BA 555Midterm Examination #2
Fall 2006Answer Key
BA 555 Practical Business Analysis
Answer Key to Midterm Examination #2
NAME (Please Print)Question 1.(8 points) Chemical Bank is attempting to determine where its assets should be invested during the current year. At present, $800,000 is available for investment in bonds, home loans, auto loans, and personal loans. The annual rate of return on each type of investment is known to be the following: bonds, 12%, home loans, 18%, auto loans, 15%, personal loans, 22%. To ensure that the banks portfolio is not too risky, the bank’s investment manager has placed the following restrictions on the bank portfolio:
- No more than 30% of the total amount invested may be in personal loans
- The amount invested in home loans cannot exceed the amount invested in auto loans
- The amount invested in personal loans cannot exceed the amount invested in bonds
Formulate an LP model for the bank to maximize the annual return on its investment portfolio.
Decision Variables (Be specific. Do not abbreviate.)B = $ amount invested in bonds
H = $ amount invested in home loans
A = $ amount invested in auto loans
P = $ amount invested in personal loans
Objective Function (Be clear. Do not forget the keyword Max or Min.)
Max 0.12 B + 0.18 H + 0.15 A + 0.22 P
Constraints (Add comments. Do not abbreviate. No need to express the constraints in the LINDO form. Do not use any decision variable that is not defined.)
! fund availability
B + H + A + P = 800000
! personal loans restriction
P <= 0.3 (B + H + A + P)
P <= B
! home loans requirement
H <= A
Question 2.(8 points) Great Outdoors Recycling (GOR) is in the process of locating drop off boxes in several locations around the city. Five retail locations (A, B, C, D, and E) with large parking lots have agreed to allow bins to be placed in their lots, but GOR would like to put as few bins in as possible and still provide access for each of the eight neighborhoods (1 through 8) in the city. Shopping patterns show that neighborhoods are served by retailers in the following pattern:
Location / Neighborhoods ServedA / 1, 2, 4, 6
B / 2, 3, 7, 8
C / 1, 2, 5, 6, 8
D / 3, 4, 5
E / 5, 7
Formulate an integer linear programming model for Great Outdoors Recycling. That is, define decision variables and state objective function and constraints.
Decision Variables (Be specific. Do not abbreviate.)A = 1 if location A is selected; = 0 if not
B = 1 if location B is selected; = 0 if not
C = 1 if location C is selected; = 0 if not
D = 1 if location D is selected; = 0 if not
E = 1 if location E is selected; = 0 if not
Objective Function (Be clear. Do not forget the keyword Max or Min.)
Min A + B + C + D + E
Constraints (Add comments. Do not abbreviate. No need to express the constraints in the LINDO form. Do not use any decision variable that is not defined.)
! to serve neighborhood 1 (and 6)
A + C >= 1
! to serve neighborhood 2
A + B + C >= 1
! to serve neighborhood 3
B + D >= 1
! to serve neighborhood 4
A + D >= 1
! to serve neighborhood 5
C + D + E >= 1
! to serve neighborhood 7
B + E >= 1
! to serve neighborhood 8
B + C >= 1
! The constraint for neighborhood 6 is the same as the one for neighborhood 1
! and can be omitted.
Question 3.(8 points) A real estate developer is planning to build an apartment building specifically for graduate students on a parcel of land adjacent to OSU. Four types of apartments can be included in the building: efficiencies, and one-, two-, three-bedroom apartments. Eachefficiency requires 500 square feet; each one-bedroom apartment requires 700 square feet; each two-bedroom apartment requires 800 square feet; and each three-bedroom apartment requires 1,000 square feet.
The developer believes that the building should include no more than 15 one-bedroom units, 22 two-bedroom units, and 10 three-bedroom units. Local zoning ordinances do not allow the developer to build more than 40 units in this particular building location, and restrict the building to a maximum of 40,000 square feet. The developer believes that, upon completion, the monthly rents from efficiency, one-, two-, and three-bedroom will be $350, $400, $550, and $800 per unit. Formulate an LP model for this problem to maximize the monthly rent.
Decision Variables (Be specific. Do not abbreviate.)E = # of efficiencies built
B1 = # of one bedrooms built
B2 = # of two bedrooms built
B3 = # of three bedrooms built
Objective Function (Be clear. Do not forget the keyword Max or Min.)
Max 350 E + 400 B1 + 550 B2 + 800 B3
Constraints (Add comments. Do not abbreviate. No need to express the constraints in the LINDO form. Do not use any decision variable that is not defined.)
! limits on the # of units
B1 <= 15
B2 <= 22
B3 <= 10
! local zoning ordinances
E + B1 + B2 + B3 <= 40
! limit on square footage
500 E + 700 B1 + 800 B2 + 1000 B3 <= 40000
Question 4.A manufacturer produces desks, bookcases, and file cabinets. The production process requires essentially three stages: fabrication, painting, and assembly. The required hours at each stage for each product and the available hours are summarized in the following table.
Required Hours (hrs.) / AvailableStage / Desk / Bookcase / File Cabinet / Hours
Fabrication / 0.1 / 0.2 / 0.5 / 450
Painting / 0.2 / 0.4 / 0.2 / 350
Assembly / 0.3 / 0.2 / 0.3 / 300
A summary of the information regarding the demand, production requirement, and profit contribution for each product is given below:
Desk / Bookcase / File CabinetMaximum demand (units) / 100 / 500 / 600
Minimum production (units) / - / - / 250
Profit contribution ($) / 75 / 80 / 36
Decision variables are defined as follows. D = units of desk to be produced, B = units of bookcase to be produced, F = units of file cabinet to be produced. For simplicity, assume these are regular decision variables and can assume real values.
Answer the following questions based on the information on the answer and sensitivity reports. Consider each part independent of the other parts.
- (2 points) How many hours of fabrication are left when producing at optimal?
ANSWER / The slack of the “fabrication” constraint is 56.67, representing unused hours of fabrication.
- (2 points) What would be the total profit contribution if the maximum demand for bookcases could be expanded by 30 units?
ANSWER / The new right-hand-side value of the “maximum demand for bookcases” is 530 and is within the range of feasibility (450, 562.5). Hence, we can use the dual price to calculate the new profit. The new profit will increase $1680 = $56 × 30.
- (2 points) Suppose that the profit contribution of desks had to be reduced due to competition. How much of a reduction could be absorbed without altering the production plan?
ANSWER / Based on the range of optimality, the profit of desks in the objective function can be reduced by $39 to $36 without altering the current optimal production plan.
- (5 points) The company’s advertising department developed three separate advertising campaigns for desks, bookcases and file cabinets, respectively, to increase the market size (i.e., to increase the demand). The total cost of each campaign along with the anticipated increase in market size are given below. Which campaign, if any, should the company adopt? Explain.
Campaign for … / Increase in Demand / Total Cost
Desk / 50 / $2,000
Bookcase / 40 / $2,000
File Cabinet / 20 / $1,000
ANSWER / The right-hand-side values of the three “max. demand” constraints can be changed by the advertising campaigns. The resulting new maximum demands are all within their perspective range of feasibility and thus the corresponding dual price can be used to calculate the new profit.
For Desk: the additional demand of 50 units will increase the profit by $39 × 50 = $1950 < the cost of the campaign $2000.
For Bookcase: the additional demand of 40 units will increase the profit by $56 × 40 = $2240 > the cost of the campaign $2000.
For File Cabinet: the additional demand of 20 units will increase the profit by $0 × 20 = $0 < the cost of the campaign $1000.
The bookcase campaign is the obvious choice.
- (5 points) At the time of developing the model, there was no limit on the availability of steel. Due to unforeseen circumstances, the availability of steel to the company is now limited to 2000 tons. Each desk, bookcase, and file cabinet requires 3, 2, and 1 ton of steel to produce respectively. How will the new shortage affect the existing optimal solutions? Explain.
ANSWER / At the current optimal production plan (100 desks, 500 bookcases, and 566.67 file cabinets), we require
100 × 3 + 500 × 2 + 566.67 × 1 = 1866.67 tons of steel.
This is still below the limit of 2000 tons and thus there is no effect to the existing optimal solutions.
The LP Model to Question 4
MAX 75 D + 80 B + 36 F
SUBJECT TO
2) 0.1 D + 0.2 B + 0.5 F <= 450(fabrication)
3) 0.2 D + 0.4 B + 0.2 F <= 350(painting)
4) 0.3 D + 0.2 B + 0.3 F <= 300(assembly)
5) D <= 100(max. demand)
6) B <= 500(max. demand)
7) F <= 600(max. demand)
8) F >= 250(min. production)
END
The LINDO Answer Report
OBJECTIVE FUNCTION VALUE
1) 67900.00
VARIABLE VALUE REDUCED COST
D 100.000000 .000000
B 500.000000 .000000
F 566.666600 .000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 56.666680 .000000
3) 16.666670 .000000
4) .000000 120.000000
5) .000000 39.000000
6) .000000 56.000000
7) 33.333370 .000000
8) 316.666600 .000000
The LINDO Sensitivity Report
OBJCOEFFICIENTRANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
D 75.000000 INFINITY 39.000000
B 80.000000 INFINITY 56.000000
F 36.000000 39.000000 36.000000
RIGHTHANDSIDERANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 450.000000 INFINITY 56.666680
3 350.000000 INFINITY 16.666670
4 300.000000 10.000010 94.999990
5 100.000000 316.666600 33.333370
6 500.000000 62.500000 50.000050
7 600.000000 INFINITY 33.333370
8 250.000000 316.666600 INFINITY
The EXCEL Model to Question 4
The EXCEL Answer and Sensitivity Reports
Hsieh, P.-H.1