(Ai)

Al \$4,939,709 - \$3,450,291 -\$232,289 \$32,110

A2 -\$1,819,428 \$4,193,005 \$1,409,471 \$1,554,215 A3 \$1,920,620 \$2,176,108 \$4,320,226 \$2,742,697

*Best choice.

Choice

*

would be the best strategy, whereas A2 would be best, if N 2 were the true state ofnature. It should be clear that we do not have a dominant strategy, so we mustcalculate the expected payoff of each alternative.

The expected payoff of the ith alternative is computed by:m

EV(A;)= LPJPr(N j)] j=i

This, of course, calculates the expected value, EV, or the weighted average payoff of Ai' where m is the number of the states of nature. The expected value of Ai is:

3

EV(Ai)= LPiJPr{N j)]i= 1

= (\$4,939,709)(.3) + (-\$3,450,291)(.4) + (-\$232,289)(3) =

\$32,109.60 or \$32,110

The EV{A2)and EV(A3)are shown in Table 10-5.

7. Select an alternative. Returning to step 1, we find that our decision criterion is toselect that alternative with the highest expect value. Alternative 3, build acombination unit, has the highest expected value at \$2,742,697. Therefore, this is thealternative that we should recommend to Mr. Dramun.

8. Interpret the results. Alternative 3, as we have seen, has an expected value of\$2,742,697. Yet, if we select alternative 3, Table 10-5 shows that we will get a payoffof \$1,920,620, \$2,176,108, or \$4,320,226. What then is the significance of theexpected value?

The expected value has several uses. Foremost, it provides a net figure that weighsthe payoffs that we can achieve under an alternative against the probabilities of

achieving each payoff. This is the only single figure that can include the impact ofboth these elements. Essentially, ifwe were to repeat this decision many times, wecould expect to average \$2,742,697 in the long run by sticking to alternative 3.Although the Selkirk project is a one-time decision, rather than a repeat decision, theexpected value calculation explicitly considers the probabilities of a future one-timeevent.

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This calculation also provides a relative comparison of the various alternatives. Aspecial advantage is that each alternative is implicitly compared on two dimensions:that of payoffs and that of probabilities of payoffs. It should be noted that, if thedecision maker is concerned about assuring some minimum payoff, or some otherobjective, a different decision criterion may be more appropriate.

Single-Stage Model-Decision Tree Approach

Decision trees provide a more graphic method for solving this problem.Decision trees represent diagrams of the alternatives, states of nature, probabilities,and partial cash flows in the problem. The decision tree approach, although

providing the same results as the payoff table approach, allows us to examine partialcash flows along the decision chain.

A generalized, single-stage decision tree is shown in Figure 10-2. Theterminals-boxes and circles-represent nodes in the decision tree. The boxnodeI-represents a decision node, whereas circles-nodes 2, 3, and 4-are event nodes. The

path Al N 1 signifies the action of selecting alternative 1 and then having state ofnature 1 occur. The net cash flow along this path is identical to P 11 in Table 10-3using the payoff table approach. Nodes should always be numbered intop-to-bottom, left-to-right order.

Rll

Figure 10-2. Generalized single-stage decision tree.

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5. The following meanings are associated with each variable from Figure 10-2:

Ai = the ith alternative

N, = the jth state of natureRij = the gross revenue resulting from following alternative Aiand state

of nature N, occursVCij = the variable costs associated with Ai and N j FC i = the fixed costs associated with Ai

Our procedure follows the following stages in addition to steps 1 through 4 ofFigure 10-1:

Step 5: Draw the decision tree.Step 6: Determine the R;/s for each alternative state of nature combination.

Step 7: Determine the partial cash flows along each branch of the tree. Step 8:Calculate the expected value for the event nodes by assessing the variable costs,

probabilities, revenues, and fixed costs.Step 9: Eliminate all but the highest-valued event node.

Step 10: Carry forward the expected value of the best event node to the decisionnode.

G ross revenues

\$22,273,670

\$16,109,191

\$12,862,541

\$16,748,100

\$16,665,030

\$17,511,537

Figure 10-3. Partial decision tree-Selkirk project.

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6.

Figure 10-3 illustrates partial completion of the Selkirk project decision treethrough step 7. The revenues (step 6) were entered directly from Table 10-1. Thevariable costs and fixed costs (step 7) can be calculated from Table 10-2 as shown.The variable costs along Ai N i can be determined from Table 10-2:

VC 11 = Rl1 x .161= \$22,273,670x .161= \$3,586,061

The remaining variable costs can be similarly calculated.The expected values for each of the event nodes must be calculated next using the

following relationship (step 8).

EV 2 = [(\$22,273,670 - \$3,586,061)x .3]+ [(\$12,273,670 - \$1,976,061)x .4]+ [(\$16,109,191 - S2,593,580)x .3] - \$13,747,900=

\$32,109.60or \$32,110

Expected values for the remaining event nodes are shown in Figure 10-4. Notice that we are evaluating nodes 2, 3, and 4 as if we were at node 1, theimmediately preceding decision node. Therefore, we must consider all fixed cashflows flowing immediately into each event node from node 1.

EV= \$1,554,215

Gross revenue

\$22,273.670

\$12,273,670

\$16,109,191

\$12,862,541

\$20,097,720

\$16,748,100

\$16,655,030

N2 \$16,968,100

\$17,511,537

Figure 10-4. Final decision tree-Selkirk project.

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The cost structure shown in Figures 10-3 and 10-4 can be confusing. All the costsstemming from the event nodes in Figure 10-3 are variable costs. These costs aresubject only to states of nature, because the magnitude of the variable .costs is afunction of volume of activity. However, the fixed costs leading into each eventnode are a function of the decision associated with the immediately precedingdecision node. Obviously the fixed costs of A 1 will be incurred (if we select thatoption) regardless of which state of nature results.

We can readily observe that node 4 has the highest expected value, so we select A3as our alternative and carry forward the value from node 4 to node 1 (step 10). This

becomes the expected value for our intended action. We also put double lines acrossunacceptable alternatives (step 9).

The decision tree approach is very valuable for analyzing multistage problems inwhich payoff tables would be difficult to construct. This approach also provides avisual reference for examining the partial cash flows in the problem.

Mult is tage Decisi on Analys is Model _______________ _

Most problems are not as simple as the Selkirk project, so the tools must beexpanded. By sequencing decision and event nodes, decision trees can be applied tomore complex problems. This section expands the Selkirk problem to exhibit theflexibility of decision trees and decision analysis.

The Case Expanded

Upon reporting the results of your analysis to the company, you have been toldthat another analysis is required. Mr. Dramun wishes to know the effect of pricing

policy on net cash flow."Mr. Dramun has advised us," Ms. Williams stated, "that the condominium or

sales approach would necessitate more specific evaluation of the pricing strategy.We can penetrate by pricing low or we can skim the cream by pricing high. In eithercase, we must consider the effect of competitive reaction as there is already a majorcompetitive resort complex in the area." This consideration will have no impact onrental alternatives.

The Marketing Research Department estimated the variable costs and the grossrevenues for the multistage problem. These figures are shown in Table 10-6. Notethat figures for alternative two (A2) will not change from those in Tables10-1 and 10-2 andare consequently not shown inTable 10-6.

This problem illustrates a multistaged decision model. The decision maker'simmediate problem centers solely on whether to build a rental unit, a condominiumunit, or a combination of the two. However, the decision maker can consider theanticipated effect that later decisions may have on the net cash flow. Bear in mindthat our immediate problem is still whether to select Ab A 2 , or A 3•

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Table 10-6. Conditional Gross Payoff for Selkirk Project*

Estimated

Construction Demand Pricing Variable Competitor Conditional

Alternative State Alternative Costs Reaction Gross Revenue

Ai Nl Sl \$3,925,494 C1 \$20,131,887C2 27,960,954

S2 4,141,577 C1 26,842,516C2 21,250,325

N2 Sl 2,153,907 C1 11,046,303C2 15,342,088S2 2,272,470 C1 14,728,407

C2 11,659,987 N3 Sl 2,827,002 C1 14,498,272

C2 20,136,498S2 2,982,617 C1 19,331,029

C2 15,303,731 A3 Nl Sl 2,851,887 C1 15,831,779

C2 18,748,159S2 3,008,872 C1 18,381,533

C2 16,248,404 N2 Sl 2,903,957 C1 15,381,533

C2 19,089,113

S2 3,063,591 C1 18,664,910C2 16,543,898

N3 Sl 2,996,737 C1 16,635,960C2 19,700,479

S2 3,161,708 C 1 19,262,691C2 17,073,749

* A1 : Condominium; A3: combination N 1 : Favor condominium; N 2 : favor rental; N3: favor neither S 1:Penetrate; S 2: skim -the-creamC 1 : Competitor follows price; C 2 : competitor does not follow price

A decision tree may be drawn to reflect the new problem based on the general treein Figure 10-5. The following steps may be used to draw a decision tree:

1. Identify the decision to be made (Ai and S K)and the states of nature affectingthe outcome of each decision (N j and Cd.

2. Build the structure of the decision tree by identifying the most immediatedecision to be made followed by those states of nature that directly affect thosedecisions. The decision maker must decide initially from among Ab A z , and A3in Figure 10-5. States of nature N 10 N z,and N 3 affect A 10 A z , and A3• This isfollowed by the next most immediate level of decisions (Sland Sz)followed bythe conditional states of nature directly affecting Sl and Sz (i.e., C 1 and C z).This process is continued until the last decision node series is completed.

3. Number of decision and event (states of nature) nodes for subsequentidentification and reference.

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7.8.9.

4. Determine the revenues at the end of each path of the decision tree (P's)and thenet cash expenses along each stage of the tree (E's).

5. Include the simple probabilities and the conditional probabilities along eachevent branch.

6. Calculate the expected value for the highest-numbered event terminals.Eliminate the least optimal alternatives and carry the best values forward to thedecision nodes. Continue this process until a value is computed for decisionnode 1.

This procedure completes the mechanical process of developing a multistagedecision tree. The multistage problem follows the same steps as the single-stage

problem with the addition of conditional event nodes and conditional decisionnodes. Small variations in the computation procedure are used

Immediatedecision

Simple or first-levelstates of nature

Subsequentdecisions

Conditional Gross payoffsstates of

nature

Rl111

R'112

R'121

R 1212

R'22'

8 '

222

R2ll1

A2,.----,

R 2m

R2121

(E2)~

~~ ~N\'

\ \

I'l 2

R2122

R22ll

R2222

Figure 10-5. General multistage decision tree.

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10.

by different authors, but these are compatible with our discussion and will result inthe same solution to any specific decision problem.

The Selkirk project redefined problem fits the decision tree approach by merelyexpanding the alternatives available and the states of nature. As we begin to developthe tree, we must identify which sets of decision alternatives must be selected first.One set deals with the type of product to be offered (A1, A2 , and A3)' whereas anotherset is concerned with pricing strategy for the products (51 and 52)' The productdecision must be made before the pricing strategy decision because price isdependent on the product. Hence, A 1, A 2, and A3 are the first branches of thedecision tree. The next stage of branches must be those states of nature thatimmediately affect A 1> A 2, and A 3 • Market reaction (N 1> N 2, and N 3) appears to bethe most relevant state of nature influencing Ai> A2 , and A3 . The balance of the treeencompasses successive development of conditional actions followed by relevantstates of nature.

The tree shown in Figure 10-6 discloses the relevant cash flow and probabilityinformation for the Selkirk project. The payoff on the extreme right-hand side of thetree represents gross payoffs excluding costs. Thus, they are ac-

c,

Pr ~.9 \$20,131,889

,_ Pr = ,1 \$27,961,954- \$26,842,516

;.t- \$21,250,32~ ~ \$11,046,303

- < Pr=.l \$15,342,088

\$14,728.407'r- -2 Pr=,6 \$11,659,987

\$14,498,272- -2 Pr = .1 \$20,136.498

- \$19,331,029

\$15,303,731

\$12,862,541

• s,(\$2,173,769)

\$20,097,720

\$16,748,100\$15,831,779

Pr~ ,8

\$18,748,159\$18,381,533\$16,248.404\$15,381,533

~ Pr::"':_2 \$19,089,113

\$18,664,910Pr= .7

Pr ~ ,8 -- \$16,543,898 \$16,635,960

\$19,700,479 Pr =.7

\$19,262,691

, ,

Figure 10-6. Cash flow decision tree-Selkirk project.

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tually gross revenues or net present value of cash inflow. Cash outflows are shownalong the paths of the tree. Examine the top branch of the tree. This branch discloses

the cash flows if we build a condominium (A1) and the demand favorscondominiums (N 1); then we decide to follow a penetration pricing approach (S 1)and out competitors follow our pricing actions (C 1)· Ifthis combination of decisionsand events occurs, then we would expect cash revenues of \$20,131,887 and cashoutflows of \$13,747,900 (branch AI) and \$3,925,494 (branch St). Thus, the net cashflow would be \$2,458,493, which is the balance of all the cash flows along that path.

Several matters are worth considering when calculating and evaluating thedecision tree in Figure 10-6. Although the tree discloses all relevant outcomes, notall the calculations need be made. We are solving this tree using backwardinduction. That is, we start first with the gross revenues and then work toward node1 while incrementally eliminating lesser-valued alternatives. Therefore, onlycalculations from successful alternatives are evaluated further.

Some time and effort can be saved by expressing costs at the lowestnumberednode possible. Alternative 1 would require a cash outlay of \$13,747,000. Thisamount could be subtracted from each revenue figure following nodes 11 through15. But why go to this extra effort? It can be treated as a constant and subtractedfrom node 2 net value.

Now that the tree has been drawn and calculated in Figure 10-6 for the Selkirk project, it is time to consider what we have. Node 11 can be considered first. If wereach node 11 by having selected Shall previous branches are considered to becertain. That is, node 11 is a conditional event node, conditioned on our getting tothat node in the first place. Ask yourself exactly what it means to be at node 11.

If you think about it carefully, you will conclude that this is the position that wehave reached after we have decided to select A1 , then N 1 occurs, and then we selectS 1. Therefore, we need only to consider the costs of S 1 and the revenues subsequentto node 11. The other figures will be considered later. Of course, we have not yetmade these decisions. We are merely supposing what would happen if we had madethem. Because node 11 is an event node (represented by a circle), we must computeits expected value. This will give us a figure that we can use to evaluate thesoundness of selecting S I when we are at node 5, the immediately precedingdecision node.The following formula may be used in this instance: 2

EV ll = LRi(R;) - rc..i= 1

= (\$20,131,887 x .9) + (\$27,961,954 x .1) - \$3,925,494 =

\$16,989,399.70 or approximately \$16,989,400.

Thus, the expected value of any event node is the weighted average of all netrevenues immediately following that node less all costs immediately preceding thatn ode. This recognizes that S 1> C 1> and C 2 are closely related

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11.

EV= \$19,345,624

E V = \$9,321,975

EV= \$10,614,885

EV= \$12,235,093

EV = \$13,932,033

EV = \$13,563,168

EV = \$13,879,471

EV = \$13,219,092

EV= \$14,116,611

EV= \$14,252,127Figure 10-7.

Expected value tree-Selkirk project,

and must be evaluated jointly just as S 2 must also be evaluated with eland c 2' Thelargest value between node 11 and node 12 is carried forward to node 5, and the

branch leading to the lower valued node (S 1) is eliminated from further evaluation.The evaluated values for each of the branches is shown in Figure 10-7.

Reviewing this figure we see that the expected value for node 11 is \$16,989,399 (as previously calculated). whereas node 12 has an expected value of \$19,345,624(calculated in a like manner). Therefore, if we were already at node S we wouldselect S 2 as it has a higher expected value than S 10which leads to node 11.

Interpreting the Results

Now that you have been exposed to the mechanics and philosophy of decisionanalysis, you should be able to reach a conclusion and present a recommendation.Once again, look critically at Figures 10-6 and 10-7 and

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review the request made by Mr. Dramun on page 225. It is clear that A3 produces thehighest expected payoff; therefore, we would recommend building a combination

complex and our expected payoff is \$1,942,903. But what does this mean?If the decision were to be made continuously, A 3 would average the highest

payoff. Furthermore, we would always select S 2 because S 2 always has a higherconditional payoff than S 1. This appears to be consistent with the consumer's price

perception of a quality resort complex.Perhaps this tree suggests some other alternatives. It appears that some alternative

between all condominiums and all rentals might yield the highest net payoff, yet wedo not know if a better alternative exists. This technique does not give us anoptimum answer. It only compares alternatives that have already been identified. Asearch could be employed by developing two new alternatives, A4and As:

A4: Provide 60% condominium units and 40% rental units. A5:

Provide 40% condominium units and 60% rental units.

The expected values of these alternatives could be compared with A 3 , the highest being selected. Incrementally bracketing the current "best" alternative wouldeventually lead to an improved decision that would be close to the optimum.

Addi ti onal Cons iderat io ns _____________________ _

The astute reader will undoubtedly ask, "When could decision analysis be usedfor decision making?" The best answer to this is to recognize the structure of thetechnique as outlined on page 237. Develop and specify the problem first,recognizing that application of several techniques may be appropriate. If, afterspecifying the problem, the following conditions exist, decision analysis would bean appropriate model to use:

1. The problem has a clearly defined objective and a criterion for evaluation.2. More than one significant alternative courses of action may be followed.3. Different conditions of the environment will influence the success or failure of

the various courses of action.

Unique forms of this technique may be encountered. The states of nature,themselves, may be proportions such as the proportion of the market favoringcondominiums or the proportion of a production lot that is defective. Another formof decision analysis is the competitive bidding model in which we wish to identifythe amount to bid on a contract that will balance potential profitability with thelikelihood of winning the contract.

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Postdecision Analysis

probabilities of the states of nature. If we decide to purchase additional information,we must then ask, "How do we incorporate this additional information into theexisting information?" Both these questions will be analyzed in Chapter 11.

Sur nr nar y __________________________________________This chapter has dealt with one of the more logical decision techniques.

Decision analysis fits many problem situations, but it is most powerful when the problem consists of discrete alternatives subject to various states of nature leading todefinite payoffs. That alternative giving the largest expected value is selected as

being the "best" decision.The technique is useful for comparing various alternatives in a logical, methodical

manner. The reader can see that the method has other powerful uses that have yet to be explored. These issues will be developed in Chapter 11.

Chapter 1 0 Problems

1. Precisely what is the meaning of expected value? Because expected value refersmainly to average values, of. what value is the concept to management?

2. Distinguish between relative frequency probability and subjective pro bability .

3. What, in essence, is a decision tree?4. Compare and contrast weighted average to expected value.S. What are the primary limitations of using decision analysis for decision

making?6. Would a professional gambler likely use decision analysis concepts to assess the

profitability of bets? Why or why not?7. Would a professional manager likely use decision analysis concepts for decision

making? Why or why not?8. Distinguish between the significance of single-stage versus multistage decision

models.9. What is the expected value of A where:

A = .2{xt}+ .8(X2)and

Xl = \$100X2 = \$700

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10. You are considering the purchase of an eating facility. Assume that the facilitycould end up as one of four types of restaurants, depending upon demand. The

likelihood of each facility is:

Greasy spoon: Pr = .2Hamburger joint: Pr = .3Diner: Pr = .4Classy casino: Pr = .1

The first-year payoff at each facility is:

Greasy spoon = \$200,000Hamburger joint = break-evenDiner = \$1,500

Classy casino = \$2,500,000

What is the minimum investment you would be willing to risk on the venturegiven each possible outcome?

11. Define decision analysis.12. Did Mr. Dramun make the correct decision with regard to the Selkirk project?

Why and why not?13. Describe the exact meaning of "maximize expected net cash flow."14. What conditions must exist before decision analysis is an applicable decision

tool?15. Evaluate Figure 10-1. Is this system complete or are additional elements

necessary?

16. Consider the following payoff table:

If

\$100 \$900850 250-20 -20

Pr(Nt) =.1

Pr(N 2) = .6Pr(N 3) = .3

\$600 50700

what action should be taken?17. Johnny Q, obviously a cover name, handles minor bets for a local dog track. On

one particular Saturday, Johnny Q was covering four races, one of which was of particular interest. The last race, which intrigued you, involved the followingdogs:

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a. If you bet \$1,000, what would be your expected payoff if you placed your bet on:

1. Texas Tomboy2. M. B. Allison3. It

b. Who do you expect to win the race? Why?c .. What odds would you expect before you would place a bet on Never Again?

18. The local trucking union is considering a strike right in the midst of your peak production period. You have four months remaining in your fiscal year. If astrike will result in a loss of \$1,000,000 and no strike will result in a profit of\$276,000 during the balance of the fiscal year, at what probability are youindifferent between a strike and a settlement before a strike?

19. Prices have been of particular interest to you in recent months. The federalgovernment has threatened action if your average prices exceed 7.5 percent.However, some finesse is possible. If you increase prices by 6 percent, thelikelihood of government action is .05, whereas the likelihood of actionincreases to .6 if prices are increased by 8 percent. A 9 percent increase brings a90 percent chance of action. If each one percentage point of price increase bringsan increased profit of \$150,000 while government action brings a fine of\$300,000, what pricing action should be taken? Use a decision tree in youranalysis.

20. A television manufacturer is going to introduce a new portable TV that operateson batteries. The president of the firm is considering two advertising campaigns,one of which he will utilize for the product introduction. The campaign costs foreach program along with the anticipated results are as follows:

Campaign

EconomyExtravagant

Cost

\$1,750,0002,300,000

Increase UnitSales by

.12

.21

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Without a campaign the following first-year results are expected:

Unit Sales

o10,00020,00030,00040,00050,000

Likelihood ofUnit Sales

.10

.15

.25

.25

.20

.05

The unit contribution to profit and overhead is \$110. Using decision treeanalysis, decide what action to take.

21. The president ofthe company in problem 20 is also considering a product

refinement that would cost \$25,000,000, 30 percent of which would have to berecovered the first year. However, unit sales could be expected to be increased