nonlinear programming
- 2. Learning Objectives Understand and solve Nonlinear Programming Problems Understand the difference between linear programming and nonlinear programming Formulate Nonlinear Programming Problems and solve using Excel
- 3. What is Nonlinear? Nonlinear Programming has the same format as a linear programming model but the objective function or constraints, or both, are nonlinear functions Nonlinear Programming problems are given a separate name because they are solved in a different manner than linear programming problems When problems fit the general linear programming format but include nonlinear functions, they are referred to as nonlinear programming problems.
- 4. 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
- 5. Nonlinear Programming In a realistic problem, the solution space may be like a mountain range, with many peaks and valleys, and the maximum or minimum solution could be at the top of any peak or at the bottom of any valley. 2018 2019 2020 2021 2022 125 100 75 50 25 0
- 6. 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). 2018 2019 2020 2021 2022 125 100 75 50 25 0
- 7. The Problem 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 2018 2019 2020 2021 2022 125 100 75 50 25 0
- 8. Nonlinear Profit Analysis Profit Function with no constraints Objective function with Multiple Constraints Profit Function with objective function and 1 constraint
- 10. To demonstrate the solution procedure, we will use a profit function based on break-even analysis Z = vp - Cf - vCv Recall that in break-even analysis the profit function, Z, is formulated as where v = sales volume (i.e., demand) p = Price Cf = Fixed Cost Cv = Variable Cost
- 11. Item 1 Item 2 Item 3 Item 4 40 30 20 10 0 One important but somewhat unrealistic assumption of this break-even model is that the volume, or demand, is independent of the price (i.e., volume remains constant, regardless of the price of the product).
- 12. 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: v = 1,500 - 24.6p The figure illustrated the fact that as price increases, demand decreases, up to a particular price level ($60.98) that will result in no sales volume.
- 13. Nonlinear Profit Analysis The linear relationship, v=1,500 - 24.6p, is illustrated in Figure 10.1
- 14. Z = vp - Cf - vCv Now we will insert our new relationship for volume (v) into our original profit equation Z = (1,500 - 24.6p)p - Cf - (1,500 - 24.6p)Cv = 1,500p - 24.6p²- Cf - 1,500Cv - 24.6pcv
- 15. Z = 1,500p - 24.6p² - 10,000 - 1,500(8) + 24.6p(8) Substituting values for fixed cost (Cf = $10,000) and variable cost (Cv = $8) into this new profit function results in the following equation: Z = 1,696.8p - 24.6p² - 22,000
- 16. 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 The greatest profit will occur at the point where the profit curve is at its highest.
- 17. At the point the slope of the curve will equal to zero, as shown in Figure 10.3.
- 18. Derivative The slope of the curve for the given function is called the derivative of a function.
- 19. Z = 1,696.8p - 24.6p² - 22,000 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:
- 20. 0 = 1,696.8 - 49.2p Given this derivative, the slope of the profit curve at its highest point is defined by the following relationship:
- 21. 0 = 1,696.8 - 49.2p 49.2p = 1,696.8 p = 1,696.8 / 49.2 p= $34.49 Now we can solve this relationship for the optimal price, p, which will maximize total profit:
- 22. The optimal volume of denim jeans to produce is computed by substituting this price into our previously developed linear relationship for volume: v = 1,500 - 24.6p = 1,500 - 24.6 (34.49) = 651.6 pair of jeans
- 23. The maximum total profit is computed as follows: Z = 1,696.8p - 24.6p² - 22,000 = 1,696.8 (34.49) - 24.6 (34.49)² - 22,000 = $ 7,259.45
- 24. The maximum profit, optimal price, and optimal volume are shown graphically in Figure 10.4
- 25. 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.
- 26. The use of calculus to find optimal values for variables is often referred to as classical optimization.
- 28. In the preceding section, the profit analysis model was developed as an extension of the breakeven model. Recall that the total profit function was Z = vp - Cf - vCv where v = volume p = price Cf = Fixed cost Cv = Variable cost and the demand function (i.e., volume as a function of price) was v = 1,500 - 24.6p
- 29. By substituting this demand function into our total profit equation, we developed a nonlinear function: Then, by substituting values for ($10,000) and ($8) into this function, we obtained We then differentiated this function, set it equal to zero, and solved for the value of p ($34.49), which corresponded to the maximum point on the profit curve (where the slope equaled zero). Z= 1,500p - 24.6p²- Cf - 1,500Cv - 24.6pcv Z = 1,696.8p - 24.6p² - 22,000
- 30. This type of model is referred to as an unconstrained optimization model. If we add one or more constraints to this model, it becomes a constrained optimization model. A constrained optimization model is more commonly referred to as a nonlinear programming model.
- 31. Now we will transform this unconstrained optimization model into a nonlinear programming model by adding the constraint p ≤ $20 In other words, because of market conditions, we are restricting the price to a maximum of $20. This constraint results in a feasible solution space, as shown in Figure 10.6.
- 33. The difficulty with nonlinear programming is that the solution is not always on the boundary of the feasible solution space formed by the constraint. For example, consider the addition of the following constraint to our original nonlinear objective function: p ≤ $40
- 34. This constraint also creates a feasible solution space, as shown in Figure 10.7. Point C represents a greater profit than point B, and it is also in the feasible solution space.