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LP Formulation Practice Set 1

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2 Ardavan Asef-Vaziri June-2013LP-Formulation 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. Problem 1. Optimal Product Mix 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

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3 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables x 1 : volume of product 1 x 2 : volume of product 2 x 3 : volume of product 3 Objective Function Max Z = 50 x x x 3 Constraints Resources 9 x 1 +3 x 2 +5 x x 1 +4 x x x Market x 3 20 Nonnegativity x 1 0, x 2 0, x 3 0 Problem Formulation

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4 Ardavan Asef-Vaziri June-2013LP-Formulation 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 fabricatio n 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. a)Formulate the problem as a Linear Programming problem. b)Solve it using excel. c)What are the final values? d)What is the optimal value of the objective function? Problem 2

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5 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables x H : volume of microwave oven type H x W : volume of microwave oven type W Objective Function Max Z = 40 x H +30 x W Constraints Resources 4 x H +2 x W x H +6 x W 450 Nonnegativity x H 0, x W 0 Problem Formulation

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6 Ardavan Asef-Vaziri June-2013LP-Formulation 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. a)Formulate the problem as a Linear Programming problem. b)Solve it using excel. c)What are the final values? d)What is the optimal value of the objective function? Problem 3

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7 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables x 1 : volume of deluxe mix x 2 : volume of standard mix Objective Function Max Z = [ (1/3)-1.5(2/3)] x 1 + [ (1/2)-1.5(1/2)] x 2 Max Z = 1.7x x 2 Constraints Resources (2/3) x 1 +(1/2) x 2 90 (1/3) x 1 +(1/2) x 2 60 Nonnegativity x 1 0, x 2 0 Problem Formulation

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8 Ardavan Asef-Vaziri June-2013LP-Formulation Resource Usage per Unit Produced ResourceProduct AProduct BAmount of resource available Q212 R122 S334 Profit/Uni t $3000$2000 The following table summarizes the key facts about two products, A and B, and the resources, Q, R, and S, required to produce them. Problem 4 a)Formulate the problem as a Linear Programming problem. b)Solve it using excel. c)What are the final values? d)What is the optimal value of the objective function?

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9 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables x A : volume of product A x B : volume of product B Objective Function Max Z = 3000 x A x B Constraints Resources 2 x A +1 x B 2 1 x A +2 x B 2 3 x A +3 x B 4 Nonnegativity x A 0, x B 0 Problem Formulation

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10 Ardavan Asef-Vaziri June-2013LP-Formulation 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. a)Formulate the problem as a Linear Programming problem. b)Solve it using excel. c)What are the final values? d)What is the optimal value of the objective function ? Problem 5

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11 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables x 1 : number of 27 TVs x 2 : number of 20 TVs Objective Function Max Z = 120 x x 2 Constraints Resources 20 x x Market x 1 40 x 2 10 Nonnegativity x 1 0, x 2 0 Problem Formulation

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12 Ardavan Asef-Vaziri June-2013LP-Formulation Ralph 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 Serving IngredientSteakPotatoesDaily Requirements (grams) Carbohydrates Protein Fat Cost per serving$4$2 Formulate the problem as an LP model. Solve it using excel. What are the final values? What is the optimal value of the objective function? Problem 6

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13 Ardavan Asef-Vaziri June-2013LP-Formulation Decision Variables x 1 : serving of steak x 2 : serving of potato Objective Function Min Z = 4 x 1 +2x 2 Constraints Resources 5 x x x 1 +5 x x 1 +2 x 2 60 Nonnegativity x 1 0, x 2 0 Problem Formulation

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14 Ardavan Asef-Vaziri June-2013LP-Formulation A 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? Problem 7 State the decision variables. x = the number of acres of wheat to plant y = the number of acres of rye to plant Write the objective function. maximize 500x +300y

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15 Ardavan Asef-Vaziri June-2013LP-Formulation Problem 7 Write the constraints. x+y 10(max acreage) x+y 7(min acreage) 200x + 100y 1200(cost) x + 2y 12 (time) x 0, y 0(non-negativity)

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16 Ardavan Asef-Vaziri June-2013LP-Formulation You are given the following linear programming model in algebraic form, where, X 1 and X 2 are the decision variables and Z is the value of the overall measure of performance. Maximize Z = X 1 +2 X 2 Subject to Constraints on resource 1: X1 + X2 5 (amount available) Constraints on resource 2: X1 + 3X2 9 (amount available) And X1, X2 0 Problem 8

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17 Ardavan Asef-Vaziri June-2013LP-Formulation Identify the objective function, the functional constraints, and the non-negativity constraints in this model. Objective Function Maximize Z = X 1 +2 X 2 Functional constraints X1 + X2 5, X1 + 3X2 9 Is (X 1,X 2 ) = (3,1) a feasible solution? , 3 + 3(1) 9 yes; it satisfies both constraints. Is (X 1,X 2 ) = (1,3) a feasible solution? , 1 + 3(9) > 9 no; it violates the second constraint. Problem 8

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18 Ardavan Asef-Vaziri June-2013LP-Formulation You are given the following linear programming model in algebraic form, where, X 1 and X 2 are the decision variables and Z is the value of the overall measure of performance. Maximize Z = 3X 1 +2 X 2 Subject to Constraints on resource 1: 3X1 + X2 9 (amount available) Constraints on resource 2: X1 + 2X2 8 (amount available) And X1, X2 0 Problem 9

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19 Ardavan Asef-Vaziri June-2013LP-Formulation Identify the objective function, Maximize Z = 3X 1 +2 X 2 the functional constraints, 3X1 + X2 9 and X1 + 2X2 8 the non-negativity constraints X1, X2 0 Is (X 1,X 2 ) = (2,1) a feasible solution? 3(2) and 2 + 2(1) 8 yes; it satisfies both constraints Is (X 1,X 2 ) = (2,3) a feasible solution? 3(2) and 2 + 2(3) 8 yes; it satisfies both constraints Is (X 1,X 2 ) = (0,5) a feasible solution? 3(0) and 0 + 2(5) > 8 no; it violates the second constraint Problem 9

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20 Ardavan Asef-Vaziri June-2013LP-Formulation The Quality Furniture Corporation produces benches and tables. The firm has two main resources Resources labor and redwood for use in the furniture. During the next production period 1200 labor hours are available under a union agreement. A stock of 5000 pounds of quality redwood is also available. Problem 10. Product mix problem : Narrative representation

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21 Ardavan Asef-Vaziri June-2013LP-Formulation Consumption and profit Each bench that Quality Furniture produces requires 4 labor hours and 10 pounds of redwood Each picnic table takes 7 labor hours and 35 pounds of redwood. Total available 1200, 5000 Completed benches yield a profit of $9 each, and tables a profit of $20 each. Formulate the problem to maximize the total profit. Problem 10. Product mix problem : Narrative representation

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22 Ardavan Asef-Vaziri June-2013LP-Formulation x 1 = number of benches to produce x 2 = number of tables to produce Maximize Profit = ($9) x 1 +($20) x 2 subject to Labor: 4 x x hours Wood:10 x x pounds and x 1 0, x 2 0. We will now solve this LP model using the Excel Solver. Problem 10. Product Mix : Formulation

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23 Ardavan Asef-Vaziri June-2013LP-Formulation Problem 10. Product Mix : Excel solution

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24 Ardavan Asef-Vaziri June-2013LP-Formulation Electro-Poly is a leading maker of slip-rings. A new order has just been received. Model 1 Model 2Model 3 Number ordered3,0002, Hours of wiring/unit21.53 Hours of harnessing/unit121 Cost to Make$50$83$130 Cost to Buy$61$97$145 The company has 10,000 hours of wiring capacity and 5,000 hours of harnessing capacity. Problem 11. Make / buy decision : Narrative representation

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25 Ardavan Asef-Vaziri June-2013LP-Formulation x 1 = Number of model 1 slip rings to make x 2 = Number of model 2 slip rings to make x 3 = Number of model 3 slip rings to make y 1 = Number of model 1 slip rings to buy y 2 = Number of model 2 slip rings to buy y 3 = Number of model 3 slip rings to buy The Objective Function Minimize the total cost of filling the order. MIN:50x x x y y y 3 Problem 11. Make / buy decision : decision variables

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26 Ardavan Asef-Vaziri June-2013LP-Formulation Demand Constraints x 1 + y 1 = 3,000} model 1 x 2 + y 2 = 2,000} model 2 x 3 + y 3 = 900} model 3 Resource Constraints 2x x 2 + 3x 3 <= 10,000 } wiring 1x x 2 + 1x 3 <= 5,000 } harnessing Nonnegativity Conditions x 1, x 2, x 3, y 1, y 2, y 3 >= 0 Problem 11. Make / buy decision : Constraints

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27 Ardavan Asef-Vaziri June-2013LP-Formulation Problem 11. Make / buy decision : Excel

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28 Ardavan Asef-Vaziri June-2013LP-Formulation Do we really need 6 variables ? x 1 + y 1 = 3,000 ===> y 1 = 3,000 - x 1 x 2 + y 2 = 2,000 ===> y 2 = 2,000 - x 2 x 3 + y 3 = 900 ===> y 3 = x 3 The objective function was MIN:50x x x y y y 3 Just replace the values MIN:50x x x (3,000 - x 1 ) + 97 ( 2,000 - x 2 ) (900 - x 3 ) MIN: x 1 -14x 2 -15x 3 We can even forget , and change the the O.F. into MIN - 11x 1 -14x 2 -15x 3 or MAX + 11x 1 +14x 2 +15x 3 Problem 11. Make / buy decision : Constraints

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29 Ardavan Asef-Vaziri June-2013LP-Formulation Resource Constraints 2x x 2 + 3x 3 <= 10,000 } wiring 1x x 2 + 1x 3 <= 5,000 } harnessing Demand Constraints x 1 <= 3,000} model 1 x 2 <= 2,000} model 2 x 3 <= 900} model 3 Nonnegativity Conditions x 1, x 2, x 3 >= 0 Problem 11. Make / buy decision : Constraints MAX + 11x 1 +14x 2 +15x 3

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30 Ardavan Asef-Vaziri June-2013LP-Formulation MIN:50x1 + 83x x3 + 61y1 + 97y y3 Demand Constraints x 1 + y 1 = 3,000} model 1 x 2 + y 2 = 2,000} model 2 x 3 + y 3 = 900} model 3 Resource Constraints 2x x 2 + 3x 3 <= 10,000 } wiring 1x x 2 + 1x 3 <= 5,000 } harnessing Nonnegativity Conditions x 1, x 2, x 3, y 1, y 2, y 3 >= 0 Problem 11. Make / buy decision : Constraints y1 = 3,000- x1 y2 = 2,000-x2 y3 = 900-x3 MIN:50x1 + 83x x3 + 61(3,000- x1) + 97(2,000-x2) + 145(900-x3) y1 = 3,000- x1>=0 y2 = 2,000-x2>=0 y3 = 900-x3>=0 x1 <= 3,000 x2 <= 2,000 x3 <= 900

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31 Ardavan Asef-Vaziri June-2013LP-Formulation Problem 12. Marketing : narrative A department store want to maximize exposure. There are 3 media; TV, Radio, Newspaper each ad will have the following impact MediaExposure (people / ad)Cost TV Radio News paper Additional information 1-Total budget is $100, 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.

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32 Ardavan Asef-Vaziri June-2013LP-Formulation Problem 12. Marketing : formulation Decision variables x 1 = Number of ads in TV x 2 = Number of ads in R x 3 = Number of ads in N Max Z = 20 x x 2 +9x 3 15 x 1 + 6x 2 + 4x x 1 4 x 2 10 x 3 7 x 1 + x 2 + x 3 15 x 1, x 2, x 3 0

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33 Ardavan Asef-Vaziri June-2013LP-Formulation 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. GloveMaterial (sq-feet)Labor (hrs)Profit Men20.58 Women Children Total 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/hr Part time workers work 20 hrs/w and are paid $10/hr We should have at least 20 full time workers. The number of full time workers must be at least twice of that of part times.

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34 Ardavan Asef-Vaziri June-2013LP-Formulation Problem 13. Decision variables X 1 : Volume of production of Mens gloves X 2 : Volume of production of Womens gloves X 3 : Volume of production of Childrens gloves Y 1 : Number of full time employees Y 2 : Number of part time employees

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35 Ardavan Asef-Vaziri June-2013LP-Formulation Problem 13. Constraints Row material constraint 2X X2 + X Full time employees Y1 20 Relationship between the number of Full and Part time employees Y1 2 Y2 Labor Required.5X X X 3 40 Y Y 2 Objective Function Max Z = 8X X 2 + 6X Y Y 2 Non-negativity X 1, X 2, X 3, Y 1, Y 2 0

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36 Ardavan Asef-Vaziri June-2013LP-Formulation Problem 13. Excel Solution

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