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**LP Formulation Practice Set 1**

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**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

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**Problem Formulation Decision Variables x1 : volume of product 1**

Objective Function Max Z = 50 x1 +20 x2 +25 x3 Constraints Resources 9 x1 +3 x2 +5 x3 500 5 x1 +4 x 350 3 x x3 150 Market x3 20 Nonnegativity x1 0, x2 0 , x3 0

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Problem 2 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?

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**Problem Formulation Decision Variables**

xH : volume of microwave oven type H xW : volume of microwave oven type W Objective Function Max Z = 40 xH +30 xW Constraints Resources 4 xH +2 xW 600 2 xH +6 xW 450 Nonnegativity xH 0, xW 0

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Problem 3 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?

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**Problem Formulation Decision Variables x1 : volume of deluxe mix**

x2 : volume of standard mix Objective Function Max Z = [ (1/3)-1.5(2/3)] x1 + [ (1/2)-1.5(1/2)] x2 Max Z = 1.7x x2 Constraints Resources (2/3) x1 +(1/2) x2 90 (1/3) x1 +(1/2) x2 60 Nonnegativity x1 0, x2 0

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**Resource Usage per Unit Produced**

Problem 4 The 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 Produced Resource Product A Product B Amount of resource available Q 2 1 R S 3 4 Profit/Unit $3000 $2000 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?

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**Problem Formulation Decision Variables xA : volume of product A**

xB : volume of product B Objective Function Max Z = 3000 xA xB Constraints Resources 2 xA +1 xB 2 1 xA +2 xB 2 3 xA +3 xB 4 Nonnegativity xA 0, xB 0

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**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?

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**Problem Formulation Decision Variables x1 : number of 27’ TVs**

Objective Function Max Z = 120 x1 +80 x2 Constraints Resources 20 x1 +10 x2 500 Market x 40 x2 10 Nonnegativity x1 0, x2 0

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**Grams of Ingredient per Serving**

Problem 6 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 Ingredient Steak Potatoes Daily Requirements (grams) Carbohydrates 5 15 ≥ 50 Protein 20 ≥ 40 Fat 2 ≤ 60 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?

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**Problem Formulation Decision Variables x1 : serving of steak**

x2 : serving of potato Objective Function Min Z = 4 x1 +2x2 Constraints Resources 5 x1 +15 x2 ≥ 50 20 x1 +5 x2 ≥ 40 15 x1 +2 x2 ≥ 60 Nonnegativity x1 0, x2 0

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Problem 7 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? 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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**Problem 7 Write the constraints. x+y ≤ 10 (max acreage)**

x+y ≤ 10 (max acreage) x+y ≥ 7 (min acreage) 200x + 100y ≤ (cost) x + 2y ≤ (time) x ≥ 0, y ≥ 0 (non-negativity)

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Problem 8 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

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Problem 8 Identify the objective function, the functional constraints, and the non-negativity constraints in this model. Objective Function Maximize Z = X1 +2 X2 Functional constraints X1 + X2 ≤ 5, X1 + 3X2 ≤ 9 Is (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.

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Problem 9 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

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**Problem 9 Identify the objective function, Maximize Z = 3X1 +2 X2**

the functional constraints, 3X1 + X2 ≤ 9 and X1 + 2X2 ≤ 8 the non-negativity constraints X1 , X2 ≥ 0 Is (X1 ,X2) = (2,1) a feasible solution? 3(2) + 1 ≤ 9 and 2 + 2(1) ≤ 8 yes; it satisfies both constraints Is (X1 ,X2) = (2,3) a feasible solution? 3(2) + 3 ≤ 9 and 2 + 2(3) ≤ 8 yes; it satisfies both constraints Is (X1 ,X2) = (0,5) a feasible solution? 3(0) + 5 ≤ 9 and 0 + 2(5) > 8 no; it violates the second constraint

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**Problem 10. Product mix problem : Narrative representation**

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.

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**Problem 10. Product mix problem : Narrative representation**

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.

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**Problem 10. Product Mix : Formulation**

x1 = number of benches to produce x2 = number of tables to produce Maximize Profit = ($9) x1 +($20) x2 subject to Labor: 4 x1 + 7 x2 hours Wood: 10 x x2 pounds and x1 0, x2 0. We will now solve this LP model using the Excel Solver.

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**Problem 10. Product Mix : Excel solution**

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**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 3 Number ordered 3,000 2, Hours of wiring/unit Hours of harnessing/unit 1 2 1 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.

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**Problem 11. Make / buy decision : decision variables**

x1 = Number of model 1 slip rings to make x2 = Number of model 2 slip rings to make x3 = Number of model 3 slip rings to make y1 = Number of model 1 slip rings to buy y2 = Number of model 2 slip rings to buy y3 = Number of model 3 slip rings to buy The Objective Function Minimize the total cost of filling the order. MIN: 50x1 + 83x x3 + 61y1 + 97y y3

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**Problem 11. Make / buy decision : Constraints**

Demand Constraints x1 + y1 = 3,000 } model 1 x2 + y2 = 2,000 } model 2 x3 + y3 = } model 3 Resource Constraints 2x x2 + 3x3 <= 10,000 } wiring 1x x2 + 1x3 <= 5,000 } harnessing Nonnegativity Conditions x1, x2, x3, y1, y2, y3 >= 0

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**Problem 11. Make / buy decision : Excel**

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**Problem 11. Make / buy decision : Constraints**

Do we really need 6 variables? x1 + y1 = 3,000 ===> y1 = 3,000 - x1 x2 + y2 = 2,000 ===> y2 = 2,000 - x2 x3 + y3 = ===> y3 = x3 The objective function was MIN: 50x1 + 83x x3 + 61y1 + 97y y3 Just replace the values MIN: 50x1 + 83x x (3,000 - x1 ) + 97 ( 2,000 - x2) + 145 ( x3 ) MIN: x1 -14x2 -15x3 We can even forget , and change the the O.F. into MIN - 11x1 -14x2 -15x3 or MAX + 11x1 +14x2 +15x3

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**Problem 11. Make / buy decision : Constraints**

MAX + 11x1 +14x2 +15x3 Resource Constraints 2x x2 + 3x3 <= 10,000 } wiring 1x x2 + 1x3 <= 5,000 } harnessing Demand Constraints x1 <= 3,000 } model 1 x2 <= 2,000 } model 2 x3 <= } model 3 Nonnegativity Conditions x1, x2, x3 >= 0

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**Problem 11. Make / buy decision : Constraints**

y1 = 3,000- x1 y2 = 2,000-x2 y3 = x3 MIN: 50x1 + 83x x3 + 61y1 + 97y y3 Demand Constraints x1 + y1 = 3,000 } model 1 x2 + y2 = 2,000 } model 2 x3 + y3 = } model 3 Resource Constraints 2x x2 + 3x3 <= 10,000 } wiring 1x x2 + 1x3 <= 5,000 } harnessing Nonnegativity Conditions x1, x2, x3, y1, y2, y3 >= 0 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 = x3>=0 x1 <= 3,000 x2 <= 2,000 x3 <= 900

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**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 Media Exposure (people / ad) Cost TV Radio News paper Additional information 1-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.

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**Problem 12. Marketing : formulation**

Decision variables x1 = Number of ads in TV x2 = Number of ads in R x3 = Number of ads in N Max Z = 20 x1 + 12x2 +9x3 15x1 + 6x2 + 4x3 100 x 4 x 10 x3 7 x1 + x x3 15 x1, x2, x3 0

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**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) Profit Men 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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**Problem 13. Decision variables**

X1 : Volume of production of Men’s gloves X2 : Volume of production of Women’s gloves X3 : Volume of production of Children’s gloves Y1 : Number of full time employees Y2 : Number of part time employees

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**Problem 13. Constraints Row material constraint**

2X X2 + X3 Full time employees Y1 20 Relationship between the number of Full and Part time employees Y1 2 Y2 Labor Required .5X X X3 40 Y Y2 Objective Function Max Z = 8X1 + 10X2 + 6X Y Y2 Non-negativity X1 , X2 , X3 , Y1 , Y2 0

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**Problem 13. Excel Solution**

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Thank you and welcome Linear Programming (LP) Modeling Application in manufacturing And marketing By M. Dadfar, PhD.

Thank you and welcome Linear Programming (LP) Modeling Application in manufacturing And marketing By M. Dadfar, PhD.

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