LP Formulation Practice Set 1. 2 Ardavan Asef-Vaziri June-2013LP-Formulation Management is...

LP FormulationPractice Set 1

2Ardavan 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

Total hours avialableProduct1 Product2 Product3

9 3 5 5005 4 0 3503 0 2 150

$50 $20 $25 Profit

Hours used per unit


Decision Variables x1 : volume of product 1

x2 : volume of product 2x3 : volume of product 3Objective Function Max Z = 50 x1 +20 x2 +25 x3

ConstraintsResources9 x1 +3 x2 +5 x3 500 5 x1 +4 x2 + 350 3 x1 + +2 x3 150 Market x3 20Nonnegativityx1 0, x2 0 , x3 0

Problem 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 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.

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


Decision 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


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.


b) Solve it using excel.

c) What are the final values?

d) What is the optimal value of the objective function?

Problem 3


Decision 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


Resource Usage per Unit Produced

Resource Product A Product B Amount of resource available

Q 2 1 2

R 1 2 2

S 3 3 4

Profit/Unit $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


b) Solve it using excel.c) What are the final values? d) What is the optimal value of the objective function?


Decision 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


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


Decision 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


Problem 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

Ingredient Steak Potatoes Daily Requirements (grams)

Carbohydrates 5 15 ≥ 50

Protein 20 5 ≥ 40

Fat 15 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?

Problem 6


Decision Variables x1 : serving of steak

x2 : serving of potato

Objective Function Min Z = 4 x1 +2x2

ConstraintsResources5 x1 +15 x2 ≥ 5020 x1 +5 x2 ≥ 4015 x1 +2 x2 ≥ 60


Problem 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


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)


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

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, 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


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

Problem 9


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

the 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

constraint

Problem 9


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


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


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 1200 hours

Wood: 10 x1 + 35 x2 5000 pounds

and x1 0, x2 0.

We will now solve this LP model using the Excel Solver.

Problem 10. Product Mix : Formulation


Problem 10. Product Mix : Excel solution


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 2,000 900

Hours of wiring/unit 2 1.5 3

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.

Problem 11. Make / buy decision : Narrative representation


x1 = Number of model 1 slip rings to make



y1 = Number of model 1 slip rings to buy



The Objective Function

Minimize the total cost of filling the order.

MIN: 50x1 + 83x2 + 130x3 + 61y1 + 97y2 + 145y3

Problem 11. Make / buy decision : decision variables


Demand Constraints

x1 + y1 = 3,000 } model 1

x2 + y2 = 2,000 } model 2

x3 + y3 = 900 } model 3

Resource Constraints

2x1 + 1.5x2 + 3x3 <= 10,000 } wiring

1x1 + 2.0x2 + 1x3 <= 5,000 } harnessing

Nonnegativity Conditions

x1, x2, x3, y1, y2, y3 >= 0

Problem 11. Make / buy decision : Constraints


Problem 11. Make / buy decision : Excel


Do we really need 6 variables? x1 + y1 = 3,000 ===> y1 = 3,000 - x1

x2 + y2 = 2,000 ===> y2 = 2,000 - x2

x3 + y3 = 900 ===> y3 = 900 - x3

The objective function was MIN: 50x1 + 83x2 + 130x3 + 61y1 + 97y2 + 145y3

Just replace the valuesMIN: 50x1 + 83x2 + 130x3 + 61 (3,000 - x1 ) + 97 (

2,000 - x2) +

145 (900 - x3 )

MIN: 507500 - 11x1 -14x2 -15x3

We can even forget 507500, and change the the O.F. into

MIN - 11x1 -14x2 -15x3 or

MAX + 11x1 +14x2 +15x3




2x1 + 1.5x2 + 3x3 <= 10,000 } wiring

1x1 + 2.0x2 + 1x3 <= 5,000 } harnessing

Demand Constraints

x1 <= 3,000 } model 1

x2 <= 2,000 } model 2

x3 <= 900 } model 3

Nonnegativity Conditions

x1, x2, x3 >= 0


MAX + 11x1 +14x2 +15x3


MIN: 50x1 + 83x2 + 130x3

+ 61y1 + 97y2 + 145y3

Demand Constraints

x1 + y1 = 3,000 } model 1

x2 + y2 = 2,000 } model 2

x3 + y3 = 900 } model 3


2x1 + 1.5x2 + 3x3 <= 10,000 }

wiring

1x1 + 2.0x2 + 1x3 <= 5,000 }

harnessingNonnegativity Conditions

x1, x2, x3, y1, y2, y3 >= 0


y1 = 3,000- x1

y2 = 2,000-x2

y3 = 900-x3

MIN: 50x1 + 83x2 + 130x3 + 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 <= 3,000x2 <= 2,000x3 <= 900


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) CostTV 20000 15000Radio 12000 6000News paper 9000 4000Additional 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.


Problem 12. Marketing : formulation

Decision variablesx1 = 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

x1 4 x2 10

x3 7 x1 + x2 + x3 15

x1, x2, x3 0


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) ProfitMen 2 0.5 8Women 1.5 0.75 10Children 1 0.67 6Total 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.


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 gloves

Y1 : Number of full time employeesY2 : Number of part time employees


Problem 13. Constraints

Row material constraint2X1 + 1.5X2 + X3 5000Full time employeesY1 20Relationship between the number of Full and Part time employeesY1 2 Y2Labor Required.5X1 + .75X2 + .67X3 40 Y1 + 20Y2

Objective FunctionMax Z = 8X1 + 10X2 + 6X3 - 520 Y1 - 200 Y2

Non-negativityX1 , X2 , X3 , Y1 , Y2 0


Problem 13. Excel Solution

2 1.5 1 0 <= 50001 0 >= 201 -2 0 >= 0

0.5 0.75 0.67 -40 -20 0 <= 08 10 6 -520 -200 0

X1 X2 X3 Y1 Y2

2 1.5 1 5000 <= 50001 25 >= 201 -2 0 >= 0

0.5 0.75 0.67 -40 -20 0 <= 08 10 6 -520 -200 4500

2500 0 0 25 12.5X1 X2 X3 Y1 Y2

LP Formulation Practice Set 1. 2 Ardavan Asef-Vaziri June-2013LP-Formulation Management is...

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