Management Science
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MNS407 Management Science, Summer 1 2023
407 Module 1: Homework#1
(Homework)
INSTRUCTOR
Karen Joiner
National University, La Jolla
CA
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This homework covers material in Chapters 1 to 3 as well
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7/6/23, 7:50 AM 407 Module 1: Homework#1 - MNS407 Management Science, Summer 1 2023 | WebAssign
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Recall the production model:
Suppose the firm in this example considers a second product that has a unit profit of $5 and requires 2 hours of labor time for each unit produced. Use y as the number of units of second product produced.
(a) Show the mathematical model when both products are considered simultaneously.
Max
s.t.
x ≥ 0, y ≥ 0
(b) Identify the controllable and uncontrollable inputs for this model. (Select all that apply.)
(i) controllable
x
y
labor hours
labor availability
unit profit
(ii) uncontrollable
x
y
labor hours
labor availability
Max 10x s.t.
5x ≤ 40 x ≥ 0
1. [–/0.8 Points] ASWMSCI15 1.E.008.DETAILS
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unit profit
(c) Draw the flowchart of the input-output process for this model. (Submit a file with a maximum size of 1 MB.)
Choose File No file chosen
This answer has not been graded yet.
(d) What are the optimal solution values of x and y? (Let P be total profit.)
x =
y =
P = $
(e) Is the model developed in part (a) a deterministic or a stochastic model? Explain.
The uncontrollable inputs are ---Select--- and ---Select--- , so the model is ---Select--- .
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For most products, higher prices result in a decreased demand, whereas lower prices result in an increased demand. Let
d = annual demand for a product in units p = price per unit
Assume that a firm accepts the following price–demand relationship as being realistic:
where p must be between $20 and $80.
(a) How many units can the firm sell at the $20 per-unit price? At the $80 per-unit price?
units at $20 per-unit =
units at $80 per-unit =
(b) What happens to annual units demanded for the product if the firm increases the per-unit price from $25 to $26? From $43 to $44? From $67 to $68? What is the suggested relationship between the per- unit price and annual demand for the product in units?
(i) What happens to annual units demanded for the product if the firm increases the per-unit price from $25 to $26?
units at $25 per-unit =
units at $26 per-unit =
(ii) What happens to annual units demanded for the product if the firm increases the per-unit price from $43 to $44?
units at $43 per-unit =
units at $44 per-unit =
(iii) What happens to annual units demanded for the product if the firm increases the per-unit price from $67 to $68?
units at $67 per-unit =
units at $68 per-unit =
(iv) What is the suggested relationship between the per-unit price and annual demand for the product in units?
This suggests that the relationship is ---Select--- and that annual demand ---Select--- by units for every $1 in price increase.
d = 1,000 − 12p
2. [–/0.8 Points] ASWMSCI15 1.E.011.DETAILS
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(c) Show the mathematical model for the total revenue (TR) in terms of p, which is the annual demand multiplied by the unit price.
TR =
(d) Based on other considerations, the firm's management will only consider price alternatives of $28, $40, and $52. Use your model from part (b) to determine the price alternative that will maximize the total revenue.
TR at $28 per-unit = $
TR at $40 per-unit = $
TR at $52 per-unit = $
Given the constraints, the Total Revenue is maximized at $ per-unit.
(e) What are the expected annual demand and the total revenue corresponding to your recommended price?
d =
TR = $
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Micromedia offers computer training seminars on a variety of topics. In the seminars each student works at a personal computer, practicing the particular activity that the instructor is presenting. Micromedia is currently planning a two-day seminar on the use of Microsoft Excel in statistical analysis. The projected fee for the seminar is $620 per student. The cost for the conference room, instructor compensation, lab assistants, and promotion is $11,520. Micromedia rents computers for its seminars at a cost of $130 per computer per day.
(a) Develop a model for the total cost (C) to put on the seminar. Let x represent the number of students who enroll in the seminar.
C =
(b) Develop a model for the total profit (P) if x students enroll in the seminar.
P =
(c) Micromedia has forecast an enrollment of 50 students for the seminar. How much profit will be earned if their forecast is accurate?
P(50) = $
(d) Compute the break-even point.
x =
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3. [–/0.8 Points] ASWMSCI15 1.E.013.DETAILS
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Par, Inc., is a small manufacturer of golf equipment and supplies whose management has decided to move into the market for medium- and high-priced golf bags. After a thorough investigation of the steps involved in manufacturing a golf bag, management determined that each golf bag produced will require the following operations: cutting and dyeing the material, sewing, finishing, and inspection and packaging. A mathematical model has been developed that shows the constraints of production, time, and cost. It can be used to determine the number of standard bags and the number of deluxe bags to produce in order to maximize total profit contribution.
Let S = number of standard bags.
Let D = number of deluxe bags.
Refer to the Par, Inc., problem described above and in Section 2.1. Suppose that Par, Inc., management encounters the following situations.
If each of these situations is encountered separately, what is the optimal solution and the total profit contribution?
(a) The accounting department revises its estimate of the profit contribution for the deluxe bag to $18 per bag.
The optimal solution occurs at with a profit of $
Max 10S + 9D s.t.
S + 1D ≤630 Cutting and Dyeing
S + D ≤600 Sewing
1S + D ≤708 Finishing
S + D≤135 Inspection and Packaging
S, D ≥ 0
7 10 1 2
5 6 2 3
1 10
1 4
(S, D) =
4. [–/0.8 Points] ASWMSCI15 2.E.015.DETAILS
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.
(b) A new low-cost material is available for the standard bag, and the profit contribution per standard bag can be increased to $20 per bag. (Assume that the profit contribution of the deluxe bag is the original $9 value.)
The optimal solution occurs at with a profit of $
.
(c) New sewing equipment is available that would increase the sewing operation capacity to 700 hours. (Assume that is the appropriate objective function.)
The optimal solution occurs at with a profit of $
.
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(S, D) =
10A + 9B
(S, D) =
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Southern Oil Company produces two grades of gasoline: regular and premium. The profit contributions are $0.40 per gallon for regular gasoline and $0.70 per gallon for premium gasoline. Each gallon of regular gasoline contains 0.3 gallons of grade A crude oil, and each gallon of premium gasoline contains 0.6 gallons of grade A crude oil. For the next production period, Southern has 16,500 gallons of grade A crude oil available. The refinery used to produce the gasoline has a production capacity of 50,000 gallons for the next production period. Southern Oil's distributors have indicated that demand for the premium gasoline for the next production period will be at most 20,000 gallons.
(a) Formulate a linear programming model that can be used to determine the number of gallons of regular gasoline and the number of gallons of premium gasoline that should be produced in order to maximize total profit contribution. (Let R be the gallons of regular gasoline and let P be the gallons of premium gasoline.)
Max
s.t. Grade A crude oil available
Production capacity
Demand for premium
(b) What is the optimal solution?
R =
P =
profit = $
(c) What is the value of the slack variable in the Grade A crude oil constraint?
R, P ≥ 0
5. [–/0.8 Points] ASWMSCI15 2.E.041.DETAILS
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Interpret this value.
After reaching the optimal solution, there is still this amount of grade A crude oil to be used.
After reaching the optimal solution, all available grade A crude oil has been used.
In order to reach the optimal solution, this amount of grade A crude oil is required.
What is the value of the slack variable in the Production capacity constraint?
Interpret this value.
After reaching the optimal solution, the refinery is still able to produce this amount of gasoline.
After reaching the optimal solution, the total production capacity is used.
In order to reach the optimal solution, the refinery must produce this amount of additional gasoline.
What is the value of the slack variable in the Demand for premium constraint?
Interpret this value.
After reaching the optimal solution, the produced amount of premium gasoline is this much less than the maximum demand.
After reaching the optimal solution, the maximum demand for premium gasoline has been reached.
In order to reach the optimal solution, this amount of additional premium gasoline needs to be produced in order to meet demand.
(d) What are the binding constraints? (Select all that apply.)
profit
grade A crude oil available
production capacity
demand for premium
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For the linear program
Max
s.t.
find the optimal solution using the graphical solution procedure. What is the value of the objective function at the optimal solution?
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5A + 9B
1A + 2B ≤ 8 7A + 5B ≤ 35
A, B ≥ 0
at (A, B) =
6. [–/0.8 Points] ASWMSCI15 2.E.010.DETAILS
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Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $63, $95, and $135, respectively. The production requirements per unit are as follows:
Number of Fans
Number of Cooling Coils
Manufacturing Time (hours)
Economy 1 1 8
Standard 1 2 12
Deluxe 1 4 14
For the coming production period, the company has 240 fan motors, 340 cooling coils, and 2,600 hours of manufacturing time available. How many economy models (E), standard models (S), and deluxe models (D) should the company produce in order to maximize profit? The linear programming model for the problem is as follows:
The computer solution is shown below.
Optimal Objective Value = 18320.00000
Variable Value Reduced Cost
E 140.00000 0.00000
S 100.00000 0.00000
D 0.00000 −24.00000
Constraint Slack/Surplus Dual Value
1 0.00000 31.00000
2 0.00000 32.00000
3 280.00000 0.00000
Variable Objective
Coefficient
Allowable
Increase
Allowable
Decrease
Max 63E + 95S + 135D s.t.
1E + 1S + 1D ≤ 240 Fan motors 1E + 2S + 4D ≤ 340 Cooling coils 8E + 12S + 14D ≤ 2,600 Manufacturing time
E, S, D ≥ 0
7. [–/0.8 Points] ASWMSCI15 3.E.013.DETAILS
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E 63.00000 12.00000 15.50000
S 95.00000 31.00000 8.00000
D 135.00000 24.00000 Infinite
Constraint RHS
Value
Allowable
Increase
Allowable
Decrease
1 240.00000 70.00000 70.00000
2 340.00000 70.00000 100.00000
3 2600.00000 Infinite 280.00000
(a) Identify the range of optimality for each objective function coefficient. (If there is no upper or lower limit, enter NO LIMIT.)
E
to
S
to
D
to
(b) Suppose the profit for the economy model is increased by $6 per unit, the profit for the standard model is decreased by $2 per unit, and the profit for the deluxe model is increased by $4 per unit. What will the new optimal solution be?
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E units
S units
D units
profit $
(c) Identify the range of feasibility for the right-hand-side values. (If there is no upper or lower limit, enter NO LIMIT.)
constraint 1
to
constraint 2
to
constraint 3
to
(d) If the number of fan motors available for production is increased by 100, will the dual value for that constraint change? Explain.
Yes, the dual value will change because 100 is greater than the allowable increase of 12.
Yes, the dual value will change because 100 is greater than the allowable increase of 70.
No, the dual value will not change because 100 is less than the allowable increase of 310.
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No, the dual value will not change because there is no upper limit to how much the constraint can increase.
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Identify the feasible region for the following set of constraints.
0.5A + 0.25B≥35 1A + 5B≥255
0.25A + 0.5B≤45 A, B ≥ 0
ℹ
ℹ
8. [–/0.8 Points] ASWMSCI15 2.E.007.DETAILS
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ℹ
ℹ
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Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcher's model. Letting
R = number of regular gloves C = number of catcher's mitts
leads to the following formulation:
The computer solution is shown below.
Optimal Objective Value = 3550.00000
Variable Value Reduced Cost
R 500.00000 0.00000
C 150.00000 0.00000
Constraint Slack/Surplus Dual Value
1 275.00000 0.00000
2 0.00000 4.50000
3 0.00000 22.00000
Variable Objective
Coefficient
Allowable
Increase
Allowable
Decrease
R 5.00000 5.50000 1.50000
C 7.00000 3.00000 3.66667
Constraint RHS
Value
Allowable
Increase
Allowable
Decrease
1 1000.00000 Infinite 275.00000
2 300.00000 100.00000 166.66667
3 100.00000 55.00000 25.00000
Max 5R + 7C s.t.
R + C ≤ 1,000 Cutting and sewing
R + C ≤ 300 Finishing
R + C ≤ 100 Packaging and shipping
R, C ≥ 0
3 2
1 2
1 3
1 8
1 4
9. [–/0.8 Points] ASWMSCI15 3.E.005.DETAILS
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(a) What is the optimal solution, and what is the value of the total profit contribution?
regular glove
catcher's mitt
value
(b) Which constraints are binding? (Select all that apply.)
cutting and sewing
finishing
packaging and shipping
(c) What are the dual values for the resources? Interpret each.
constraint 1 Each additional hour of cutting and sewing will improve profits by $275.00.
Each additional hour of cutting and sewing will improve profits by $4.50.
Each additional hour of cutting and sewing will improve profits by $22.00.
Additional hours of cutting and sewing will not improve profits.
constraint 2 Each additional hour of finishing will improve profits by $275.00.
Each additional hour of finishing will improve profits by $4.50.
Each additional hour of finishing will improve profits by $22.00.
Additional hours of finishing will not improve profits.
constraint 3 Each additional hour of packing and shipping will improve profits by $275.00.
Each additional hour of packing and shipping will improve profits by $4.50.
Each additional hour of packing and shipping will improve profits by $22.00.
Additional hours of packing and shipping will not improve profits.
(d) If overtime can be scheduled in one of the departments, where would you recommend doing so?
cutting and sewing
finishing
packaging and shipping
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Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:
Rental Class
Super Saver Deluxe Business
Room Type I $30 $35 —
Type II $20 $30 $40
Type I rooms do not have high-speed Internet access and are not available for the Business rental class.
Round Tree's management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 160 rentals in the Super Saver class, 60 rentals in the Deluxe class, and 50 rentals in the Business class. Round Tree has 100 Type I rooms and 120 Type II rooms.
(a) Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types. Is the demand by any rental class not satisfied? Explain.
If demand materializes as forecast, there will be rooms not reserved in the ---Select--- class.
(b) How many reservations can be accommodated in each rental class?
Super Saver
Deluxe
Business
(c) Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5, should this incentive be offered?
Increasing demand for Deluxe class accommodations will ---Select--- profit. Since the predicted change in profit is ---Select--- the added cost of the breakfast, the incentive ---Select--- be offered.
(d) With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a Type I or a Type II room? Why?
10. [–/0.8 Points] ASWMSCI15 3.E.021.DETAILS
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---Select--- , because this will increase profit by $ .
(e) Could the linear programming model be modified to plan for the allocation of rental demand for the next night? What information would be needed and how would the model change?
We would need a forecast of demand for each rental class on the next night to use as the right- hand sides of the first three constraints.
We would need to know if Type 1 rooms can be used as Business class rooms the next night and add a variable to the objective function.
We would need to know whether the profit per night of each type of room and rental class will change and use these as objective coefficients.
We would need to know how many rooms of Type I and Type II there will be on the next night to use as the right-hand sides of the last two constraints.
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