In Chapter 4, we introduced you to the topic of uncertainty and how it can affect decision making. Although uncertainty plays a critical role by limiting the amount of information available to managers, another factor is their psychological orientation. For instance, the optimistic manager will typically follow a maximax choice (maximizing the maximum possible payoff), the pessimist will often pursue a maximin choice (maximizing the minimum possible payoff), and the manager who desires to minimize his “regret” will opt for a minimax choice. Let’s briefly look at these different approaches using an example.

Consider the case of a marketing manager at Visa International in New York. He has determined four possible strategies (we’ll label these S1, S2, S3, and S4) for promoting the Visa card throughout the northeastern United States. However, he is also aware that one of his major competitors, American Express, has three competitive strategies (CA1, CA2, and CA3) for promoting its own card in the same region. In this case, we’ll assume that the Visa executive has no previous knowledge that would allow him to place probabilities on the success of any of his four strategies. With these facts, the Visa card manager formulates the matrix in Exhibit QM–1 to show the various Visa strategies and the resulting profit to Visa, depending on the competitive action chosen by American Express.

In this example, if our Visa manager is an optimist, he’ll choose S4 because that could produce the largest possible gain ($28 million). Note that this choice maximizes the maximum possible gain (maximax choice). If our manager is a pessimist, he’ll assume only the worst can occur. The worst outcome for each strategy is as follows: $\text{S1}=\text{\$11 million},\text{S2}=\text{\$9 million},\text{ S3}=\text{\$15 million},\text{and S4}=\text{\$14 million}\text{.}$ Following the maximin choice, the pessimistic manager would maximize the minimum payoff—in other words, he’d select S3.

In the third approach, managers recognize that once a decision is made it will not necessarily result in the most profitable payoff. What could occur is a “regret” of profits forgone (given up)—regret referring to the amount of money that could have been made had a different strategy been used. Managers calculate regret by subtracting all possible payoffs in each category from the maximum possible payoff for each given—in this case, for each competitive action. For our Visa manager, the highest payoff, given that American Express engages in CA1, CA2, or CA3, is $24 million, $21 million, or $28 million, respectively (the highest number in each column). Subtracting the payoffs in Exhibit QM–1 from these figures produces the results in Exhibit QM–2.

The maximum regrets are $\text{S1}=\text{\$17 million},\text{S2}=\text{\$15 million},\text{ S3}=\text{\$13 million},\text{and S4}=\text{\$7 million}\text{.}$ The minimax choice minimizes the maximum regret, so our Visa manager would choose S4. By making this choice, he’ll never have a regret of profits forgone of more than $7 million. This result contrasts, for example, with a regret of $15 million had he chosen S2 and American Express had taken CA1.

Decision trees are a useful way to analyze hiring, marketing, investment, equipment purchases, pricing, and similar decisions that involve a progression of decisions. They’re called decision trees because, when diagrammed, they look a lot like a tree with branches. Typical decision trees encompass expected value analysis by assigning probabilities to each possible outcome and calculating payoffs for each decision path.

Exhibit QM–3 illustrates a decision facing Becky Harrington, the midwestern region site selection supervisor for Barry’s Brews. Becky supervises a small group of specialists who analyze potential locations and make store site recommendations to the midwestern region’s director. The lease on the company’s store in Winter Park, Florida, is expiring, and the property owner has decided not to renew it. Becky and her group have to make a relocation recommendation to the regional director. Becky’s group has identified an excellent site in a nearby shopping mall in Orlando. The mall owner has offered her two comparable locations: one with 12,000 square feet (the same as she has now) and the other a larger, 20,000-square-foot space. Becky’s initial decision concerns whether to recommend renting the larger or smaller location. If she chooses the larger space and the economy is strong, she estimates the store will make a $320,000 profit. However, if the economy is poor, the high operating costs of the larger store will mean that the profit will be only $50,000. With the smaller store, she estimates the profit at $240,000 with a good economy and $130,000 with a poor one.

Exhibit QM–3

As you can see from Exhibit QM–3, the expected value for the larger store is $\text{\$239,000}[(.70\times 320)+(.30\times 50)]\text{.}$ The expected value for the smaller store is $\text{\$207,000}[(.70\times 240)+(.30\times 130)]\text{.}$ Given these projections, Becky is planning to recommend the rental of the larger store space. What if Becky wants to consider the implications of initially renting the smaller space and then expanding if the economy picks up? She can extend the decision tree to include this second decision point. She has calculated three options: no expansion, adding 4,000 square feet, and adding 8,000 square feet. Following the approach used for Decision Point 1, she could calculate the profit potential by extending the branches on the tree and calculating expected values for the various options.

How many units of a product must an organization sell in order to break even—that is, to have neither profit nor loss? A manager might want to know the minimum number of units that must be sold to achieve his or her profit objective or whether a current product should continue to be sold or should be dropped from the organization’s product line. Break-even analysis is a widely used technique for helping managers make profit projections.2

Break-even analysis is a simplistic formulation, yet it is valuable to managers because it points out the relationship among revenues, costs, and profits. To compute the break-even point (BE), the manager needs to know the unit price of the product being sold (P), the variable cost per unit (VC), and the total fixed costs (TFC).

An organization breaks even when its total revenue is just enough to equal its total costs. But total cost has two parts: a fixed component and a variable component. Fixed costs are expenses that do not change, regardless of volume, such as insurance premiums and property taxes. Fixed costs, of course, are fixed only in the short term because, in the long run, commitments terminate and are, thus, subject to variation. Variable costs change in proportion to output and include raw materials, labor costs, and energy costs.

The break-even point can be computed graphically or by using the following formula:

This formula tells us that (1) total revenue will equal total cost when we sell enough units at a price that covers all variable unit costs and (2) the difference between price and variable costs, when multiplied by the number of units sold, equals the fixed costs.

When is break-even analysis useful? To demonstrate, assume that, at Jose’s Bakersfield Espresso, Jose charges $1.75 for an average cup of coffee. If his fixed costs (salary, insurance, etc.) are $47,000 a year and the variable costs for each cup of espresso are $0.40, Jose can compute his break-even point as follows: $\$47,000/(1.75-.40)=34,815$ (about 670 cups of espresso sold each week), or when annual revenues are approximately $60,926. This same relationship is shown graphically in Exhibit QM–4.

Exhibit QM–4

How can break-even analysis serve as a planning and decision-making tool? As a planning tool, break-even analysis could help Jose set his sales objective. For example, he could establish the profit he wants and then work backward to determine what sales level is needed to reach that profit. As a decision-making tool, break-even analysis could also tell Jose how much volume has to increase in order to break even if he is currently operating at a loss, or how much volume he can afford to lose and still break even if he is currently operating profitably. In some cases, such as the management of professional sports franchises, break-even analysis has shown the projected volume of ticket sales required to cover all costs to be so unrealistically high that management’s best choice is to sell or close the business.

Matt Free owns a software development company. One product line involves designing and producing software that detects and removes viruses. The software comes in two formats: Windows and Mac versions. He can sell all of these products that he can produce, which is his dilemma. The two formats go through the same production departments. How many of each type should he make to maximize his profits?

A close look at Free’s operation tells us he can use a mathematical technique called linear programming to solve his resource allocation dilemma. As we will show, linear programming is applicable to his problem, but it cannot be applied to all resource allocation situations. Besides requiring limited resources and the objective of optimization, it requires that there be alternative ways of combining resources to produce a number of output mixes. A linear relationship between variables is also necessary, which means that a change in one variable will be accompanied by an exactly proportional change in the other. For Free’s business, this condition would be met if it took exactly twice the time to produce two diskettes—irrespective of format—as it took to produce one.

Many different types of problems can be solved with linear programming. Selecting transportation routes that minimize shipping costs, allocating a limited advertising budget among various product brands, making the optimum assignment of personnel among projects, and determining how much of each product to make with a limited number of resources are just a few. To give you some idea of how linear programming is useful, let’s return to Free’s situation. Fortunately, his problem is relatively simple, so we can solve it rather quickly. For complex linear programming problems, computer software has been designed specifically to help develop solutions.

First, we need to establish some facts about the business. He has computed the profit margins to be $18 for the Windows format and $24 for the Mac. He can, therefore, express his objective function as $\text{maximum profit}=\$18\text{R}+\$24\text{S,}$ where R is the number of Windows-based CDs produced and S is the number of Mac CDs. In addition, he knows how long it takes to produce each format and the monthly production capacity for virus software: 2,400 hours in design and 900 hours in production (see Exhibit QM–5). The production capacity numbers act as constraints on his overall capacity. Now Free can establish his constraint equations:

Of course, because a software format cannot be produced in a volume less than zero, Matt can also state that R > 0 and S > 0. He has graphed his solution as shown in Exhibit QM–6. The beige shaded area represents the options that do not exceed the capacity of either department. What does the graph mean? We know that total design capacity is 2,400 hours. So if Matt decides to design only the Windows format, the maximum number he can produce is $600(2,400\text{hours}÷\text{hours of design for each Windows version})\text{.}$ If he decides to produce all Mac versions, the maximum he can produce is $400(2,400\text{hours}÷\text{6 hours of design for Mac})\text{.}$ This design constraint is shown in Exhibit QM–6 as line BC. The other constraint Matt faces is that of production. The maximum of either format he can produce is 450 because each takes two hours to copy, verify, and package. This production constraint is shown in the exhibit as line DE.

Exhibit QM–6

Free’s optimal resource allocation will be defined at one of the corners of this feasibility region (area ACFD). Point F provides the maximum profits within the constraints stated. At point A, profits would be zero because neither virus software version is being produced. At points C and D, profits would be $9,600 (400 units @ $24) and $8,100 (450 units @ $18), respectively. At point F, profits would be $9,900 (150 Windows units @ $18 + 300 Mac units @ $24).3

You are a supervisor for a branch of Bank of America outside of Cleveland, Ohio. One of the decisions you have to make is how many of the six teller stations to keep open at any given time. Queuing theory, or what is frequently referred to as waiting line theory, could help you decide.

A decision that involves balancing the cost of having a waiting line against the cost of service to maintain that line can be made more easily with queuing theory. These types of common situations include determining how many gas pumps are needed at gas stations, tellers at bank windows, toll takers at toll booths, or check-in lines at airline ticket counters. In each situation, management wants to minimize cost by having as few stations open as possible yet not so few as to test the patience of customers. In our teller example, on certain days (such as the first of every month and Fridays), you could open all six windows and keep waiting time to a minimum, or you could open only one, minimize staffing costs, and risk a riot.

The mathematics underlying queuing theory is beyond the scope of this book, but you can see how the theory works in our simple example. You have six tellers working for you, but you want to know whether you can get by with only one window open during an average morning. You consider 12 minutes to be the longest you would expect any customer to wait patiently in line. If it takes 4 minutes, on average, to serve each customer, the line should not be permitted to get longer than three deep $(\text{12 minutes}÷\text{4 minutes per customer}=\text{3 customers}).$ If you know from past experience that, during the morning, people arrive at the average rate of two per minute, you can calculate the probability $(P)$ of customers waiting in line as follows:

where $n=\text{3 customers},\text{arrival rate}=\text{2 per minute},\text{and service rate}=\text{4 minutes}$ per customer.

Putting these numbers into the foregoing formula generates the following:

What does a $P$ of .0625 mean? It tells you that the likelihood of having more than three customers in line during the average morning is 1 chance in 16. Are you willing to live with four or more customers in line 6 percent of the time? If so, keeping one teller window open will be enough. If not, you will have to assign more tellers to staff more windows.

When you order checks from a bank, have you noticed that the reorder form is placed about two-thirds of the way through your supply of checks? This practice is a simple example of a fixed-point reordering system. At some preestablished point in the process, the system is designed to “flag” the fact that the inventory needs to be replenished. The objective is to minimize inventory carrying costs while at the same time limiting the probability of stocking out of the inventory item. In recent years, retail stores have increasingly been using their computers to perform this reordering activity. Their cash registers are connected to their computers, and each sale automatically adjusts the store’s inventory record. When the inventory of an item hits the critical point, the computer tells management to reorder.

One of the best-known techniques for mathematically deriving the optimum quantity for a purchase order is the economic order quantity (EOQ) model (see Exhibit QM–7). The EOQ model seeks to balance four costs involved in ordering and carrying inventory: the purchase costs (purchase price plus delivery charges less discounts), the ordering costs (paperwork, follow-up, inspection when the item arrives, and other processing costs), carrying costs (money tied up in inventory, storage, insurance, taxes, etc.), and stock-out costs (profits forgone from orders lost, the cost of reestablishing goodwill, and additional expenses incurred to expedite late shipments). When these four costs are known, the model identifies the optimal order size for each purchase.

Exhibit QM–7

The objective of the economic order quantity (EOQ) model is to minimize the total costs associated with the carrying and ordering costs. As the amount ordered gets larger, average inventory increases and so do carrying costs. For example, if annual demand for an inventory item is 26,000 units, and a firm orders 500 each time, the firm will place 52 $[26,000/500]$ orders per year. This order frequency gives the organization an average inventory of 250 $[500/2]$ units. If the order quantity is increased to 2,000 units, fewer orders (13) $[26,000/2,000]$ will be placed. However, average inventory on hand will increase to 1,000 $[2,000/2]$ units. Thus, as holding costs go up, ordering costs go down, and vice versa. The optimum economic order quantity is reached at the lowest point on the total cost curve. That’s the point at which ordering costs equal carrying costs—or the economic order quantity (see point Q in Exhibit QM–7).

To compute this optimal order quantity, you need the following data: forecasted demand for the item during the period (D); the cost of placing each order (OC); the value or purchase price of the item (V); and the carrying cost (expressed as a percentage) of maintaining the total inventory (CC). Given these data, the formula for EOQ is as follows:

Let’s work an example of determining the EOQ. Take, for example, Barnes Electronics, a retailer of high-quality sound and video equipment. The owner, Sam Barnes, wishes to determine the company’s economic order quantities of high-quality sound and video equipment. The item in question is a Sony compact voice recorder. Barnes forecasts sales of 4,000 units a year. He believes that the cost for the sound system should be $50. Estimated costs of placing an order for these systems are $35 per order and annual insurance, taxes, and other carrying costs at 20 percent of the recorder’s value. Using the EOQ formula, and the preceding information, he can calculate the EOQ as follows:

The inventory model suggests that it’s most economical to order in quantities or lots of approximately 168 recorders. Stated differently, Barnes should order about $24[4,000/168]$ times a year. However, what would happen if the supplier offers Barnes a 5 percent discount on purchases if he buys in minimum quantities of 250 units? Should he now purchase in quantities of 168 or 250? Without the discount, and ordering 168 each time, the annual costs for these recorders would be as follows:

With the 5 percent discount for ordering 250 units, the item cost $[\$50\times (\$50\times .05)]$ would be $47.5.

The annual inventory costs would be as follows:

These calculations suggest to Barnes that he should take advantage of the 5 percent discount. Even though he now has to stock larger quantities, the annual savings amounts to nearly $10,000. A word of caution, however, needs to be added. The EOQ model assumes that demand and lead times are known and constant. If these conditions can’t be met, the model shouldn’t be used. For example, it generally shouldn’t be used for manufactured component inventory because the components are taken out of stock all at once, in lumps, or odd lots, rather than at a constant rate. Does this caveat mean that the EOQ model is useless when demand is variable? No. The model can still be of some use in demonstrating trade-offs in costs and the need to control lot sizes. However, more sophisticated lot sizing models are available for handling demand and special situations. The mathematics for EOQ, like the mathematics for queuing theory, go far beyond the scope of this text.