Graphical Methods in Linear Programming

Transcription

1 Appendix 2 Graphical Methods in Linear Programming We can use graphical methods to solve linear optimization problems involving two variables. When there are two variables in the problem, we can refer to them as x 1 and x 2, and we can do most of the analysis on a two-dimensional graph. Although the graphical approach does not generalize to a large number of variables, the basic concepts of linear programming can all be demonstrated in the two-variable context. When we run into questions about more complicated problems, we can ask, what would this mean for the two-variable problem? Then, we can look for answers in the two-variable case, using graphs. Another advantage of the graphical approach is its visual nature. Graphical methods provide us with a picture to go with the algebra of linear programming, and the picture can anchor our understanding of basic definitions and possibilities. For these reasons, the graphical approach provides useful background for working with linear programming concepts. A2.1. AN EXAMPLE Consider the planning and scheduling problem facing a manufacturer of microwave ovens with two models in its line the standard and the deluxe. Each oven is assembled from component parts and subassemblies that are produced in the mechanical and electronics departments. The following table shows the number of production hours per oven required in each department and the capacities of the three production departments, in monthly hours. Standard (h/oven) Deluxe (h/oven) Capacity (h/mo) Assembly Department Mechanical Department Electronics Department Optimization Modeling with Spreadsheets, Second Edition. Kenneth R. Baker # 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 377

2 378 Appendix 2 Graphical Methods in Linear Programming The sales department believes that there will be demand for as many ovens as the company can produce. The accounting department has determined that the variable profit contributions are $50 for each standard and $40 for each deluxe. The problem is to determine a production plan to maximize monthly profit contribution. Our first step in analyzing this problem is to express it algebraically. The problem of devising an output plan boils down to finding the best number of standard ovens and deluxe ovens for the firm to produce in the coming month. Thus, we let x 1 ¼ number of standard ovens x 2 ¼ number of deluxe ovens Once we determine the values of x 1 and x 2, the problem will be solved. Furthermore, the criterion is to maximize the profit contribution generated by our plan. In particular, we can write our objective function as Maximize z ¼ 50x 1 þ 40x 2 where z represents the value of the objective function. Having specified decision variables and an objective function, we turn our attention to the constraints of the problem, the limited capacities in the assembly, mechanical, and electronics departments. In assembly, the number of hours consumed by a production schedule cannot exceed the 560 hours available. We can write this requirement algebraically as 4x 1 þ 4x Similarly, for the mechanical and electronics departments, we require 3x 1 þ 2x x 1 þ 4x In standard form, an algebraic statement of our full model is Maximize z ¼ 50x 1 þ 40x 2 subject to 4x 1 þ 4x (A) 3x 1 þ 2x (M) 2x 1 þ 4x (E) Finally, implicit in our definition of the two decision variables is the requirement that they must both remain nonnegative (x 1 0 and x 2 0). We begin the graphical analysis with the constraints. For a graphical approach, we can work with equations more readily than inequalities, so we consider the equations corresponding to each of the constraints in turn. For the assembly department constraint (A), the line 4x 1 þ 4x 2 ¼ 560 defines the locus of all points at which the department is fully utilized. That is, the line represents the set of product mix combinations

3 A2.1. An Example 379 (x 1, x 2 ) that consume all 560 available hours in assembly. To plot the line, note that the x 1 intercept is 140 (obtained by setting x 2 ¼ 0 and solving the equation for x 1 ). Similarly, the x 2 intercept is 140. Plotting the two intercepts, (140, 0) and (0, 140) on a graph, and connecting the two points with a straight line, we construct the plot shown in Figure A2.1, where the label A on the graph is used to associate this line with the assembly hours constraint. The line plotted in Figure A2.1 represents all combinations (x 1, x 2 ) that consume exactly 560 hours of assembly time. Our model, however, looks for combinations that consume no more than 560 hours. Combinations that consume fewer that 560 hours are also admissible, and these correspond to points (x 1, x 2 ) that lie below the line. In fact, if we consider only the points that are admissible in the assembly constraint and that also meet the nonnegativity requirements, then we are left with the shaded triangle shown in Figure A2.1. Next, we plot a line corresponding to mechanical department hours, or 3x 1 þ 2x 2 ¼ 400, as shown in Figure A2.2 with the label M. Points on this line represent combinations of standard and deluxe ovens that consume exactly 400 hours of mechanical time, and points below the line consume fewer than 400 hours. The line corresponding to the mechanical department has a slope of 3/2, in comparison to the assembly department line, which has a slope of 1. Only points that lie below both constraint lines (and in the nonnegative region) are admissible decisions, as indicated by the shading. Finally, we plot a line corresponding to the electronic department limit, 2x 1 þ 4x 2 ¼ 400, shown in Figure A2.3 with the label E. This line has a slope of 1/2, and again we shade the region that lies below all three of the lines, as shown in figure. The shaded region now represents the set of all points that are admissible decisions: They satisfy all of the constraints in our problem. The shaded area in Figure A2.3 is called the feasible region. It is a five-sided polygon containing, along its boundary or inside, all of the points that correspond to Figure A2.1. Sketch of first constraint.

4 380 Appendix 2 Graphical Methods in Linear Programming Figure A2.2. Sketch of second constraint. feasible decisions in our model. Our next step is to find the best value of the objective function that can be achieved by a point in this feasible region. To pursue this search, consider what happens when we set the objective function (z) equal to a fixed value. For example, suppose z ¼ Then all points (x 1, x 2 ) that achieve a value of 2000 in the objective function lie on the line 50x 1 þ 40x 2 ¼ Furthermore, all points that achieve a value of 2000 and that are feasible in the constraints lie along this line and within the feasible region. Consider a second line corresponding to z ¼ Like the first objective function line, this one has a slope of 5/4, so it is parallel to the first. However, it has different intercepts and lies above and to the right of the first line, as shown in Figure A2.4. From these two lines, we can imagine an entire family of lines, each with a slope of 5/4 and each corresponding Figure A2.3. Sketch of third constraint.

5 A2.1. An Example 381 Figure A2.4. Sketch of objective function lines. to a particular value of z. Relatively larger values of z correspond to lines in this family that lie farther above and to the right. We wish to find the largest value of z attainable in this family and within the feasible region. A look at the figure indicates that this value occurs at the intersection of the assembly and mechanical constraints. We can make this result more precise by solving for the point at which the assembly and mechanical constraints intersect. This point is (120, 20), as shown in Figure A2.5, corresponding to a product mix of 120 standard ovens and 20 deluxe ovens. The corresponding profit total is $6800, corresponding to the objective function line for z ¼ 6800, labeled OF in the figure. Using graphical methods, we have found the best value of the objective function and the decisions that generate it. Figure A2.5. Sketch of optimal point.

6 382 Appendix 2 Graphical Methods in Linear Programming A2.2. GENERALITIES The oven-manufacturing problem was merely an example, but it does illustrate the principles of graphical solution methods for optimization. In the example, all constraints are of the LT (less-than) variety. This means that points on the graph are feasible if they lie on or below the corresponding constraint line. On the other hand, had we encountered GT (greater-than) constraints, the feasible points would have been found on or above the corresponding constraint line. The other possibility, EQ (equal-to) constraints, would force us to consider only points lying directly on the constraint line. In our example, the criterion was to maximize the objective function. By sketching the implications for two lines, each corresponding to a particular value of the objective function (sometimes called an iso-value line), we can begin to see a family of related objective function lines, leading to a maximum feasible value at one corner of the feasible region. Our objective function contained all positive coefficients, so the process of maximization led us to lines ever higher and to the right on our graph. Had we been interested in minimization (of a function with all positive coefficients), we would have been led to lines lower and to the left of a given starting point. For other combinations involving negative coefficients, the idea is to plot the graph of two or three iso-value lines, in order to see where on the graph the optimum will ultimately be found. An examination of iso-value lines could reveal that there is no limit, in some direction, to the value of the objective function. This would be the case if we were analyzing an unbounded problem. In other circumstances, attempting to delineate the feasible region itself will reveal an infeasible problem, in which the constraints are mutually contradictory. These two exceptional cases can therefore be identified while carrying out the graphical analysis. The graphical method is valuable because it produces a picture of the optimization process. That picture may make it easier to interpret what occurs during an optimization procedure. However, the graphical method is more difficult when there are three dimensions and impossible when there are more than three dimensions, so we can use it only for relatively simple cases. Nevertheless, two-dimensional examples illustrate most of the principles of linear programming. As an example, suppose we consider well-posed linear programs in which the feasible region exists and in which the objective function is not unbounded. The theory tells us that an optimum can be found at one of the corners of the feasible region. This property is very useful, because it means that we don t have to search for an optimal point in the interior of the region (an area that contains an infinite number of points.) Instead, we can limit our search to the boundary, and just to the corner points on that boundary, which are finite in number. This is what most linear programming codes do: They search systematically among the corner points on the boundary of the feasible region, stopping only when an optimum has been located. A second theoretical result tells us that we can identify an optimal corner point by showing that its objective function value is better than those of its neighboring corner points. Each of the corner points in our graph has two neighbors, and either of these can be reached by moving along one of the boundaries of the feasible region. When we

7 A2.2. Generalities 383 start our search at one of the corner points, only two things can happen. Either the objective function is better than at both neighboring corners (in which case, we have found the optimum), or one of the neighbors has a better value. In the latter case, we move to that point and then evaluate the neighboring possibilities from the new location. In linear programming problems, we are essentially guaranteed that this search procedure ultimately leads to the optimum. Indeed, this approach lies at the heart of the simplex algorithm, which is the most popular method in use today for finding solutions to linear programming problems. (For an algebraic glimpse of the simplex method, see Appendix 3.)