Introduction to Management Science Authors: Taylor B.W. Published year: 2006 Pages: 134-135/358

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## Chapter 10. Nonlinear Programming

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More attention has been devoted to linear programming in this text than to any other single topic. It is a very versatile technique that can be and has been applied to a wide variety of problems. Besides chapters devoted specifically to linear programming models and applications, we have also presented several variations of linear programming, integer and goal programming, and unique applications of linear programming for transportation and assignment problems. In all these cases, all the objective functions and constraints were linear; that is, they formed a line or plane in space. However, many realistic business problems have relationships that can be modeled only with nonlinear functions. When problems fit the general linear programming format but include nonlinear functions, they are referred to as nonlinear programming problems.

Nonlinear programming problems are given a separate name because they are solved in a different manner than are linear programming problems. In fact, their solution is considerably more complex than that of linear programming problems, and it is often difficult, if not impossible , to determine an optimal solution, even for a relatively small problem. In linear programming problems, solutions are found at the intersections of lines or planes, and though there may be a very large number of possible solution points, the number is finite, and a solution can eventually be found. However, in nonlinear programming there may be no intersection or corner points; instead, the solution space can be an undulating line or surface, which includes virtually an infinite number of points. For a realistic problem, the solution space may be like a mountain range, with many peaks and valleys, and the maximum or minimum solution point could be at the top of any peak or at the bottom of any valley. What is difficult in nonlinear programming is determining if the point at the top of a peak is just the highest point in the immediate area (called a local optimal , in calculus terms) or the highest point of all (called the global optimal ).

The solution techniques for nonlinear programming problems generally involve searching the solution surface for peaks or valleysthat is, high points or low points. The problem encountered by these methods is that they sometimes have trouble determining whether the high point they have identified is just a local optimal solution or the global optimal solution. Thus, finding a solution is often difficult and can involve very complex mathematics that are beyond the scope of this text.

In this chapter we present the basic structure of nonlinear programming problems and use Excel to solve simple models.

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### Nonlinear Profit Analysis

We begin our presentation of nonlinear programming models by determining the optimal value for a single nonlinear function. To demonstrate the solution procedure, we will use a profit function based on break-even analysis. In Chapter 1 we used break-even analysis to begin our study of model building, so it seems appropriate that we return to this basic model to complete our study of model building.

Recall that in break-even analysis the profit function, Z , is formulated as

Z = vp c f vc v

where

v = sales volume (i.e., demand)

p = price

c f = fixed cost

c v = variable cost

One important but somewhat unrealistic assumption of this break-even model is that volume, or demand, is independent of price (i.e., volume remains constant, regardless of the price of the product). It would be more realistic for the demand to vary as price increased or decreased. For our Western Clothing Company example from Chapter 1, let us suppose that the dependency of demand on price is defined by the following linear function:

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v = 1,500 24.6 p

This linear relationship is illustrated in Figure 10.1. The figure illustrates the fact that as price increases , demand decreases, up to a particular price level (.98) that will result in no sales volume.

##### Figure 10.1. Linear relationship of volume to price

Now we will insert our new relationship for volume ( v ) into our original profit equation:

Substituting values for fixed cost ( c f = ,000) and variable cost ( c v = ) into this new profit function results in the following equation:

Because of the squared term , this equation for profit is now a nonlinear, or quadratic, function that relates profit to price, as shown in Figure 10.2.

##### Figure 10.2. The nonlinear profit function

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In Figure 10.2, the greatest profit will occur at the point where the profit curve is at its highest. At that point the slope of the curve will equal zero, as shown in Figure 10.3.

##### Figure 10.3. Maximum profit for the profit function

In calculus the slope of a curve at any point is equal to the derivative of the mathematical function that defines the curve. The derivative of our profit function is determined as follows :

The slope of a curve at any point is equal to the derivative of the curve's function .

Given this derivative, the slope of the profit curve at its highest point is defined by the following relationship:

0 = 1,696.8 49.2 p

The slope of a curve at its highest point equals zero .

Now we can solve this relationship for the optimal price, p , which will maximize total profit:

The optimal volume of denim jeans to produce is computed by substituting this price into our previously developed linear relationship for volume:

The maximum total profit is computed as follows:

The maximum profit, optimal price, and optimal volume are shown graphically in Figure 10.4.

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##### Figure 10.4. Maximum profit, optimal price, and optimal volume

An important concept we have yet to mention is that by extending the break-even model this way, we have converted it into an optimization model. In other words, we are now able to maximize an objective function (profit) by determining the optimal value of a variable (price). This is exactly what we did in linear programming when we determined the values of decision variables that optimized an objective function. The use of calculus to find optimal values for variables is often referred to as classical optimization .

Classical optimization is the use of calculus to determine the optimal value of a variable .

 Introduction to Management Science Authors: Taylor B.W. Published year: 2006 Pages: 134-135/358