EMGT 5412 Operations Management Science Nonlinear Programming: Introduction Dincer Konur Engineering Management and Systems Engineering 1

Outline Introduction to Nonlinear functions Two Non-linear models –Economic Order Quantity –Simple Linear Regression Cases from the Book –Excel Solver and Nonlinear Optimization –Decreasing marginal returns –Separable programming Chapter 8 2

Outline Introduction to Nonlinear functions Two Non-linear models –Economic Order Quantity –Simple Linear Regression Cases from the Book –Excel Solver and Nonlinear Optimization –Decreasing marginal returns –Separable programming 3

Introduction to Nonlinear Functions In linear programming –All of the functions are linear –Constant marginal returns Constant per unit profit Constant per unit cost Constant per unit resource usage In non-linear programming –Functions are nonlinear –Varying marginal returns Proportionality of the linear functions 4

Introduction to Nonlinear Functions Proportionality Assumption of Linear Programming –The contribution of each activity to the value of the objective function is proportional to the level of the activity. In other words, the term in the objective function involving this activity consists of a coefficient times the decision variable. Nonlinear programming problems arise when any activity has a non-proportional relationship where the contribution of the activity to the measure of performance is not proportional to the level of the activity. 5

Introduction to Nonlinear Functions Linear relations 6

Introduction to Nonlinear Functions Nonlinear relations Decreasing Marginal Returns Piecewise Linear with Decreasing Marginal Returns 7

Introduction to Nonlinear Functions Nonlinear relations Decreasing Marginal Returns Except for Discontinuities Increasing Marginal Returns 8

Introduction to Nonlinear Functions We cannot use Simplex method We may not have those nice corner points as in the case of linear programming 9

Introduction to Nonlinear Functions Two important class of nonlinear functions: –Convex Increasing marginal returns –Concave Decreasing marginal returns 10

Introduction to Nonlinear Functions Convex vs. Concave 11

Outline Introduction to Nonlinear functions Two Non-linear models –Economic Order Quantity –Simple Linear Regression Cases from the Book –Excel Solver and Nonlinear Optimization –Decreasing marginal returns –Separable programming 12

Economic Order Quantity Model A very commonly used inventory control model –Check your smart phones: there is an EOQ application Economic Order Quantity Model: –You know how many items you will sell in a year –You know the cost of holding one item in inventory for one year –You want to determine how many to ship each time when you replenish your inventory –There is a fixed cost with each shipment 13

Economic Order Quantity Model Economic Order Quantity Model: –What is the order quantity that will minimize your costs per unit time? 14

Economic Order Quantity Model The inventory level… 15

Economic Order Quantity Model The optimization problem is… 16

Economic Order Quantity Model G(Q) is a convex function –And we have no constraints in our problem When you minimize a convex function with no constraints, the optimal solution will be given by the point where the first order derivative is 0 –First Order Optimality Condition: The solution to {min G(Q)}, when G(Q) is convex, is achieved at Q* such that dG(Q)/dQ = 0 17

Economic Order Quantity Model Then the Economic Order Quantity –The optimal solution is 18

Simple Linear Regression Linear Regression: –You have a set of observations (data points) –And you want to use these data to explain the relation between two variables One variable is explanatory variable X The other is the dependent variable Y You believe that you have –Using you observations, find the values of B 0 and B 1 such that the total squared errors are minimized. 19

Simple Linear Regression Least squares method 20

Simple Linear Regression The optimization –F is convex… then the optimal solution 21

Outline Introduction to Nonlinear functions Two Non-linear models –Economic Order Quantity –Simple Linear Regression Cases from the Book –Excel Solver and Nonlinear Optimization –Decreasing marginal returns –Separable programming 22

Spreadsheet Nonlinear Optimization Consider the following model in algebraic form: Maximize Profit = 0.5x 5 – 6x x 3 – 39x x subject to x ≤ 5 x ≥ 0 Nonlinear function 23

Spreadsheet Nonlinear Optimization Profit function of the model 24

Spreadsheet Nonlinear Optimization Starting with x=0 Starting with x=3 Starting with x=4.7 25

Spreadsheet Nonlinear Optimization Local maximum vs. global maximum Local maximum Global maximum 26

Spreadsheet Nonlinear Optimization Excel Solver’s nonlinear method –Climbing the mountain for maximization until it reaches the first peak or constraint –Going down for minimization until it reaches the first bottom or constraint –No guarantee that we will find the global optimum UNLESS!!!!! –We are maximizing a concave function –We are minimizing a convex function These functions have single peak and single bottom 27

Decreasing Marginal Returns Wyndor Glass Co. Case –D: number of doors –W: number of windows –Constraints: D ≤ 4 (Plant 1 availability) 2W ≤ 12 (Plant 2 availability) 3D + 2W ≤ 18(Plant 3 availability) 28

Decreasing Marginal Returns $375 is the gross profit for one door $25D 2 is the marketing cost for D doors 29

Decreasing Marginal Returns 30

Decreasing Marginal Returns 31

Decreasing Marginal Returns Spreadsheet model –Global optimum… 32

Decreasing Marginal Returns 33

Separable Programming Wyndor Glass. Co with overtime –Wyndor Glass has accepted a special order for hand-crafted goods to be made in plants 1 and 2 throughout the next four months. –Filling this order will require borrowing certain employees from the work crews of regular products. –The remaining workers will need to work overtime to utilize the full production capacity of each plant’s machinery for the regular products. –The original constraints of Hours Used ≤ Hours Available are still valid. However, the objective function will need to be modified because of the additional cost of using overtime work. –In particular, because of the additional cost, the profit per unit will be reduced for those units that require overtime. 34

Separable Programming We have the following data Maximum Weekly Production Profit per Unit Produced Product Regular TimeOvertimeTotal Regular TimeOvertime Doors314$300$200 Windows

Separable Programming 36

Separable Programming For each activity that violates the proportionality assumption, separate its profit graph into parts, with a line segment in each part. Then, instead of using a single decision variable to represent the level of each such activity, introduce a separate new decision variable for each line segment on that activity’s profit graph. Since the proportionality assumption holds for these new decision variables, formulate a linear programming model in terms of these variables. For the Wyndor problem, these new decision variables are –D R = Number of doors produced per week on regular time –D O = Number of doors produced per week on overtime –W R = Number of windows produced per week on regular time W O = Number of windows produced per week on overtime 37

Separable Programming Mathematical formulation Maximize profit=300D R +200D O +500W R + 100W O subject to D R <=3, D O <=1, W R <=3, W O <=3 D R +D O <=4 2(W R + W O )<=12 3(D R +D O )+ 2(W R + W O )<=18 D R >=0, D O >=0, W R >=0, W O >=0 NOTE: This will work since we have decreasing marginal profit… We know that if D R <=3, then D O =0 38

Separable Programming 39

Separable Programming Read Wyndor problem with both overtime costs and nonlinear marketing costs (pg ) –Converting decreasing marginal profits to piecewise decreasing marginal profits (this is approximation) 40

Other solution methods KKT conditions for convex models Dynamic programming Heuristic methods –Genetic algorithms –Grid search 41

Further study Read Chapter 8 Practice problems.. –8.9, 8.11, 8.14, 8.16 Further reading… –Portfolio selection case in Chapter 8 (pg ) 42