DSS610 Quiz 4 Linear programming
(NOTE: There is an Excel Spreadsheet available for these problems but for the ones asking for “Standard Form” hardcopy/manual layout and computation will be expected besides using the spreadsheet and Solver. But Solver is available for you to use to check your manual answers.)
1. Southern Sporting Good Company makes basketballs and footballs which have profits of $12 and $16 respectively. Each product is produced from two resources ru
er and leather. The resource requirements for each product and the total resources available are as follows:
Resource Requirements per Unit
Product
Ru
er (lb.)
Leather (ft2.)
Basketball
3
4
Football
2
5
Total resources available
500 lb.
800 ft2
a. State the optimal solution (standard form)
. What would be the effect on the optimal solution if the profit for a basketball changes from $12 to $13? What would be the effect if the profit for a football changed from $16 to $15?
c. What would be the effect on the optimal solution if 500 additional pounds of ru
er could be obtained? What would be the effect if 500 additional square feet of leather could be obtained?
2. A company produces two products, A and B, which have profits of $9 and $7, respectively. Each unit of product must be processed on two assembly lines where the required production times are as follows:
Hours/Unit
Product
Line 1
Line 2
A
12
4
B
4
8
Total Hours
60
40
a. Formulate a linear programming model to determine the optimal product mix that will maximize profit.
. Transform this model into standard form.
3. Solve problem 2 using the computer (Solver)
a. State the optimal solution
. What would be the effect on the optimal solution if the production time on line 1 was reduced to 40 hours?
4. For the linear programming model formulated in problem 2 and solved in problem 3, answer the following:
a. What are the sensitivity ranges for the objective function coefficients?
. Determine the shadow prices for additional hours of production time on line 1 and line 2 hours.
5. Formulate and solve the model for the following problem: (Solver)
Irwin Textile Mills produce two types of cotton cloth – denim and corduroy. Corduroy is a heavier grade of cotton cloth and as such requires 7.5 pounds of raw cotton per yard. Whereas denim requires 5 pounds of raw cotton per yard. A yard of corduroy requires 3.2 hours of processing time; a yard of denim requires 3.0 hours. Although the demand for denim is practically unlimited, the maximum demand for corduroy is 510 yards per month. The manufacturer has 6500 pounds of cotton and 3000 hours of processing time available each month. The manufacturer wants to know how many of each type of cloth to produce to maximize profit.
Answer the following questions:
a. How much extra cotton and processing time are left over at the optimal solution? Is the demand for corduroy met?
. What is the effect on the optimal solution if the profit per yard of denim is increased from $2.25 to $3.00? What is the effect of the profit per yard of corduroy is increased from $3.10 to $4.00?
c. What would be the effect on the optimal solution if Irwin Mills could contain only 6000 pound of cotton per month?
6. Continuing the model from problem 5, answer the following:
a. If Irwin Mills can obtain additional cotton or processing time, but not both, which should it select? How much? Explain your answer.
. Identify the sensitivity ranges for the objective function coefficients and for the constraint quantity values. Then explain the sensitivity range for the demand for corduroy.
7. United Aluminum Company of Cincinnati produces three grades (high, medium, and low) of aluminum at two mills. Each mill has a different production capacity (in tons per day) for each grade as follows:
Aluminum Grade
Mill
1
2
High
6
2
Medium
2
2
Low
4
10
The company has contracted with a manufacturing firm to supply at least 12 tons of high-grade aluminum, and 5 tons of low-grade aluminum. It costs United $6000 per day to operate mill 1 and $7000 per day to operate mill 2. The company wants to know the number of days to operate each mill in order to meet the contract at minimum cost.
Formulate a linear programming model for this problem in “Standard Form”.
8. Solve the linear programming model formulated in problem 7 by using the computer and answer the following:
a. Identify and explain the shadow prices for each of the aluminum grade contract requirements.
. Identify the sensitivity ranges for the objective function coefficients and the constraint quantity values.
c. Would the solution values change if the contract requirements for high-grade aluminum were increased from 12 tons to 20 tons? If yes, what would the new solution values be?
9. The manager of a Burger Doodle franchise wants to determine how many sausage biscuits and ham biscuits to prepare each morning for
eakfast customers to maximize profit. The two types of biscuits require the following resources:
Biscuit
Labor (hr.)
Sausage (lb.)
Ham (lb.)
Flour (lb.)
Sausage
0.010
0.10
----
0.04
Ham
0.024
----
0.15
0.04
The franchise has 6 hours of labor available each morning. The manager has a contract with a local grocer for 30 pounds of sausage and 30 pounds of ham each morning. The manager also purchases 16 pounds of flour. The profit for a sausage biscuit is $0.60; the profit for a ham biscuit is $0.50. Formulate a linear programming model for this problem (Solver).
10. Solve the linear programming model developed for problem 9 by using the computer and answer the following questions:
a. Identify and explain the shadow process for each of the resource constraints
. Which of the resources constrains profit the most?
c. Identify the sensitivity ranges for the profit of a sausage biscuit and the amount of sausage available. Explain these sensitivity ranges.
P1
Sporting Goods
(a) Resource Requirements per Unit (b-1) Resource Requirements per Unit
Product Ru
er (lb.) Leather (ft2) Product Ru
er (lb.) Leather (ft2)
Basketball 3 4 Basketball 3 4
Football 2 5 Football 2 5
input Constraints
Total resources available 500 800 Total resources available 500 800
Basketball Football Basketball Football
Profits ($) 12 16 Profits ($) 13 16
Basketball Decision variables Basketball
Football Football
Maximized profits Objective function Maximized profits
(b-2) Resource Requirements per Unit (C-1) Resource Requirements per Unit
Product Ru
er (lb.) Leather (ft2) Product Ru
er (lb.) Leather (ft2)
Basketball 3 4 Basketball 3 4
Football 2 5 Football 2 5
input Constraints
Total resources available 500 800 Total resources available 1000 800
Basketball Football Basketball Football
Profits ($) 12 15 Profits ($) 12 16
Basketball Decision variables Basketball
Football Football
Maximized profits Objective function Maximized profits
Please use computer method to solve the problem (C-2) Resource Requirements per Unit
Please enter your solution in Yellow cells Product Ru
er (lb.) Leather (ft2)
Basketball 3 4
Football 2 5
Total resources available 500 1300
Basketball Football
Profits ($) 12 16
Decision variables Basketball
Football
Objective function Maximized profits
P2
A & B products
Hours/ Unit
Product Line 1 Line2
A 12 4
B 4 8
Constraints Note: SUMPRODUCT(Col1, Col2) is a easier way to multiply
Total Hours 60 40 two rows or two columns
A B Standard Linear Programming: Maximixe 9 * A + 7 * B
Profits ($) 9 7 Subject to 12 * A + 4 * B <= 60
4 * A + 8 * B <= 40
Product A decision variables A >= 0
Product B B >= 0
Maximized profits Objective function
P3
A & B products
(a) Hours/ Unit (b) Hours/ Unit
Product Line 1 Line2 Product Line 1 Line2
A 12 4 A 12 4
B 4 8 B 4 8
Constraints
Total Hours 60 40 Total Hours 40 40
A B A B
Profits ($) 9 7 Profits ($) 9 7
Product A decision variables Product A
Product B Product B
Maximized profits Objective function Maximized profits
(c-1) Hours/ Unit
Product Line 1 Line2
A 12 4
B 4 8
Total Hours 60 40
A B
Profits ($) 9 15
Product A
Product B
Maximized profits
(c-2) Hours/ Unit
Product Line 1 Line2
A 12 4
B 4 8
Total Hours 60 40
A B
Profits ($) 9 20
Product A
Product B
Maximized profits
P4
Please run sensitivity analysis on P10 and enter the results (changes in coefficients of the objective function, and shadow prices) in YELLOW cells
Sensitivity analysis Min Max
Product A Coefficients in the objective function (profits for A and B)
Product B
Shadow price Additional Profits
Line 1
Line 2
P5
Irwin textile mills
(a) Corduroy denim
Profits for each product 3.1 2.25
Resources Available Resources Left ove
Cotton 7.5 5 <= 6500
Labor 3.2 3 <= 3000
Demands
Corduroy <= 510 Demand met
denim <= 1000000 unlimited
Maximized profits
(b) Corduroy denim Corduroy denim
Profits for each product 3.1 3 Profits for each product 4 2.25
Resources Available Resources Resources Available Resources
Cotton 7.5 5 <= 6500 Cotton 7.5 5 <= 6500
Labor 3.2 3 <= 3000 Labor 3.2 3 <= 3000
Demands Demands
Corduroy <= 510 Corduroy <= 510
denim <= 1000000 denim <= 1000000
Maximized profits Maximized profits
C Corduroy denim
Profits for each product 3.1 2.25
Resources Available Resources
Cotton 7.5 5 <= 6000
Labor 3.2 3 <= 3000
Demands
Corduroy <= 510
denim <= 1000000
Maximized profits
P6
Please run sensitivity analysis on P5 and answer the following questions.
a. If Irwin Mills can obtain additional cotton or processing time, but not both, which should it select? How much? Explain your answer.
Answer:
b. Identify the sensitivity ranges for the objective function coefficients and for the constraint quantity values.
Then explain the sensitivity range for the demand for corduroy.
Min Max
Corduroy Demand ranges
P7 & P8
United Aluminum company
Aluminum Grade Mill
1 2 constraints
High 6 2 >= 12
Medium 2 2 >= 8 Please note, this constraint is missing in the problem
Low 4 10 >= 5
Aluminum Grade Mill
1 2
Cost ($) 6000 7000
Mill #1
Mill #2
Minimize Cost ($)
a. Identify and explain the shadow prices for each of the aluminum grade contract requirements.
Shadow Price
High
Medium
Low
b. Identify the sensitivity ranges for the objective function coefficients and the constraint quantity values.
Min Max
Mill #1
Mill #2
c. Aluminum Grade Mill
1 2 constraints
High 6 2 >= 20
Medium 2 2 >= 8
Low 4 10 >= 5
Aluminum Grade Mill
1 2
Cost ($) 6000 7000
Mill #1
Mill #2
Minimize Cost ($)
P9
Burger Doodle franchise
Biscuit Labor (hr.) Sausage (lb.) Ham (lb.) Flour (lb.)
Sausage 0.01 0.1 --- 0.04
Ham 0.024 --- 0.15 0.04
Constraints
6 30 30 16
Sausage Ham
Profits ($) 0.6 0.5
Sausage
Ham
Maximize Profits