2 Problem 1. Optimal Product Mix Management is considering devoting some 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 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
3 Problem Formulation Decision Variables x1 : volume of product 1 Objective FunctionMax Z = 50 x1 +20 x2 +25 x3ConstraintsResources9 x1 +3 x2 +5 x3 5005 x1 +4 x 3503 x x3 150Marketx3 20Nonnegativityx1 0, x2 0 , x3 0
4 Problem 2An 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?
5 Problem Formulation Decision Variables xH : volume of microwave oven type HxW : volume of microwave oven type WObjective FunctionMax Z = 40 xH +30 xWConstraintsResources4 xH +2 xW 6002 xH +6 xW 450NonnegativityxH 0, xW 0
6 Problem 3A 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?
7 Problem Formulation Decision Variables x1 : volume of deluxe mix x2 : volume of standard mixObjective FunctionMax Z = [ (1/3)-1.5(2/3)] x1 + [ (1/2)-1.5(1/2)] x2Max Z = 1.7x x2ConstraintsResources(2/3) x1 +(1/2) x2 90(1/3) x1 +(1/2) x2 60Nonnegativityx1 0, x2 0
8 Resource Usage per Unit Produced Problem 4The following table summarizes the key facts about two products, A and B, and the resources, Q, R, and S, required to produce them.Resource Usage per Unit ProducedResourceProduct AProduct BAmount of resource availableQ21RS34Profit/Unit$3000$2000Formulate 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?
9 Problem Formulation Decision Variables xA : volume of product A xB : volume of product BObjective FunctionMax Z = 3000 xA xBConstraintsResources2 xA +1 xB 21 xA +2 xB 23 xA +3 xB 4NonnegativityxA 0, xB 0
10 Problem 5 Formulate the problem as a Linear Programming problem. 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?
11 Problem Formulation Decision Variables x1 : number of 27’ TVs Objective FunctionMax Z = 120 x1 +80 x2ConstraintsResources20 x1 +10 x2 500Marketx 40x2 10Nonnegativityx1 0, x2 0
12 Grams of Ingredient per Serving Problem 6Ralph Edmund has decided to go on a steady diet of only streak and potatoes s (plus some liquids and vitamins supplements). 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.Grams of Ingredient per ServingIngredientSteakPotatoesDaily Requirements (grams)Carbohydrates515≥ 50Protein20≥ 40Fat2≤ 60Cost per serving$4$2Formulate the problem as an LP model. Solve it using excel. What are the final values? What is the optimal value of the objective function?
14 Problem 7A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres. However, he has only $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of rye, how many acres of each should be planted to maximize profits?State the decision variables. x = the number of acres of wheat to planty = the number of acres of rye to plant Write the objective function.maximize 500x +300y
16 Problem 8You 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 X2Subject toConstraints on resource 1: X1 + X2 ≤ 5 (amount available)Constraints on resource 2: X1 + 3X2 ≤ 9 (amount available)AndX1 , X2 ≥ 0
17 Problem 8Identify the objective function, the functional constraints, and the non-negativity constraints in this model.Objective Function Maximize Z = X1 +2 X2Functional constraints X1 + X2 ≤ 5, X1 + 3X2 ≤ 9Is (X1 ,X2) = (3,1) a feasible solution?3 + 1 ≤ 5, (1) ≤ 9 yes; it satisfies both constraints.Is (X1 ,X2) = (1,3) a feasible solution?1 + 3 ≤ 5, (9) > 9 no; it violates the second constraint.
18 Problem 9You 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 X2Subject toConstraints on resource 1: 3X1 + X2 ≤ 9 (amount available)Constraints on resource 2: X1 + 2X2 ≤ 8 (amount available)AndX1 , X2 ≥ 0
19 Problem 9 Identify the objective function, Maximize Z = 3X1 +2 X2 the functional constraints,3X1 + X2 ≤ 9 and X1 + 2X2 ≤ 8the non-negativity constraintsX1 , X2 ≥ 0Is (X1 ,X2) = (2,1) a feasible solution?3(2) + 1 ≤ 9 and 2 + 2(1) ≤ 8 yes; it satisfies both constraintsIs (X1 ,X2) = (2,3) a feasible solution?3(2) + 3 ≤ 9 and 2 + 2(3) ≤ 8 yes; it satisfies both constraintsIs (X1 ,X2) = (0,5) a feasible solution?3(0) + 5 ≤ 9 and 0 + 2(5) > 8 no; it violates the second constraint
20 Problem 10. Product mix problem : Narrative representation The Quality Furniture Corporation producesbenches and tables.The firm has two main resourcesResourceslabor and redwood for use in the furniture.During the next production period1200 labor hours are available under a union agreement.A stock of 5000 pounds of quality redwood is also available.
21 Problem 10. Product mix problem : Narrative representation Consumption and profitEach bench that Quality Furniture produces requires4 labor hours and 10 pounds of redwoodEach picnic table takes 7 labor hours and 35 pounds of redwood.Total available 1200, 5000Completed benches yield a profit of $9 each,and tables a profit of $20 each.Formulate the problem to maximize the total profit.
22 Problem 10. Product Mix : Formulation x1 = number of benches to producex2 = number of tables to produceMaximize Profit = ($9) x1 +($20) x2subject toLabor: 4 x1 + 7 x2 hoursWood: 10 x x2 poundsand x1 0, x2 0.We will now solve this LP model using the Excel Solver.
24 Problem 11. Make / buy decision : Narrative representation Electro-Poly is a leading maker of slip-rings.A new order has just been received.Model 1 Model 2 Model 3Number ordered 3,000 2,Hours of wiring/unitHours of harnessing/unit 1 2 1Cost to Make $50 $83 $130Cost to Buy $61 $97 $145The company has 10,000 hours of wiring capacity and 5,000 hours of harnessing capacity.
25 Problem 11. Make / buy decision : decision variables x1 = Number of model 1 slip rings to makex2 = Number of model 2 slip rings to makex3 = Number of model 3 slip rings to makey1 = Number of model 1 slip rings to buyy2 = Number of model 2 slip rings to buyy3 = Number of model 3 slip rings to buyThe Objective FunctionMinimize the total cost of filling the order.MIN: 50x1 + 83x x3 + 61y1 + 97y y3
31 Problem 12. Marketing : narrative A department store want to maximize exposure.There are 3 media; TV, Radio, Newspapereach ad will have the following impactMedia Exposure (people / ad) CostTVRadioNews paperAdditional information1-Total budget is $100,000.2-The maximum number of ads in T, R, and N are limited to 4, 10, 7 ads respectively.3-The total number of ads is limited to 15.
32 Problem 12. Marketing : formulation Decision variablesx1 = Number of ads in TVx2 = Number of ads in Rx3 = Number of ads in NMax Z = 20 x1 + 12x2 +9x315x1 + 6x2 + 4x3 100x 4x 10x3 7x1 + x x3 15x1, x2, x3 0
33 Problem 13. ( From Hillier and Hillier) Men, women, and children gloves.Material and labor requirements for each type and the corresponding profit are given below.Glove Material (sq-feet) Labor (hrs) ProfitMenWomenChildrenTotal available material is 5000 sq-feet.We can have full time and part time workers.Full time workers work 40 hrs/w and are paid $13/hrPart time workers work 20 hrs/w and are paid $10/hrWe should have at least 20 full time workers.The number of full time workers must be at least twice of that of part times.
34 Problem 13. Decision variables X1 : Volume of production of Men’s glovesX2 : Volume of production of Women’s glovesX3 : Volume of production of Children’s glovesY1 : Number of full time employeesY2 : Number of part time employees
35 Problem 13. Constraints Row material constraint 2X X2 + X3 Full time employeesY1 20Relationship between the number of Full and Part time employeesY1 2 Y2Labor Required.5X X X3 40 Y Y2Objective FunctionMax Z = 8X1 + 10X2 + 6X Y Y2Non-negativityX1 , X2 , X3 , Y1 , Y2 0