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1. Introduction to Linear Programming Ardavan Asef-Vaziri Systems and Operations Management

2. The Lego Production Problem You have a set of legos 8 small bricks 6 large bricks These are your “raw materials”. You have to produce tables and chairs out of these legos. These are your “products”.

3. The Lego Production Problem Weekly supply of raw materials: 8 Small Bricks 6 Large Bricks Products: Chair Table Profit = 15 cents per Chair Profit = 20 cents per Table

4. Problem Formulation X1 is the number of Chairs X2 is the number of Tables Large brick constraint X1+2X2  6 Small brick constraint 2X1+2X2  8 Objective function is to Maximize 15X1+20 X2 X1 ≥ 0 X2 ≥ 0

5. Linear Programming • We can make Product1 and Product2. • There are 3 resources; Resource1, Resource2, Resource3. • Product1 needs one hour of Resource1, nothing of Resource2, and three hours of resource3. • Product2 needs nothing from Resource1, two hours of Resource2, and two hours of resource3. • Available hours of resources 1, 2, 3 are 4, 12, 18, respectively. • Contribution Margin of product 1 and Product2 are \$300 and \$500, respectively. • Formulate the Problem • Solve the problem using solver in excel

6. Problem Formulation Objective Function Z = 3 x1 +5 x2 Constraints Resource 1 x1 4 Resource 2 2x2  12 Resource 3 3 x1 + 2 x2  18 Nonnegativity x1  0, x2  0

7. Feasible, Infeasible, and Optimal Solution • Given the following problem • Maximize Z = 3x1 + 5x2 • Subject to: the following constraints x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 • x1, x2 ≥ 0 • What combination of x1 and x2 could be the optimal solution? • A) x1 = 4, x2 = 4 • B) x1 = -3, x2 = 6 • C) x1 = 3, x2 = 4 • D) x1 = 0, x2 = 7 • E) x1 = 2, x2 = 6 Infeasible; Violates Constraint 3 Infeasible; Violates nonnegativity Feasible; z = 3×3+ 5×4 = 29 Infeasible; Violates Constraint 2 Feasible; z = 3×2+ 5×6 = 36 and Optimal

8. Optimal Product Mix The Omega Manufacturing Co. has discontinued the production of a certain non-profitable product line. This act created considerable excess capacity. Management is considering devoting this excess capacity to one or more of three products. The hours required from each resource for each unit of product, the available capacity (hours per week) of the the three resources, as well as the profit of each unit of product are given below. Sales department indicates that the sales potentials for products 1 and 2 exceeds maximum production rate, but the sales potential for product 3 is 20 units per week. Formulate the problem and solve it using excel

9. Practice (Page 304, Prob. 3) • An appliance manufacturer produces two models of microwave ovens: H and W. Both models require fabrication and assembly work: each H uses four hours fabrication and two hours of assembly, and each W uses two hours fabrication and six hours of assembly. There are 600 fabrication hours this week and 450 hours of assembly. Each H contributes \$40 to profit, and each W contributes \$30 to profit. • Formulate the problem as a Linear Programming problem. • Solve it using excel. • What are the final values? • What is the optimal value of the objective function?

10. Practice (Page 304, Prob. 4) • A small candy shop is preparing for the holyday season. The owner must decide how many how many bags of deluxe mix how many bags of standard mix of Peanut/Raisin Delite to put up. The deluxe mix has 2/3 pound raisins and 1/3 pounds peanuts, and the standard mix has 1/2 pound raisins and 1/2 pounds peanuts per bag. The shop has 90 pounds of raisins and 60 pounds of peanuts to work with. Peanuts cost \$0.60 per pounds and raisins cost \$1.50 per pound. The deluxe mix will sell for 2.90 per pound and the standard mix will sell for 2.55 per pound. The owner estimates that no more than 110 bags of one type can be sold. • Formulate the problem as a Linear Programming problem. • Solve it using excel. • What are the final values? • What is the optimal value of the objective function?

11. Assignment 6a:1 Due at the beginning of next class The following table summarizes the key facts about two products, A and B, and the resources, Q, R, and S, required to produce them. • Formulate the problem as a Linear Programming problem. • Solve it using excel. • What are the final values? • What is the optimal value of the objective function?

12. Assignment 6a:2 Due at the beginning of next class The Apex Television Company has to decide on the number of 27” and 20” sets to be produced at one of its factories. Market research indicates that at most 40 of the 27” sets and 10 of the 20” sets can be sold per month. The maximum number of work-hours available is 500 per month. A 27” set requires 20 work-hours and a 20” set requires 10 work-hours. Each 27” set sold produces a profit of \$120 and each 20” set produces a profit of \$80. A wholesaler has agreed to purchase all the television sets produced if the numbers do not exceed the maximum indicated by the market research. • Formulate the problem as a Linear Programming problem. • Solve it using excel. • What are the final values? • What is the optimal value of the objective function?

13. Assignment 6a:3 Due at the beginning of next class Ralph Edmund loves steaks and potatoes. Therefore, he has decided to go on a steady diet of only these two foods (plus some liquids and vitamins supplements) for all his meals. Ralph realizes that this isn’t the healthiest diet, so he wants to make sure that he eats the right quantities of the two foods to satisfy some key nutritional requirements. He has obtained the following nutritional and cost information: Ralph wishes to determine the number of daily servings (may be fractional of steak and potatoes that will meet these requirements at a minimum cost. Formulate the problem as a Linear Programming problem. Solve it using excel. What are the final values? What is the optimal value of the objective function?

14. Assignment 6a:4 Due at the beginning of next class You are given the following linear programming model in algebraic form, where, X1 and X2 are the decision variables and Z is the value of the overall measure of performance. Maximize Z = X1 +2 X2 Subject to Constraints on resource 1: X1 + X2 ≤ 5 (amount available) Constraints on resource 2: X1 + 3X2 ≤ 9 (amount available) And X1 , X2 ≥ 0

15. Assignment 6a:4 Due at the beginning of next class • Identify the objective function, the functional constraints, and the non-negativity constraints in this model. • Incorporate this model into a spreadsheet. • Is (X1 ,X2) = (3,1) a feasible solution? • Is (X1 ,X2) = (1,3) a feasible solution? • Use the Excel Solver to solve this model.

16. Assignment 6a:5 Due at the beginning of next class You are given the following linear programming model in algebraic form, where, X1 and X2 are the decision variables and Z is the value of the overall measure of performance. Maximize Z = 3X1 +2 X2 Subject to Constraints on resource 1: 3X1 + X2 ≤ 9 (amount available) Constraints on resource 2: X1 + 2X2 ≤ 8 (amount available) And X1 , X2 ≥ 0

17. Assignment 6a:5 Due at the beginning of next class • Identify the objective function, the functional constraints, and the non-negativity constraints in this model. • Incorporate this model into a spreadsheet. • Is (X1 ,X2) = (2,1) a feasible solution? • Is (X1 ,X2) = (2,3) a feasible solution? • Is (X1 ,X2) = (0,5) a feasible solution? • Use the Excel Solver to solve this model.

18. Example A Production System Manufacturing Two Products, P and Q \$90 / unit \$100 / unit Q: P: 60 units / week 110 units / week D D Purchased Part 10 min. 5 min. \$5 / unit C B C 10 min. 5 min. 25 min. B A A 15 min. 10 min. 10 min. RM1 RM2 RM3 \$20 per \$20 per \$25 per unit unit unit Time available at each work center: 2,400 minutes per week Operating expenses per week: \$6,000