1. LP Formulation Practice Set 1

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 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

4. 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?

5. 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

6. 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?

7. 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

8. 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?

9. 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

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

12. 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?

13. 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

14. 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  

15. 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)

16. 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

17. 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.

18. 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

19. 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

20. 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.

21. 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.

22. 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.

23. Problem 10. Product Mix : Excel solution

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 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.

25. 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

26. 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

27. Problem 11. Make / buy decision : Excel

28. 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

29. 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

30. 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

31. 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.

32. 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

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) 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.

34. 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

35. 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

36. Problem 13. Excel Solution