© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Linear Optimization Models Chapter 11

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Introduction Optimization problems: Can be used to support and improve managerial decision making Maximize or minimize some function, called the objective function, and have a set of restrictions known as constraints Can be linear or nonlinear 2

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Introduction Typical applications: A manufacturer wants to develop a production schedule and an inventory policy that will satisfy demand in future periods and at the same time minimize the total production and inventory costs A financial analyst would like to establish an investment portfolio from a variety of stock and bond investment alternatives that maximizes the return on investment A marketing manager wants to determine how best to allocate a fixed advertising budget among alternative advertising media such as web, radio, television, newspaper, and magazine that maximizes advertising effectiveness A company had warehouses in a number of locations. Given specific customer demands, the company would like to determine how much each warehouse should ship to each customer so that total transportation costs are minimized 3

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Introduction Linear optimization models are also known as linear programs Linear programming: A problem-solving approach developed to help managers make better decisions Numerous applications in today’s competitive business environment For instance, GE Capital uses linear programming to help determine optimal lease structuring 4

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem Problem Formulation Mathematical Model for the Par, Inc. Problem

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem Illustration: Par, Inc. - A small manufacturer of golf equipment and supplies Management has decided to move into the market for medium- and high-priced golf bags Par’s distributor to buy all the produced bags by the end of third month 6

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem Operations involved in manufacturing a golf bag: Cutting and dyeing the material Sewing Finishing (inserting umbrella holder, club separators, etc.) Inspection and packaging Table 11.1: Production Requirements Per Golf Bag 7

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem Estimated total time available for the next three months to perform different operations: Required profit contribution: Standard bag: $10/unit Deluxe bag: $9/unit DepartmentNumber of hours Cutting and Dyeing630 Sewing600 Finishing708 Inspection and Packaging135 8

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem Develop a mathematical model of the Par, Inc. problem to determine the number of standard bags and the number of deluxe bags to produce to maximize total profit contribution Problem Formulation Problem formulation or modeling: Process of translating the verbal statement of a problem into a mathematical statement (or model) 9

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem General guidelines for problem formulation: Understand the problem thoroughly Describe the objective Describe each constraint Define the decision variables Write the objective in terms of the decision variables Write the constraints in terms of the decision variables

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem Describe each constraint Constraint 1Number of hours of cutting and dyeing time used must be less than or equal to the number of hours of cutting and dyeing time available. 2Number of hours of sewing time used must be less than or equal to the number of hours of sewing time available. 3Number of hours of finishing time used must be less than or equal to the number of hours of finishing time available. 4Number of hours of inspection and packaging time used must be less than or equal to the number of hours of inspection and packaging time available. 11

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem Define the decision variables S = number of standard bags D = number of deluxe bags Write the objective in terms of the decision variables If Par makes $10 for every standard and $9 for every deluxe bag, Total profit contribution = 10S + 9D = Objective function Objective: Max 10S + 9D 12

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem 13

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem 14

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem 15

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Maximization Problem Mathematical Model for the Par, Inc. Problem (contd.) This is a linear programming model (or linear program) because the objective function and all constraint functions are linear functions of the decision variables Linear function: Mathematical function in which each variable appears in a separate term and is raised to the first power 16

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Solving the Par, Inc. Problem The Geometry of the Par, Inc. Problem Solving Linear Programs with Excel Solver

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Solving the Par, Inc. Problem To find the optimal solution to the problem modeled as a linear program: The optimal solution must have the highest objective function value The optimal solution must be a feasible solution—a setting of the decision variables that satisfies all of the constraints of the problem Search over the feasible region—a set of all possible solutions Find the solution that gives the best objective function value 18

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Solving the Par, Inc. Problem The Geometry of the Par, Inc. Problem When only two decision variables, the functions of variables are linear If constraints are inequalities, the constraint cuts the space in two The line and the area on one side of the line is the space the satisfies that constraint These subregions are called half spaces The intersection of the half spaces make up the feasible region

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.1: Feasible Region for the Par, Inc. Problem 20

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.2: The Optimal Solution to the Par, Inc. Problem 21

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Solving the Par, Inc. Problem Based on the geometry of Figure 11.2, to solve a linear optimization problem we only have to search the extreme points of the feasible region to find the optimal solution Extreme points are found where constraints intersect on the boundary of the feasible region

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Solving the Par, Inc. Problem Solving Linear Programs with Excel Solver The first step is to construct the relevant what-if model A what-if model for optimization allows the user to try different values of the decision variables and see: Whether that trial solution is feasible The value of the objective function for that trial solution Convey to Excel Solver the structure of the linear optimization model 23

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.3: What-If Spreadsheet Model for Par, Inc. 24

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.4: Solver Dialog Box and Solution to the Par, Inc. Problem 25

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Solving the Par, Inc. Problem The optimal solution: To make 540 Standard bags and 252 Deluxe bags for a profit of $7,668 Using all the cutting and dyeing time as well as all finishing time, from cells B19:B22 compared to C19:C22 The results are consistent with the results obtained in Figures 11.1 and

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.5: The Solver Answer Report for the Par, Inc. Problem 27

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Solving the Par, Inc. Problem A binding constraint is one that holds as an equality at the optimal solution The slack value for each less-than-or-equal-to constraint indicates the difference between the left-hand and right-hand values for a constraint By adding a nonnegative slack variable, we can make the constraint equality

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Minimization Problem Problem Formulation Solution for the M&D Chemicals Problem

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Minimization Problem Illustration: Production requirements for M&D Chemicals: The combined production for products A and B must total at least 350 gallons Separately a major customer’s order for 125 gallons of product A must also be satisfied Processing time: Product A: 2 hours/gallon Product B: 1 hour/gallon For the coming month, 600 hours of processing time are available 30

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Minimization Problem Production cost: Product A: $2/gallon; Product B: $3/gallon Objective: Minimizing the total production cost Problem Formulation To find the minimum-cost production schedule: Define the decision variables and the objective function Let A = number of gallons of product A B = number of gallons of product B Objective function = 2A + 3B 31

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Simple Minimization Problem Linear program for the M&D Chemicals problem: Min 2A + 3B s.t. 1A ≥ 125 Demand for product A 1A + 1B ≥ 350 Total production 2A + 1B ≤ 600 Processing time A, B ≥ 0 A surplus variable tells how much over the right-hand side the left- hand side of a greater-than-or-equal-to constraint is for a solution 32

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.6: Solver Dialog Box and Solution to the M&D Chemical Problem 33

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.7: The Solver Answer Report for the M&D Chemicals Problem 34

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Special Cases of Linear Program Outcomes Alternative Optimal Solutions Infeasibility Unbounded

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Special Cases of Linear Program Outcomes Alternative Optimal Solutions Where the optimal objective function contour line coincides with one of the binding constraint lines on the boundary of the feasible region In these situations, more than one solution provides the optimal value for the objective function 36

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Special Cases of Linear Program Outcomes Illustration using the Par, Inc. problem Original objective function: 10S + 9D Assume the profit for the standard golf bag decreased to $6.30. Revised objective function: 6.3S + 9D The optimal solution occurs at two extreme points: Extreme point 4 (S = 300, D = 420) and Extreme point 3 (S = 540, D = 252) 37

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.8: Par, Inc. Problem with an Objective Function of 6.3S + 9D (Alternative Optimal Solutions) 38

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Special Cases of Linear Program Outcomes Infeasibility Means no solution to the linear programming problem No points satisfy all the constraints and the nonnegativity conditions simultaneously Graphically, a feasible region does not exist Infeasibility occurs because: Management’s expectations are too high Too many restrictions have been placed on the problem 39

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.9: No Feasible Region for the Par, Inc. Problem with Minimum Production Requirements of 500 Standard and 360 Deluxe Bags 40

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Special Cases of Linear Program Outcomes Interpretation of Infeasibility for the Par, Inc. problem Let the management know that the resources available are not sufficient to make 500 standard bags and 360 deluxe bags Provide details to the management on: Minimum amounts of resources that must be available The amounts currently available Additional amounts that would be required to accomplish this level of production 41

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Table 11.2: Resources Needed to Manufacture 500 Standard Bags and 360 Deluxe Bags 42

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Special Cases of Linear Program Outcomes An infeasible problem when solved in Excel Solver: Will return a message indicating that no feasible solutions exists—indicating no solution to the linear programming problem will satisfy all constraints Careful inspection is necessary to identify why the problem is infeasible One of the approaches is to drop one or more constraints and re-solve the problem If we find an optimal solution for this revised problem, then the constraint(s) that were omitted, in conjunction with the others, are causing the problem to be infeasible 43

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Special Cases of Linear Program Outcomes Unbounded The situation in which the value of the solution May be made infinitely large—for a maximization linear programming May be made infinitely small—for a minimization linear programming Without violating any of the constraints 44

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Special Cases of Linear Program Outcomes Illustration: Consider the following linear program with two decision variables, X and Y: Max 20X + 10Y s.t. 1X ≥ 2 1Y ≤ 5 X, Y ≥ 0 45

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.10: Example of an Unbounded Problem 46

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Special Cases of Linear Program Outcomes Solving an unbounded problem using Excel Solver: Returns a message “Objective Cell values do not converge” In linear programming models of real problems: The occurrence of an unbounded solution means that the problem has been improperly formulated 47

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Sensitivity Analysis Interpreting Excel Solver Sensitivity Report

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Sensitivity Analysis Sensitivity analysis: The study of how the changes in the input parameters of an optimization model affect the optimal solution It helps in answering the questions: How will a change in a coefficient of the objective function affect the optimal solution? How will a change in the right-hand-side value for a constraint affect the optimal solution? The shadow price for a constraint is the change in the optimal objective function value if the right-hand side of that constraint is increased by one 49

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Sensitivity Analysis Interpreting Excel Solver Sensitivity Report Consider the M&D chemicals problem: A = number of gallons of product A B = number of gallons of product B Min 2A + 3B s.t. 1A ≥ 125Demand for product A 1A + 1B ≥ 350Total production 2A + 1B ≤ 600 Processing time A, B ≥ 0 50

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.11: Solver Sensitivity Report for the M&D Chemicals Problem 51

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Sensitivity Analysis Classical sensitivity analysis: Based on the assumption that only one piece of input data has changed It is assumed that all other parameters remain as stated in the original problem When interested in what would happen if two or more pieces of input data are changed simultaneously: The easiest way to examine the effect of simultaneous changes is to make the changes and rerun the model 52

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Investment Portfolio Selection Transportation Planning Advertising Campaign Planning

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples 54

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples 55

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Investment Portfolio Selection Portfolio selection problems involve situations in which a financial manager must select specific investments—for example, stocks and bonds—from a variety of investment alternatives Objective: Maximization of expected return or minimization of risk Constraints: Restrictions on the type of permissible investments, state laws, company policy, and so on 56

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Illustration Welte Mutual Funds, Inc., located in New York City, is looking for investment opportunities for $100,000 The firm’s top financial analyst identified five investment opportunities and projected their annual rates of return Table 11.3: Investment Opportunities for Welte Mutual Funds 57

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Welte investment guidelines: Neither industry (oil or steel) should receive more than $50,000 Amount invested in government bonds should be at least 25 percent of the steel industry investments The investment in Pacific Oil, the high-return but high-risk investment, cannot be more than 60 percent of the total oil industry investment 58

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples 59

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples 60

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples 61

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.12: The Solution for the Welte Mutual Funds Problem 62

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Transportation Planning Transportation problem arises in planning for the distribution of goods and services from several supply locations to several demand locations Quantity of goods available at each supply location (origin) is limited Quantity of goods needed at each of several demand locations (destinations) is known Objective: Minimize the cost of shipping goods from the origins to the destinations 63

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Illustration using Foster Generators problem Involves the transportation of a product from three plants to four distribution centers To determine how much of its production should be shipped from each plant to each distribution center 64

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Production capacities over the next three-month planning period for one type of generator: The three-month forecast of demand for the distribution centers: 65

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.13: The Network Representation of the Foster Generators Transportation Problem 66

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Objective is to determine: Routes to be used Quantity to be shipped via each route Minimum total transportation cost Let x ij = number of units shipped from origin i to destination j where i = 1, 2,..., m and j = 1, 2,..., n 67

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Table 11.4: Transportation Cost Per Unit for the Foster Generators Transportation Problem ($) 68

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Supply constraints x 11 + x 12 + x 13 + x 14 ≤ 5000 Cleveland supply x 21 + x 22 + x 23 + x 24 ≤ 6000 Bedford supply x 31 + x 32 + x 33 + x 34 ≤ 2500 York supply Demand constraints x 11 + x 21 + x 31 = 6000 Boston demand x 12 + x 22 + x 32 = 4000 Chicago demand x 13 + x 23 + x 33 = 2000 St. Louis demand x 14 + x 24 + x 34 = 1500 Lexington demand 69

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples A 12-variable, 7-constraint linear programming formulation of the Foster Generators transportation problem: 70

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.14: Spreadsheet Model and Solution for the Foster Generator Problem 71

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Advertising Campaign Planning: Designed to help marketing managers allocate a fixed advertising budget to various advertising media Objective: Maximize reach, frequency, and quality of exposure Restrictions: Company policy, contract requirements, and media availability 72

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Illustration: Relax-and-Enjoy Lake Development Corporation: Developing a lakeside community at a privately owned lake Primary market includes all middle- and upper-income families within approximately 100 miles of the development Employed the advertising firm of Boone, Phillips, and Jackson (BP&J) to design the promotional campaign 73

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Table 11.5: Advertising Media Alternatives for the Relax-and-Enjoy Lake Development Corporation 74

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Problem Formulation: Budget: $30,000 Restrictions imposed: At least 10 television commercials must be used At least 50,000 potential customers must be reached No more than $18,000 may be spent on television advertisements The decision to be made is how many times to use each medium 75

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Define the decision variables: DTV = number of times daytime TV is used ETV = number of times evening TV is used DN = number of times daily newspaper is used SN = number of times Sunday newspaper is used R = number of times radio is used Objective: Maximizing the total exposure quality units for the overall media selection plan 76

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Linear programming model for the Relax-and-Enjoy advertising campaign planning problem: Max 65DTV + 90ETV + 40DN + 60SN + 20R Exposure quality DTV ≤ 15 ETV ≤ 10 DN ≤ 25 Availability of media SN ≤ 4 R ≤ DTV ETV + 400DN SN +100R ≤ 30,000 Budget 77

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. General Linear Programming Notation and More Examples Linear programming model for the Relax-and-Enjoy advertising campaign planning problem (contd.): DTV + ETV ≥ 10 Television 1500DTV ETV ≤ 18,000 restrictions 1000DTV ETV DN SN + 300R ≥ 50,000 Customers reached DTV, ETV, DN, SN, R ≥ 0 78

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.15: A Spreadsheet Model and the Solution for the Relax-and-Enjoy Lake Development Corporation Problem 79

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 11.16: The Excel Sensitivity Report for the Relax-and-Enjoy Lake Development Corporation Problem 80

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Generating an Alternative Optimal Solution for a Linear Program

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Generating an Alternative Optimal Solution for a Linear Program Illustration: Consider the Foster Generators transportation problem From Figure 11.14, the optimal solution: x 11 = 1000, x 12 = 4000, x 13 = 0, x 14 = 0 x 21 = 2500, x 22 = 0, x 23 = 2000, x 24 = 1500 x 31 = 2500, x 32 = 0, x 33 = 0, x 34 = 0 Optimal cost: $39,500 For the revised model to be optimal, the solution must give a total cost of $39,500 82

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Generating an Alternative Optimal Solution for a Linear Program From Figure 11.14: x 13 = x 14 = x 22 = x 32 = x 33 = x 34 = 0 If the sum of these variables is maximized and if the optimal objective function value of the revised problem is positive A different feasible solution that is also optimal is found 83

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Generating an Alternative Optimal Solution for a Linear Program Revised model 84

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Table 11.6: An Alternative Optimal Solution to the Foster Generators Transportation Problem 85

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Generating an Alternative Optimal Solution for a Linear Program In the original solution (Figure 11.14): Boston distribution center is sourced from all three plants, whereas each of the other distribution centers is sourced by one plant Hence, the manager in the Boston distribution center has to deal with three different plant managers, whereas each of the other distribution center managers has only one plant manager The alternative solution (Table 11.6) provides a more balanced solution Managers in Boston and Chicago each deal with two plants, and those in St. Louis and Lexington, which have lower total volumes, deal with only one plant Because the alternative solution seems to be more equitable, it might be preferred Both the solutions give a total cost of $39,500 86

© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Generating an Alternative Optimal Solution for a Linear Program General approach to find an alternative optimal solution to a linear program: Step 1: Solve the linear program Step 2: Make a new objective function to be maximized; It is the sum of those variables that were equal to zero in the solution from Step 1 Step 3: Keep all the constraints from the original problem; add a constraint that forces the original objective function to be equal to the optimal objective function value from Step 1 Step 4: Solve the problem created in Steps 2 and 3; if the objective function value is positive, an alternative optimal solution is found 87