Lean Thinking

LP FormulationPractice Set 1#Ardavan Asef-Vaziri June-2013LP-FormulationManagement 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 MixSales 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

#Ardavan Asef-Vaziri June-2013LP-Formulation2Decision Variables x1 : volume of product 1 x2 : volume of product 2x3 : volume of product 3Objective Function Max Z = 50 x1 +20 x2 +25 x3ConstraintsResources9 x1 +3 x2 +5 x3 500 5 x1 +4 x2 + 350 3 x1 + +2 x3 150 Market x3 20Nonnegativityx1 0, x2 0 , x3 0Problem Formulation#Ardavan Asef-Vaziri June-2013LP-Formulation3An 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?Problem 2#Ardavan Asef-Vaziri June-2013LP-Formulation4Decision Variables xH : volume of microwave oven type H xW : volume of microwave oven type W

Objective Function Max Z = 40 xH +30 xW ConstraintsResources4 xH +2 xW 600 2 xH +6 xW 450

NonnegativityxH 0, xW 0

Problem Formulation#Ardavan Asef-Vaziri June-2013LP-Formulation5A 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?Problem 3#Ardavan Asef-Vaziri June-2013LP-Formulation6Decision Variables x1 : volume of deluxe mix x2 : volume of standard mix

Objective Function Max Z = [2.9-0.60(1/3)-1.5(2/3)] x1 + [2.55-0.60(1/2)-1.5(1/2)] x2 Max Z = 1.7x1 + 1.5 x2 ConstraintsResources(2/3) x1 +(1/2) x2 90 (1/3) x1 +(1/2) x2 60

Nonnegativityx1 0, x2 0

Problem Formulation#Ardavan Asef-Vaziri June-2013LP-Formulation7Resource Usage per Unit ProducedResource Product AProduct BAmount of resource availableQ212R122S334Profit/Unit$3000$2000The following table summarizes the key facts about two products, A and B, and the resources, Q, R, and S, required to produce them.Problem 4Formulate 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?#Ardavan Asef-Vaziri June-2013LP-Formulation8Decision Variables xA : volume of product A xB : volume of product B

Objective Function Max Z = 3000 xA +2000 xB ConstraintsResources2 xA +1 xB 21 xA +2 xB 23 xA +3 xB 4

NonnegativityxA 0, xB 0

Problem Formulation#Ardavan Asef-Vaziri June-2013LP-Formulation9The 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?

Problem 5#Ardavan Asef-Vaziri June-2013LP-Formulation10Decision Variables x1 : number of 27 TVs x2 : number of 20 TVs

Objective Function Max Z = 120 x1 +80 x2 ConstraintsResources20 x1 +10 x2 500Market x1 40 x2 10

Nonnegativityx1 0, x2 0

Problem Formulation#Ardavan Asef-Vaziri June-2013LP-Formulation11Ralph 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 50Protein205 40Fat152 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?Problem 6#Ardavan Asef-Vaziri June-2013LP-Formulation12Decision Variables x1 : serving of steak x2 : serving of potato

Objective Function Min Z = 4 x1 +2x2ConstraintsResources5 x1 +15 x2 5020 x1 +5 x2 4015 x1 +2 x2 60

Nonnegativityx1 0, x2 0 Problem Formulation#Ardavan Asef-Vaziri June-2013LP-Formulation13A 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 7State the decision variables. x = the number of acres of wheat to planty = the number of acres of rye to plantWrite the objective function. maximize 500x +300y #Ardavan Asef-Vaziri June-2013LP-Formulation14Problem 7Write 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)#Ardavan Asef-Vaziri June-2013LP-Formulation15You 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 Problem 8#Ardavan Asef-Vaziri June-2013LP-Formulation16Identify 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, 3 + 3(1) 9 yes; it satisfies both constraints. Is (X1 ,X2) = (1,3) a feasible solution?1 + 3 5, 1 + 3(9) > 9 no; it violates the second constraint.

Problem 8#Ardavan Asef-Vaziri June-2013LP-Formulation17You 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 Problem 9#Ardavan Asef-Vaziri June-2013LP-Formulation18Identify the objective function, Maximize Z = 3X1 +2 X2the functional constraints, 3X1 + X2 9 and X1 + 2X2 8 the non-negativity constraintsX1 , X2 0 Is (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 constraintProblem 9#Ardavan Asef-Vaziri June-2013LP-Formulation19The Quality Furniture Corporation produces benches 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. Problem 10. Product mix problem : Narrative representation #Ardavan Asef-Vaziri June-2013LP-FormulationConsumption and profitEach bench that Quality Furniture produces requires 4 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. Problem 10. Product mix problem : Narrative representation #Ardavan Asef-Vaziri June-2013LP-Formulationx1= number of benches to producex2= number of tables to produce

Maximize Profit = ($9) x1 +($20) x2subject to Labor: 4 x1 + 7 x2 1200 hoursWood:10 x1 + 35 x2 5000 poundsand x1 0, x2 0.

We will now solve this LP model using the Excel Solver.Problem 10. Product Mix : Formulation#Ardavan Asef-Vaziri June-2013LP-Formulation

Problem 10. Product Mix : Excel solution#Ardavan Asef-Vaziri June-2013LP-FormulationElectro-Poly is a leading maker of slip-rings.A new order has just been received. Model 1 Model 2Model 3Number ordered3,0002,000900Hours of wiring/unit21.53Hours of harnessing/unit121Cost to Make$50$83$130Cost 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#Ardavan Asef-Vaziri June-2013LP-Formulationx1 = Number of model 1 slip rings to makex2 = 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 buyThe Objective FunctionMinimize the total cost of filling the order.MIN:50x1 + 83x2 + 130x3 + 61y1 + 97y2 + 145y3 Problem 11. Make / buy decision : decision variables#Ardavan Asef-Vaziri June-2013LP-FormulationDemand Constraintsx1 + y1 = 3,000} model 1x2 + y2 = 2,000} model 2x3 + y3 = 900} model 3Resource Constraints2x1 + 1.5x2 + 3x3 y1 = 3,000 - x1 x2 + y2 = 2,000 ===> y2 = 2,000 - x2x3 + y3 = 900 ===> y3 = 900 - x3

The objective function was MIN:50x1 + 83x2 + 130x3 + 61y1 + 97y2 + 145y3Just replace the valuesMIN:50x1 + 83x2 + 130x3 + 61 (3,000 - x1 ) + 97 ( 2,000 - x2) + 145 (900 - x3 )MIN:507500 - 11x1 -14x2 -15x3We can even forget 507500, and change the the O.F. into MIN - 11x1 -14x2 -15x3 or MAX + 11x1 +14x2 +15x3 Problem 11. Make / buy decision : Constraints#Ardavan Asef-Vaziri June-2013LP-FormulationResource Constraints2x1 + 1.5x2 + 3x3