1.1: Introduction to Quantitative Analysis

This is a course about the use of quantitative methods to assist in decision making. The subject matter makes up the discipline known as decision sciences, or you might hear it called management science or operations research.

We will be covering a number of descriptive and prescriptive mathematical models that have proven useful to managers, generally since World War II, although some of the models date back to the early 1900's. Mathematical models are simply representations of reality that provide a framework for a scientific approach to the study of managerial problems.

Models also help us gain insight into relationships such as the relationship that exists between an objective and a constraint. For example, we will be using linear programming prescriptive models to represent the relationship between a profit maximization objective and one or more limited resource constraints. Suppose a company manufactures two products, X and Y. Product X has a profit contribution of $100 and Y $300. The mathematical expression for maximizing profit is simply:

Maximize Profit = 100 X + 300 Y

What should this company do to maximize profit? That's right, make an infinite number of Y's! But the real world presents constraints to every operation. Suppose that it takes 4 hours of labor to assemble an X, and 20 hours of labor to assemble each Y. If there are 140 hours of labor available in the next production period, we can expand the problem as follows:

Maximize Profit = 100 X + 300 Y

Subject to Labor Constraint: 4 X + 20 Y < 140

The labor constraint equation uses numbers to represent the technological rate of resource consumption of the two products (the coefficients of 4 and 20), as well as a less than or equal sign to put an upper limit on the 140 hour labor constraint. If we wanted to continue our strategy of making all Y's and no X's, we see that we could make 7 Y's by using all of the 140 hours of labor.

Let X = 0, then

4 (0) + 20 Y = 140 gives

20 Y = 140 or

Y = 140/20 = 7

Solving for Profit, we get:

Profit = 100 (0) + 300 (7) = $2,100

But there is another alternative worth examining. We could make all X's and no Y's. Since X's require much less labor, we can make 35 of them, at a profit of $3,500 - the better alternative. To be sure we have the best or optimal solution that maximizes our profit criteria, we would need the mathematical model to check the profit contribution of all other product mix combinations.

Obviously, this is a small problem that can be solved by observation. Imagine what the linear program problem formulation looks like for an auto manufacturer which much select between perhaps 10 models of one make of car subject to a multitude of labor, equipment, and market constraints. For these large problems we need a more formal problem solving process.

The scientific approach to the study of managerial problems incorporates a problem solving process. The text presents a seven-step process:

1. Identify and define the problem

2. Determine the set of alternative solutions

3. Determine the criterion or criteria that will be used to evaluate the alternatives.

4. Evaluate the alternatives (using an appropriate quantitative method or model).

5. Choose an alternative

6. Implement the selected alternative (the decision)

7. Evaluate the results and determine if a satisfactory solution has been obtained.

Decision-making is concerned with the first five steps and will be our focus through this course. The first three steps are concerned with structuring the problem, and the next two with analysis. When the manager adds qualitative considerations to the selection process, and follows through with implementation and evaluation, the problem solving process is complete.

We will start off the course with descriptive models that describe outcomes for selected alternatives, and in that manner assist the decision-maker in selecting between alternatives. These models include decision analysis, forecasting, project management, and queuing or waiting line models.

For example, I hope you will enjoy the module on queuing or waiting lines simply because most of us probably have "war stories" about waiting in lines some where at some time. Did you know that there are descriptive models that predict how many customers will be standing in a given line, and for how long, given that we know the customer arrival rate and system service rate inputs? In this module, we will introduce models that will prove to you why banks and Disney utilize a single line/multiple server configuration to minimize waiting time in line for their customers.

I hope you will also enjoy the forecasting module where simple mathematical techniques are used to discover patterns such as trend and seasonality in our data. In the forecasting module, we drive home how important it is to measure and report the reliability of a forecast - almost as important as the forecast itself.

The project management material helps us learn about the importance of focusing on critical activities, and on the concept of slack resources. The decision analysis material we start with helps us appreciate the value that can be placed on information as we study a model designed to help manager select between decisions when faced with several state of nature scenarios.

The first prescriptive or optimization model that we will examine will be the classic economic order quantity inventory model that has been used by managers to determine how much to order and when for the past 80 years. The model is prescriptive in the sense that we provide inputs such as demand, the cost to carry and the cost to order inventory, and the model selects the order quantity alternative that minimizes total incremental inventory cost.

We wrap up the course with linear and integer programming optimization models that select alternatives which provide the best value for the problem criteria subject to structural and environmental constraints. An example of an optimization problem was presented at the beginning of this section. Optimization models are very powerful and are used to assist decision-makers in capital budgeting, project selection, resource allocation, and scheduling and transportation problems.

Before we get started with our first module, Decision Analysis, you may want to load the CD-ROM that comes with "The Management Scientist." I took a break to load my CD-ROM on my laptop at this point.

1. First, I broke open the seal and took out the CD-ROM (meaning I just bought the book!) and placed it in the CD Drive in my personal computer.

Warning: As I proceeded with the installation steps below, I was warned to close any open applications before proceeding with the installation. Since I had Word open, I stopped the installation, closed Word, then continued with Step 2 below.

2. Next, following the installation instructions on page 2 of The Management Scientist, I clicked the Start button in the Windows Task Bar and chose Run.

3. On the Command line, I typed D:\Setup (after noting to change "D" if my CD-ROM drive has a different letter designation).

4. I then clicked the OK button (or press the Enter key).

5. I then followed the prompts of the installation.

You may want to open the program now, or wait till we get into an application in the next section of these notes. Instructions for running "The Management Scientist" program are also on page 2 of the software manual. Go to Programs from the Start button; go to "The Management Scientist" program and click on The Management Scientist or its logo. Click Continue on the opening screen and you are ready to select one of the twelve applications.

Let's now get started with our first subject, Decision Analysis.


1.2: Structuring the Decision Analysis Problem

Decision analysis, our first quantitative method in this course, is concerned with selecting an option or alternative course of action (the decision) given prior knowledge of its outcome (called a payoff) for various future scenarios (called states of nature or events). The decision-maker has control over the process of selecting an alternative course of action, but not over the states of nature, at least not in the short run. Let's illustrate these terms with an example.

Suppose a manufacturer has three alternative courses of action for the next production period. They can make their product (we will use the symbol d₁ to represent the first decision alternative), buy their product from another manufacturer and sell it to their customers (d₂), or they can do nothing in the next production period (d₃). Suppose further that the manufacture has a simple forecast for the next productions period: demand will be low (state of nature symbol s₁) or high (s₂). The final input for the structure of the decision analysis problem is the outcome or payoff resulting from each state of nature/decision combination.

A convenient structure for displaying the decision alternatives, states of nature and payoffs is a payoff table.

Table 1.2.1

	States of	Nature
Decision Alternatives	Low Demand, s1	High Demand, s2
Make Product, d1	($ 20,000)	$ 90,000
Buy Product, d2	$ 10, 000	$ 70,000
Do Nothing, d3	$ 5,000	$ 5,000

The payoff table shows, for example, that making the product leads to a profit of $90,000 should the demand turn out to be high, or a loss of $ 20,000 if demand is low. I will follow the convention in the text by giving the symbol V_ij to represent the payoff associated with decision alternative i and state of nature j.

To continue with the table, buying the product shows that the manufacturer can avoid a loss, compared to making the product, if the demand turns out to be low since the manufacturer avoids paying fixed production costs. If the demand is high, the manufacturer makes a profit, but not as much as if they made the product since they miss the production economies of scale. If the manufacturer does nothing, a small profit is earned from selling existing inventory that just meets a low level of demand (both V₃₁ and V₃₂ equal $5,000).

Another way to display the structure of the decision problem is with a decision tree.

Figure 1.2.1

				Payoffs
		-	----- s1: Low Demand	($ 20,000)
		|
-	--- d1: Make -------->	B
|		|
|		-	----- s2: High Demand	$ 90,000
|
|		-	----- s1: Low Demand	$ 10,000
|		|
A	--- d2: Buy ---------->	C
|		|
|		-	----- s2: High Demand	$ 70,000
|
|		-	----- s1: Low Demand	$ 5,000
|		|
-	--- d3: Do Nothing -->	D
		|
		-	----- s2: High Demand	$ 5,000

In this decision tree, the cell labeled A denotes a decision node, followed by its three decision branches. The cells labeled B, C and D denote state of nature nodes which are each followed by the two state of nature branches. Payoffs are shown at the terminal end of each state of nature branch.

Structuring the decision problem, as in formulation of all quantitative models, establishes the first major benefit of these formal methods of decision making. That is, the decision-maker is required to formally consider many of the important aspects of a decision problem during the problem-structuring phase. These aspects may be overlooked when the decision-maker uses "gut feel" or some other purely qualitative technique to make decisions.

After the problem is structured, we can now turn to its analysis: selecting one of the possible decision alternatives according to predetermine selection criteria. There are two major approaches in decision analysis that depend on the availability of information on the states of nature. One approach is called decision making without probabilities, and the other, decision making with probabilities.


1.3: Decision Making without Probabilities

In this approach, the decision-maker has no information concerning the relative likelihood of each of the states of nature. It is sometimes called "decision making under risk." There are three classic criteria used in decision-making strategies without probabilities, as described in the next three subsections.

Optimistic Criterion
In this strategy, the decision-maker evaluates each decision alternative in terms of the best payoff that can occur. The strategy is best illustrated in the payoff table repeated below.

Table 1.3.1

	States of	Nature	Maximum
Decision Alternatives	Low Demand, s1	High Demand, s2	Payoff
Make Product, d1	($ 20,000)	$ 90,000	$ 90,000
Buy Product, d2	$ 10, 000	$ 70,000	$ 70,000
Do Nothing, d3	$ 5,000	$ 5,000	$ 5,000

Note that an extra column is added to record the maximum payoff for each decision alternative (maximum payoff in each row of the table). The decision-maker employing the optimistic criterion then selects the decision alternative associated with the maximum of the maximum payoffs. Since $90,000 is the maximum of the maximum payoffs, d₁ is the selected decision. This strategy is sometimes called the maximax strategy. The "eternal optimist" would be one using this approach - it captures the philosophy of decision-makers that accept risk of large losses to make substantial gains.

Conservative Criterion
In this strategy, the decision-maker evaluates each decision alternative in terms of the worst payoff that can occur. The payoff table is repeated here with a new column to record the minimum payoffs for each decision alternative (minimum payoff for each row in the table).

Table 1.3.2.

 

	States of	Nature	Minimum
Decision Alternatives	Low Demand, s1	High Demand, s2	Payoff
Make Product, d1	($ 20,000)	$ 90,000	($ 20,000)
Buy Product, d2	$ 10, 000	$ 70,000	$ 10,000
Do Nothing, d3	$ 5,000	$ 5,000	$ 5,000

This strategy, which is sometimes called the maximin strategy, then selects that decision alternative associated with the maximum of the minimum payoffs. In this situation, the decision-maker would select d₂, buy the product, since $10,000 is the maximum of the minimum payoffs. Some believe this strategy is associated with "eternal pessimists", but to be fair, it is actually a conservative strategy used by decision-makers who seek to avoid large losses.

Compromise Minimax Regret Strategy
The third classic strategy for decision making under risk is called the minimax regret strategy, and is neither purely optimistic nor conservative. This approach first starts by converting the payoff table into a regret table. The regret table looks at each state of nature, one at a time, and asks, "if I knew ahead of time that state of nature s₁ will occur, what would I do?" The answer to maximize profit would be, "buy the product (d₂)", since that leads to the highest profit, $10,000. If the decision-maker selected d₂ and s₁ occurred, there would be no regret. On the other hand, if the decision-maker selected d₃, there would be a regret or opportunity loss of $5,000 ($10,000 that could have been gained minus $5,000 that was gained). Similarly, there would be a regret of $30,000 if the decision-maker selected d₁ and state of nature s₁ occurred ($10,000 that could have been gained minus the minus $20,000 loss).

The regret numbers for s₂ are prepared in a similar fashion. "If I knew ahead of time that state of nature s₂ would occur, what would I do?" The answer, again to maximize profit, is "make the product (d₁)," since that leads to the highest profit for s₂, $90,000. If the decision-maker selected d₁ and s₂ occurred, there would be no regret. On the other hand, if the decision-maker selected d₂, there would be an opportunity loss or regret of $20,000 ($90,000 that could have been gained minus $70,000 that was gained). Likewise, if the decision-maker selected d₃ there would be a regret of $85,000 ($90,000 minus $5,000). Table 1.3.3 illustrates the completed regret table.

Table 1.3.3.

	States of	Nature	Maximum
Decision Alternative	Low Demand, s1	High Demand, s2	Regret
Make Product, d1	$ 30,000	$ 0	$ 30,000
Buy Product, d2	$ 0	$ 20,000	$ 20,000
Do Nothing, d3	$ 5,000	$ 85,000	$ 85,000

Next, I create a new column, "Maximum Regret," to record the maximum regret value associated with each decision strategy (the maximum regret in each row of the regret table). This strategy, which is sometimes called the minimax regret strategy, then selects that decision alternative associated with the minimum of the maximum regrets. In this situation, the decision-maker would select d₂, buy the product, since $20,000 is the minimum of the maximum regrets. Some believe this strategy follows a "middle of the road" approach of minimizing losses.

Why don't you follow along with me while I illustrate the above strategies with The Management Scientist software? First, I will click on the Start button on the Windows Task Bar, then select Programs, then The Management Scientist, then click on The Management Scientist icon, select Continue, then click on Decision Analysis Module (Module Number 10), and we are ready to start a new problem.

Click on File, then New, then enter 3 decision alternatives, 2 states of nature, click OK, and then enter the numbers for the payoff table (omit the dollar signs and use a minus sign for the negative $20,000). Next, select Solution, then Solve, keep the default selection Maximize the Payoff, and then select Optimistic Criterion to make your first run. You can then select to print this solution, or better, save it to a file. I saved the solution to a Management Scientist Out File. The Out files can be inserted into a Word document, as I did below. In an open Word file, select Insert from the top tool bar, then File, then search for your Management Scientist Out file (I saved mine under The Management Scientist Folder so I "Looked In" that folder, changed file type to "All Files" and opened the appropriate out file to insert it below as Printout 1.3.1.

You may use this technique to e-mail computer output solutions to cases this semester. You can open a Word document for your answer report, and insert the Management Scientist Output file into the report following the above procedure. Alternatively, you can write your answer narrative in an e-mail, then insert the Management Scientist Out file into the e-mail, and forward to me. I can easily open the Management Scientist Out file as a Word document and read your output. If we ever experience e-mail problems, you can always print your computer output files and fax them to me.

Now, back to the computer output, Printout 1.3.1, below.


Printout 1.3.1

DECISION ANALYSIS

*****************

YOU HAVE INPUT THE FOLLOWING PAYOFF TABLE:

******************************************

	STATES OF	NATURE
DECISION	1	2
1	- 20000	90000
2	10000	70000
3	5000	5000

DECISION RECOMMENDATION

***********************

USING THE OPTIMISTIC CRITERION

DECISION ALTERNATIVE	CRITERION VALUE	RECOMMENDED DECISION
1	90000	YES
2	70000
3	5000

Note that the output repeats your input, then provides the criterion column for making the decision, as well as the recommended decision for the criterion selected. For practice, go ahead and run "The Management Scientist" for the conservative and minimax regret criteria. Please note that the format of your OUT file will be a little different than above as I had to insert tables to conform to html format - you will not have to do that for a Word file.

Your first case is Problem 18, McHuffter Condominiums, pp. 129-130 in the text. The material we have covered up to this point should prepare you to work on Part b. of this problem. You may also want to sketch out the decision tree (Part a.), although that is not assigned for turn-in with the other parts. As you work with "The Management Scientist" to answer the various questions of this and other problems, note that you have the opportunity to save your input file as well as the output. For small problems such as this one, that may not so important - but later on, such as in the forecasting material where there is substantial input, it might be handy to save the data input to a file.

This concludes the first approach to decision analysis: decision-making under risk. One of the critiques of the approach is that it works with limited information on the states of nature. Because of that, a second approach has been developed to add information on the states of nature concerning their relative likelihood of occurrence. That is our next section of material.


Module 1.4: Decision Making with Probabilities

In this approach, the decision-maker has information concerning the relative likelihood of each of the states of nature. It is sometimes called "decision making under uncertainty." The criterion used in decision-making strategy with probabilities is to select that decision so as to maximize the expected value of the outcome. To illustrate the approach, let's refer again to the payoff table for our make-buy example, this time adding a row for probabilities and a column for the expected value of the decision alternatives.

Table 1.4.1

 

	States of	Nature	Expected
Decision Alternative	Low Demand, s1	High Demand, s2	Value
Make Product, d1	($ 20,000)	$ 90,000	$ 51,500
Buy Product, d2	$ 10,000	$ 70,000	$ 49,000
Do Nothing, d3	$ 5,000	$ 5,000	$ 5,000
Probabilities	P (s1) = 0.35	P (s2) = 0.65

In decision analysis, it is assumed that the probabilities are long-term relative frequencies. Since they are often simply the subjective judgment of the decision- maker, the techniques is subject to the criticism of this limitation. But this criticism can be levied against any quantitative approach - the output is only as good as the input. To counter the criticism, we will add a sensitivity analysis step after the initial solution. For now, let's learn the expected value approach.

The expected value (EV) for a decision alternative is the sum of the [probabilities of the states of nature times the payoffs]. For the "make product" decision:

EV (d₁) = [P (s₁) * V₁₁] + [P (s₂) * V₁₂]

= [0.35 * -20,000] + [0.65 * 90,000]
= $51,500

The EV of $51,500 represents the long run outcome of repeated "make product" experiments. That is, if we could theoretically conduct the "make product" decision 100 times, 35 times we would lose $20,000, and 65 times we would make $90,000. The weighted average of these outcomes is $51,500. In reality, we do not conduct the experiment 100 times - we make the decision once and we are either going to lose $20,000 or make $90,000. However, and this is very important, we use the expected value approach to assist us in making the decision.

I should add, at this point, that to abide by the laws of probability, each probability must be a real number between 0 and 1, and the sum of the probabilities for the states of nature must sum to one. For this to happen, the states of nature must be mutually exclusive and exhaustive - that is, there cannot also be a state of nature called, for example, medium demand. If there was such a state of nature, it would have to be added to the payoff table and accounted for with a third probability.

The expected values for the second and third decision alternatives are calculated in a similar fashion. These are shown in Table 1.4.1. Following the criterion of selecting that decision alternative which maximizes the expected value of the outcome, we select decision alternative d₁ as our optimal or best strategy.

The expected value computations can be shown adjacent to the state of nature nodes in the decision tree, as illustrated in Figure 1.4.1.

Figure 1.4.1

				Payoffs
		-	----- s1: Low Demand: P(s1) = 0.35	($ 20,000)
		|
-	--- d1: Make ------->	B	EVB = 0.35 * -20,000 + 0.65 * 90,000 = $51,000
|		|
|		-	----- s2: High Demand: P(s2)= 0.65	$ 90,000
|
|		-	----- s1: Low Demand: P(s1)= 0.35	$ 10,000
|		|
A	--- d2: Buy ---\\----->	C	EVC = 0.35 * 10,000 + 0.65 * 70,000 = $49,000
|		|
|		-	----- s2: High Demand: P(s2)= 0.65	$ 70,000
\\
|		-	----- s1: Low Demand: P(s1)= 0.35	$ 5,000
|		|
-	--- d3: Do Nothing -->	D	EVD = 0.35 * 5,000 + 0.65 * 5,000 = $5,000
		|
		-	----- s2: High Demand: P(s2) = 0.65	$ 5,000

The decision tree shows the expected value computations by each of the state of nature nodes labeled B, C and D. Since the expected value for the "make" decision" is the maximum, that decision branch is selected as the optimum strategy. The "buy" and "do nothing" branches are pruned, as indicated by the crosshatched lines. The decision tree does not provide additional information, it simply presents a picture of the decision strategy with probabilities.


Using The Management Scientist for Decision Making with Probabilities
Let's return to the software for a moment and rerun this example problem using the expected value criterion for selecting a decision alternative.

After you open "The Management Scientist" program, click on File, then New, then enter 3 decision alternatives, and 2 states of nature just as before. This time, select State of Nature Probabilities, and then click OK. Enter the numbers for the payoff table and the probabilities for the two states of nature. Next, select Solution, then Solve, and keep the default selection Maximize the Payoff. As before, you can print this solution, or better, save it to a file. I saved the solution to an Out file, then inserted it to this open Word document as before.


Printout 1.4.1

DECISION ANALYSIS

*****************

YOU HAVE INPUT THE FOLLOWING PAYOFF TABLE:

******************************************

	STATES OF	NATURE
DECISION	1	2
1	- 20000	90000
2	10000	70000
3	5000	5000
PROBABILITIES OF STATES	0.35	0.65

 DECISION RECOMMENDATION

***********************

 USING THE EXPECTED VALUE CRITERION

DECISION ALTERNATIVE	CRITERION VALUE	RECOMMENDED DECISION
1	51,500.00	YES
2	49,000.00
3	5000.00

EXPECTED VALUE OF PERFECT INFORMATION IS 10,500.00


The printout information confirms the computations we made earlier in this section. For now, ignore the Expected Value of Perfect Information shown at the bottom of the Decision Recommendation. We will cover that in Section 1.5. Before that, we need to go over a very important topic for any initial solution to any quantitative method - sensitivity analysis.

Sensitivity Analysis
When you apply quantitative approaches to solve management problems it is always a good idea to ask, "how sensitive is my solution to changes in data input." This is especially true when the input consists of subjective estimates. When the solution remains the optimal solution given large changes in selected important inputs, there is a high level of confidence in the solution. On the other hand, when the optimal solution strategy changes for very small changes to the one or more inputs, the decision-maker should be cautious before implementation. Perhaps some time should be spent in refining the input.

Suppose the decision-maker examines the solution and realizes that the "buy" decision lost out to the "make" decision by only $2,500. Further suppose the decision-maker has little confidence in the payoff for the "buy product" decision under the "high demand" state of nature. The question is, "at what "buy product, high demand" payoff would I be indifferent between the "make" and "buy" decisions, given all other applicable data input items remain the same?" To answer this question, we note that the point of indifference occurs where the "make" and the "buy" EV's are equal. Mathematically, this is expressed as the equation:

EV_B = EV_C

Now writing out the computational formulas for the EV's:

[P (s₁) * V₁₁] + [P (s₂) * V₁₂] = [P (s₁) * V₂₁] + [P (s₂) * V22]

Now, substitute the data input values except for the unknown "buy product, high demand" payoff, V₂₂:

[0.35 * -20,000] + [0.65 * 90,000] = [0.35 * 10,000] + [0.65 * V₂₂]

Next, solve the equation for V₂₂:

0.65 * V₂₂ = 51,500 - [0.35 * 10,000]

V₂₂ = 48,000 / 0.65

V₂₂ = 73,846

So, if the decision-maker erred by just 5.5% (estimates the "buy product, high demand" payoff to be $70,000 when it is really $73,846), the decision-maker would have selected the wrong strategy.

As a rule of thumb, if a decision strategy changes based on a five or less percent change to an input, we say that the solution is very sensitive to change and it might be wise to consider investing in more accurate data input values.

Probabilities are sometimes also subjective estimates when there has been no history or experience with a particular state of nature. By observation, we can see that as long as P (s₁) remains at or below 0.35 the decision-maker will favor the optimal "make product" decision since a low probability for s₁ gives little relative weight to the $20,000 loss. But as P (s₁) increases (resulting in a decrease for P (s₂)), the expected values approach a point of indifference or a break-even point. The question is, at what P (s₁) would the decision-maker be indifferent between the "buy" and "make" decisions, all other data input remaining the same?

Again, we begin by setting EV_B equal to EV_C to represent break-even:

EV_B = EV_C

Next, we write out the computational formulas for the EV's:

[P (s₁) * V₁₁] + [P (s₂) * V₁₂] = [P (s₁) * V₂₁] + [P (s₂) * V₂₂]

Now, substitute the data input values except for the unknown state of nature probabilities:

[P (s₁) * -20,000] + [P (s₂) * 90,000] =

[P (s₁) * 10,000] + [P (s₂) * 70,000]

You may recall from an algebra class that we cannot solve one equation when there are two unknowns. However, recall that the sum of the probabilities of the states of nature must equal one. That is, with two states of nature:

P (s₁) + P (s₂) = 1

We can rewrite this as:

P (s₂) = 1 - P (s₁)

Then, we can substitute [1 - P (s₁)] for each P (s₂) in the break-even equation and continue solving for P (s1).

{P (s₁) * -20,000} + {[1 - P (s₁)] * 90,000} =

{P (s₁) * 10,000} + {[1 - P (s₁) * 70,000]}

Now simplify the equation:

- 20,000 P (s₁) + 90,000 - 90,000 P (s₁) =

10,000 P (s₁) + 70,000 - 70,000 P (s₁)

Finally, collect terms and solve for P (s1):

-110,000 P (s₁) + 90,000 = - 60,000 P (s₁) + 70,000

- 50,000 P (s₁) = - 20,000

P (s₁) = -20,000 / - 50,000 = 0.40

Conclusion: as long as P (s1) stays less than or equal to 0.40, the "make" decision strategy will remain the optimal strategy.

I was once involved in a decision analysis problem with a group of medical doctors who owned their practice. The problem of interest was whether to pay of a loan on a five-year schedule, or draw down cash and pay off the loan now. We did a lot of sensitivity analysis on the probabilities attached to states of nature representing the up or down swing of the economy in the next five years. I know that working formula after formula on the white board in their office would have been boring to the doctors. So instead, we used the computer software package to simply give us results for all of the desired changes. The point is, whether you use "brute force" by changing the input values in the payoff and probability tables in the software package, or run a number of break-even formulas, it is a good idea to always check how sensitive your solution is to changes in the inputs.

What if you find out that your solution is sensitive to changes in the inputs. In that case, it may be appropriate to obtain additional information concerning your input numbers to make them more accurate or to increase your confidence in their original value. However, while information has value, it is sometimes expensive to obtain. Our last topic in decision analysis focuses on the value of information.

I hope you are enjoying the first quantitative method of the course. This technique has found countless applications in assisting managers make decisions. I have personally helped companies use decision analysis in equipment loan, construction, and enterprise resource planning situations. I continue to believe that the formal steps in structuring the problem, which forces the decision-maker to consider decision alternatives, states of nature, probabilities and payoffs, is almost as valuable as coming up with the decision itself. As well, every project I have worked on found considerable benefit from attention to sensitivity analysis. Sorry for rambling - back to the subject.


1.5 The Value of Information

We made the point that information has value, but it also costs money and time to obtain. So the question becomes, how much would be willing to pay for additional information. The upper limit is established by the first concept, the Value of Perfect Information.

The Value of Perfect Information
Let's look at the payoff table once again.

Table 1.5.1

	States of	Nature	Expected
Decision Alternative	Low Demand, s1	High Demand, s2	Value
Make Product, d1	($ 20,000)	$ 90,000	$ 51,500
Buy Product, d2	$ 10,000	$ 70,000	$ 49,000
Do Nothing, d3	$ 5,000	$ 5,000	$ 5,000
Probabilities	P (s1) = 0.35	P (s2) = 0.65

Suppose we knew ahead of the production period which state of nature would occur - sort of like having a crystal ball. If we knew ahead of time that s₁ would occur, we would select d₂ as our optimal strategy to maximize our payoffs. If we knew ahead of time that s₂ would occur, we would select d₁ to maximize our payoffs. Over time, the expected value with this perfect information (abbreviated as EV w PI) would be:

EV w PI = [P (s₁) * V₂₁] + [P (s₂) * V₁₂]

= [ 0.35 * 10,000 ] + [ 0.65 * 90,000 ] = $62,000

Recall that without this perfect information, the expected value was $51,500 by going with d₁. The difference between the expected value with perfect information and the expected value without perfect information is called the Expected Value of Perfect Information (EVPI):

EVPI = EV w PI - EV w/o PI = 62,000 - 51,500 = $10,500

EVPI places an upper bound on how much we should be willing to pay someone to get additional information on the future states of nature in order to improve our decision making strategy.

For example, we might be willing to pay a consultant for a market research study to help us learn more about the demand states of nature. We don't expect the research study to result in perfect information, but we hope it will represent a good proportion of the $10,500. We also know that we do not want to pay more than $10,500 for the new information.

New information about the states of nature takes the form of revisions to the probability estimates in decision analysis. This information may be obtained through market research studies, product testing or some other sampling process. In decision analysis, this topic is called the Expected Value of Sample Information.

The Expected Value of Sample Information (EVSI)
Recall that we have probabilities on the states of nature, P (s₁) and P (s₂). These are more technically called prior probabilities. If we plan to obtain sample information, the results will be expressed as revisions to these prior probabilities.

Suppose a market research consultant offers to conduct a demand analysis that will result in two indicators. I₁ will be the indicator used to represent a prediction of low demand in the next production period. I₁ then corresponds to the state of nature we labeled s₁, the difference is that one is a prediction and one is an actual occurrence. I₂ will be the indicator used to represent a prediction of high demand in the next production period. This indicator corresponds to our state of nature s₂.

This indicator information will be used to calculate conditional probabilities. P (s₁ given I₁) will be the probability that s₁ will occur given or conditioned on the consultant's I₁ prediction. P (s₁ given I₁) is generally written as P (s₁ | I₁). Since there are two states of nature and two indicators, we will need three additional conditional probabilities, P (s₁ | I₂), P (s₂ | I₁) and P (s₂ | I₂).

In order to compute these conditional probabilities and then EVSI, we need to know the consultant's track record. That is, how has the consultant done in prior market demand studies? In terms of decision analysis, we want to know the accuracy of indicator predictions given what state of nature actually occurred. Suppose the following information is available from the consultant.

Table 1.5.2.

	Market	Research
State of Nature	Predict Low Demand, I1	Predict High Demand, I2
Low Demand, s1	P ( I1 | s1 ) = 0.90	P ( I2 | s2 ) = 0.10
High Demand, s	P ( I1 | s2 ) = 0.20	P ( I 2 | s2 ) = 0.80

We are almost there, but note that these conditional probabilities are the reverse of what we need. To get from P (I₁ | s_j) to P (s_j | I₁), we need to set up a table and do some calculations. Same for Indicator 2. Let's do I₁ first.

Table 1.5.3.

States of Nature, sj	Prior Probabilities, P (sj)	Consultants Track Record, P ( I1| sj )	Joint Probabilities, P ( I1 and sj)	Conditional Probabilities, P ( sj | I1 )
s1	P (s1) = 0.35	P (I1|s1) = 0.90	0.35*0.90 = 0.315	P( s1| I1)=.315 / 0.445 =0 .708
s2	P (s2) = 0.65	P (I1|s2) = 0.20	0.65*0.20 = 0.13	P( s2| I1 ) =0.13/0.445 = 0.292
			P(I1)=.315 + .13 =0.445	Note: 0.708 + 0.292 = 1.00

The first three columns in Table 1.5.3 are from input information. The joint probabilities column is computed by multiplying the prior probabilities times the consultant track record probabilities in each row. The conditional probabilities in the last column are computed by dividing the joint probabilities by P(I₁).

Note how consultant research information can change the estimates on the probabilities. Now the probability of low demand becomes 0.71 given that the consultant predicts low demand, and taking into account the consultant's track record. This revision reflects prior accuracy for the low demand case. Note that the probability of high demand, s₂ goes from 0.65 to 0.29 given that the consultant predicts that demand will be low.

I believe the above steps illustrate the power of the complete decision analysis technique. The steps illustrate how information has value to justify its expense. Now, we do similar calculations for Indicator two in Table 1.5.4.

Table 1.5.4.
 

States of Nature, sj	Prior Probabilities, P (sj)	Consultants Track Record, P ( I2| sj )	Joint Probabilities, P ( I2 and sj)	Conditional Probabilities, P ( sj | I2 )
s1	P (s1) = 0.35	P (I2|s1) = 0.10	0.35*0.10 = 0.035	P( s1| I2)=.035 / 0.555 = 0 .063
s2	P (s2) = 0.65	P (I2|s2) = 0.80	0.65*0.80 = 0.52	P( s2| I2 )=0.52 / 0.555 = 0.937
			P(I2)=.035 + .52 =0.555	Note: 0.063 + 0.937 = 1.00

Although we will use the computer to produce the above conditional probabilities as well as the following results, it is instructive to picture the decision problem incorporating the sample information. This is done in the decision trees in Figure 1.5.1, 1.5.2 and 1.5.3.

Figure 1.5.1

					Payoffs
		-	----- I1: Predict Low Demand: P(I1)= 0.445	C	$ 27,500
		|
-	-- Do Mkt Research->	B	EVB = 0.445 * 27,500 + 0.555 * 83,100 = $58,400
|		|
|		-	----- I2: Predict High Demand: P(I2 )= 0.555	D	$ 83,100
|
|
|
A	-- No Mkt Research->		EV w/o Perfect Information = $51,500 (Make Product)

Figure 1.5.1 shows a new decision to our problem: the decision of whether or not to engage the services of the market research consultant. If the manufacturer does not engage the services, the expected value has already been determined to be $51,500 by selecting d₁, "make " the product. Note that I have added the expected value that would be obtained should the decision maker engage the services of the marketing consultant, $58,400. The first event after engaging the services of the consultant is the consultant's predictions.

The consultant's predictions are then followed by the manufacturer's decisions to "make,", "buy," or "do nothing;" which are then followed by the actual states of nature, low and high demand. Figure 1.5.2 illustrates the decision tree leading from the node labeled C in Figure 1.5.1, the consultant's prediction of low demand.

Figure 1.5.2.

				Payoffs
		-	----- s1: Low Demand: P(s1|I1) = 0.707	($ 20,000)
		|
-	--- d1: Make ------->	E	EVE = 0.707*-20,000 + 0.292*90,000 = $11,900
|		|
|		-	----- s2: High Demand: P(s2|I1) = 0.292	$ 90,000
\\
|		-	----- s1: Low Demand: P(s1|I1) = 0.707	$ 10,000
|		|
C	--- d2: Buy -------->	F	EVF = 0.707*10,000 + 0.292*70,000 = $27,500
|		|
|		-	----- s2: High Demand: P(s2|I1) = 0.292	$ 70,000
\\
|		-	----- s1: Low Demand: P(s1|I1) = 0.707	$ 5,000
|		|
-	--- d3: Do Nothing -->	G	EVG = 0.707*5,000 + 0.292*5,000 = $5,000
		|
		-	----- s2: High Demand: P(s2|I1) = 0.292	$ 5,000

This part of the decision tree provides back-up for the calculation of the $27,500 expected value for node 3. That value represents the highest expected value and is associated with the "buy" decision. Thus, if the consultant is hired to conduct the market research, and predicts low demand, the best decision is to "buy" the product. When working out decision trees by hand, we always start at the right of the tree, move left, pruning branches coming from decision nodes as we go. You should observe that every state of nature node gets an expected value computation, the sum of the products of probabilities times payoffs. Every decision node simply gets the highest expected value from its branches.

Next we examine the computations for the node labeled D in Figure 1.5.1., the event that occurs if we engage the consultant and the consultant's prediction is high demand. This is shown in Figure 1.5.3.

Figure 1.5.3.

				Payoffs
		-	----- s1: Low Demand: P(s1|I2) = 0.063	($ 20,000)
		|
-	--- d1: Make ------->	H	EVE = 0.063*-20,000 + 0.937*90,000 = $83,100
|		|
|		-	----- s2: High Demand: P(s2|I2) = 0.937	$ 90,000
|
|		-	----- s1: Low Demand: P(s1|I2) = 0.063	$ 10,000
|		|
D	--- d2: Buy ----\\-->	I	EVF = 0.063*10,000 + 0.937*70,000 = $66,200
|		|
|		-	----- s2: High Demand: P(s2|I2) = 0.937	$ 70,000
\\
|		-	----- s1: Low Demand: P(s1|I2)= 0.063	$ 5,000
|		|
-	--- d3: Do Nothing -->	J	EVG = 0.063*5,000 + 0.967*5,000 = $5,000
		|
		-	----- s2: High Demand: P(s2|I2) = 0.967	$ 5,000

This part of the decision tree provides back-up for the calculation of the $83,100 expected value for node D. That value represents the highest expected value and is associated with the "make" decision. Thus, if the consultant is hired to conduct the market research, and predicts high demand, the best decision is to "make" the product.

Looking back at Figure 1.5.1, we see that the expected value with the market research consultant report, named the expected value with sample information (EV w SI), is $58,400. The expected value without the sample information (EV w/o SI) (same as the expected value without perfect information) is $51,500.

To get the expected value of sample information, we just make the subtraction:

EVSI = EV w SI - EV w/o SI = 58,400 - 51,500 = $6,900

Like EVPI, this is a powerful piece of information. We now know how much we would be willing to spend to obtain market research from the consultant - that is a great bargaining chip to have at negotiation time!

Efficiency of the Sample Information

Pause and Reflect
Recall that the expected value of perfect information was $10,500 - that placed a theoretical upper bound on how much we would be willing to spend for perfect information. But forecasts and consultants are not perfect. If we can get the "track record" of a forecaster consultant, we can then determine how they stack up to the theoretical EVPI.

To compute the efficiency of the sample information, we simply divide the EVSI by EVPI and convert the decimal to a percent:

Efficiency = EVSI / EVPI = 6,900 / 10,500 = .657 = 65.7 %

The closer this number is to 100%, the better.

I am not suggesting that every decision analysis exercise includes this last procedure of computing EVSI and its efficiency. Often the track record of the sample information producer may not be available. But when it is (and it should ALWAYS be available if you do the market research "in-house", this is a powerful tool to put some limits on how much you would want to spend to acquire more information.

I appreciate that the computation of EVSI is tedious - it has taken me hours to produce the decision trees in power point, convert them to html images and insert them into to these pages (I am not complaining - I always wanted to learn how to convert power point slides to html text images and now I know!). Fortunately, "The Management Scientist" does all of this work for us. We will close this first module with a demonstration of the computer output for EVPI and EVSI.

Using "The Management Scientist" for Computing EVPI and EVSI
Open up the software program as before. This time, when you select the Decision Analysis Module, then file, and enter a new problem, click State of Nature Probabilities after you enter the number of decisions and number of states of nature. After you click State of Nature Probabilities, also click Compute Decision Strategies, followed by entering 2 when you are asked for the Number of Indicator Outcomes. The data entry screen then asks for the payoffs, the probabilities and the "track record" conditional probabilities about the consultant.

When you select solve, the following file can either be printed or saved to a Management Scientist Out file.

Printout 1.5.1

DECISION ANALYSIS

*****************

YOU HAVE INPUT THE FOLLOWING PAYOFF TABLE:

******************************************

	STATES OF	NATURE
DECISION	1	2
1	- 20000	90000
2	10000	70000
3	5000	5000
PROBABILITIES OF STATES	0.35	0.65

DECISION RECOMMENDATION

***********************

USING THE EXPECTED VALUE CRITERION

DECISION ALTERNATIVE	CRITERION VALUE	RECOMMENDED DECISION
1	51,500.00	YES
2	49,000.00
3	5000.00

EXPECTED VALUE OF PERFECT INFORMATION IS 10,500.00

DEVELOPING A DECISION STRATEGY BASED ON SAMPLE INFORMATION

**********************************************************

YOU HAVE INPUT THE FOLLOWING PROBABILITIES FOR INDICATOR OUTCOMES:

******************************************************************

GIVEN STATE	INDICATOR OUTCOME 1	INDICATOR OUTCOME 2
1	0.90	0.10
2	0.20	0.80

OPTIMAL DECISION STRATEGY

*************************

IF INDICATOR	BEST DECISION	EXPECTED VALUE	INDICATOR OUTCOME
1	2	27,528.09	0.445
2	1	83,063.06	0.555

EXPECTED VALUE OF THE ABOVE STRATEGY 58,350.00

EXPECTED VALUE OF THE SAMPLE INFORMATION 6,850.00

EFFICIENCY OF THE SAMPLE INFORMATION 65.2%

You should be able to pick out all of the terms and their values in the computer output from our discussion. Please note that I rounded my expected value computations to the nearest 100 so that my numbers will not be as accurate as the computer output. I should also note that there are special decision analysis software packages, even one created by Microsoft. The special packages have the same capabilities as our text package, plus many more features. For example, the special packages will "draw" the decision tree, allow multiple branching such as state of nature branches followed by more state of nature branches, and handle much larger problems than 10 decision nodes. The special packages also cost in the $100's and only do decision analysis. Our package may have limitations but it includes 11 other modules!


Well, that's it! You should now be ready to work the first case assignment, Problem 18, parts b.-g. in the text. Please e-mail any questions that you may have. Remember, for full credit, the problem is due January 27, 2001.

That finishes our first quantitative method. I believe decision analysis is a good place to start since it illustrates the five-step decision-making process in a picture called the decision tree. The tree shows the decision alternatives to the problem, the factors that effect the alternatives (states of nature and their probabilities), the outcomes of decision alternatives (payoffs) and the criterion for selection (maximize expected value). Please note that our illustration involved maximize expected value - if you were working a minimize problem (such as select alternatives so as to minimize costs) you would simply indicate that criterion at the appropriate input screen.