## Figure

Optimal Input Combination with Limited Resources

Given limited resources, output is maximized at point R because this point lies on the higher isoquant that intersects the feasible space.

Units of L employed

Optimal Input Combination with Limited Resources

Given limited resources, output is maximized at point R because this point lies on the higher isoquant that intersects the feasible space.

Units of L employed

feasible space

Graphical region that is both technically and economically feasible and includes the optimal solution

Production possibilities for this problem are determined by noting that, in addition to limitations on inputs L and K, the firm must operate within the area bounded by production process rays A and D. Combining production possibilities with input constraints restricts the firm to operation within the shaded area on 0PRS in Figure 9.4. This area is known as the feasible space in the programming problem. Any point within this space combines L and K in a technically feasible ratio without exceeding availability limits on L and K. Because the firm is trying to maximize production of Q subject to constraints on the use of L and K, it should operate at the feasible space point that touches the highest possible isoquant. This is point R in Figure 9.4, where Q = 3.

Although it is possible to solve the foregoing problem by using carefully drawn graphs, it is typically easier to combine graphic analysis with analytical techniques to obtain accurate solutions efficiently. For example, consider Figure 9.4 again. Even if the isoquant for Q = 3 were not drawn, it would be apparent from the slopes of the isoquants for 2 or 4 units of output that the optimal solution to the problem must be at point R. It is obvious from the graph that maximum production is obtained by operating at the point where both inputs are fully employed. Because R lies between production processes C and D, the output-maximizing input combination uses only those two production processes. All 20 units of L and 11 units of K will be employed, because point R lies at the intersection of these two input constraints.

Using this information from the graph, it is possible to quickly and easily solve for the optimal quantities to be produced using processes C and D. Recall that each unit of output produced using process C requires 7.5 units of L. Thus, the total L required in process C equals 7.5 X QC. Similarly, each unit produced using process D requires 5 units of L, so the total L used in process D equals 5 X QD. At point R, 20 units of L are being used in processes C and D together, and the following must hold:

A similar relation can be developed for the use of K. Each unit of output produced from process C requires 3 units of K, whereas process D uses 5 units of K to produce each unit of output. The total use of K in processes C and D equals 11 units at point R, so

Equations 9.1 and 9.2 both must hold at point R. Output quantities from processes C and D at that location are determined by solving these equations simultaneously. Subtracting Equation 9.2 from Equation 9.1 to eliminate the variable QD isolates the solution for QC:

7.5Qc + 5Qd = 20 minus 3.0QC + 5QD = 11 4.5Qc = 9 Qc = 2

Substituting 2 for QC in Equation 9.2 determines output from process D:

Total output at point R is 3 units, composed of 2 units from process C and 1 unit from process D.

The combination of graphic and analytic techniques allows one to obtain precise linear programming solutions with relative ease.

MANAGERIAL APPLICATION 9.2

Managers of small to medium-sized companies often plug hypothetical financial and operating data into spreadsheet software programs and then recalculate profit figures to see how various changes might affect the bottom line. A major problem with this popular "What if?" approach to decision analysis is the haphazard way in which various alternatives are considered. Dozens of time-consuming recalculations are often necessary before suggestions emerge that lead to a clear improvement in operating efficiency. Even then, managers have no assurance that more profitable or cost-efficient decision alternatives are not available.

The frustrations of "What if?" analysis are sure to become a thing of the past with the increasing popularity of new Solver LP programs, included as a basic feature of spreadsheet software, like Microsoft Excel. Solver LP tools are capable of solving all but the toughest problems and are extremely user-friendly for those with little LP training or computer experience. More powerful, but still easy to use, LP software is provided by Lindo Systems,

Inc. Lindo is the leading supplier of LP optimization software to business, government, and academia. Lindo software is used to provide critical answers to thousands of businesses, including over one-half of the Fortune 500. What'sBest! is an innovative LP program and a popular modeling tool for business problems. First released in 1985, What'sBest! soon became the industry leader, specializing in tackling large-scale, real-world problems. Like more basic Solver LP programs, What'sBest! software is designed for the PC environment.

It is stunning to note how quickly inexpensive, powerful, and easy-to-use LP software for the PC has come forth. As new generations of user-friendly LP software emerge, appreciation of the value of the LP technique as a practical and powerful tool for decision analysis will continue to flourish as a powerful and practical tool for managerial decision making.

See: Home page information for Lindo, Lingo, and What'sBest! software can be found on the Internet (http://www.lindo.com).