1. A logistics specialist for Charm City Inc. must distribute cases of parts from 3 factories to 3 assembly plants. The monthly supplies and demands, along with the per-case transportation costs are:
Assembly Plant
1 2 3 Supply
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A 6 10 14 200
Factory B 2 2 6 400
C 2 8 7 200
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Demand 220 320 200
The specialist wants to distribute at least 100 cases of parts from factory B to assembly plant 2.
(a) Formulate a linear programming problem to minimize total cost for this transportation problem.
(b) Solve the linear programming formulation from part (a) by using either Excel or QM for Windows. Find and interpret the optimal solution and optimal value. Please also include the computer output with your submission.
The following questions are mathematical modeling questions. Please answer by defining decision variables, objective function, and all the constraints. Write all details of the formulation. Please do NOT solve the problems after formulating.
2. A congressman's district has recently been allocated $45 million for projects. The congressman has decided to allocate the money to four ongoing projects. However, the congressman wants to allocate the money in a way that will gain him the most votes in the upcoming election. The details of the four projects and votes per dollar for each project are given below.
Project Votes/dollar
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Parks 0.07
Education 0.08
Roads 0.09
Health Care 0.11
Family Welfare 0.08
In order to also satisfy some local influential citizens, he must meet the following guidelines.
- None of the projects can receive more than 30% of the total allocation.
- The amount allocated to education cannot exceed the amount allocated to health care.
- The amount allocated to roads must be equal to or more than the amount spent on parks.
- All of the money must be allocated.
Formulate a linear programming model for the above situation by determining
(a) The decision variables
(b) Determine the objective function. What does it represent?
(c) Determine all the constraints. Briefly describe what each constraint represents.
3. An ad campaign for a trip to Greece will be conducted in a limited geographical area and can use TV time, radio time, newspaper ads, and magazine ads. Information about each medium is shown below.
|
Medium
|
Cost Per Ad
|
Number Reached
|
|
TV
|
8500
|
12000
|
|
Radio
|
1800
|
4000
|
|
Newspaper
|
2400
|
5500
|
|
Magazine
|
2200
|
4500
|
The number of TV ads cannot be more than 4. Each of the media must have at least two ads. The total number of Magazine ads and Newspaper ads must be more than the total number of Radio ads and TV ads. There must be at least a total of 12 ads. The advertising budget is $50,000. The objective is to maximize the total number reached.
Formulate a linear programming model for the above situation by determining
(a) The decision variables
(b) Determine the objective function. What does it represent?
(c) Determine all the constraints. Briefly describe what each constraint represents.
4. The Charm City Vacuum Company wants to assign three salespersons to three sales regions. Given their experiences, the salespersons are able to cover the regions in different amounts of time. The amount of time (days) required by each salesperson to cover each region is shown in the following table:
Region (days)
Salesperson I II III
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A 11 18 12
B 11 15 14
C 10 14 16
However, because of his health reason, salesperson C does not want to be assigned to region II.
The Company wants to assign either salesperson A or salesperson C to region I. The objective is to minimize total time of covering the three sales regions.
(a) The decision variables
(b) Determine the objective function. What does it represent?
(c) Determine all the constraints. Briefly describe what each constraint represents.
5. The Charm City Inc. must select among a series of new investment alternatives. The potential investment alternatives, the net present value of the future stream of returns, the capital requirements, and the available capital funds over the next three years are given below:
Net Present Capital Requirements ($)
Alternative Value ($) Year 1 Year 2 Year 3
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Warehouse expansion 30,850 32,000 12,000 38,000
Test market new product 92,300 58,000 41,000 45,000
Advertising campaign 40,000 25,000 12,500 11,800
Research & Development 82,000 53,000 13,000 44,000
Purchase new equipment 33,000 12,500 4,500 8,900
_____________________________________________________________
Capital funds available 110,500 65,000 88,750
The company wants to select at least 3 alternatives. In addition, the company also wants to select at least two alternatives from the warehouse expansion, research & development and purchase new equipment alternatives.
Develop a capital budgeting problem to maximize the total net present value in this situation.
Please answer by defining decision variables, objective function, and all the constraints. Write all details of the formulation. Please do NOT solve the problem after formulating.
6. Jodi wants to lease a new car and start a part time business to give people car rides. She has contacted three automobile dealers for pricing information. Each dealer offered Jodi a closed-end 36-month lease with no down payment due at the time of signing. Each lease includes a monthly charge and a mileage allowance. Additional miles receive a surcharge on a per-mile basis. The three dealers provided the details about the monthly lease cost, the mileage allowance, and the cost for additional miles.
Jodi is not sure how many miles she will drive over the next three years for this business but she believes it is reasonable to assume that she will drive 10,000 miles per year, 14,000 miles per year, or 18,000 miles per year. With this assumption, Jodi estimated her total profit for the three lease options. The three lease options and the associated profits for each are given below:
Dealer 10000 Miles 14000 Miles 18000 Miles
A $ 7000 $10500 $13500
B $ 8500 $11500 $11000
C $10000 $ 9500 $ 9800
Determine the optimal decision to lease the car from a dealer and the profit associated with it by using the following decision criteria.
a. Maximax
b. Maximin
c. Equal likelihood
d. Minimax regret criterion.
7. For the problem given in Question 2, the probabilities are given by P(10000 miles) = 0.5, P(14000 miles) = 0.3 and P(18000 miles) = 0.2.
a. Compute the expected value for each decision and select the best one.
b. Compute the expected regret value for each decision and select the best one.
c. Calculate and interpret the expected value of perfect information.
8. A single-server queuing system with an infinite calling population and a first-come, first-served queue discipline has the following arrival and service rates:
λ = 12.1 customers per hour
µ = 14.5 customers per hour
Determine P0, P1, P4, L, Lq, W, Wq, and U.
9. A bank has one drive-up teller. The teller can serve at the rate of 11.5 bank customer in an hour. Customers arrive at the drive-up window on an average every 7.5 minutes. The bank is currently analyzing the possibility of adding a second drive-up window at an annual cost of $15,000. It is assumed that arriving cars would be equally divided between both windows. It is estimated that each minute's reduction in customer waiting time would increase the bank's revenue by $2,000 annually. Should the second drive-up window be installed?
10. Cakes baked by The Charm City Bakery are transported from the ovens to be packaged by one of five wrappers. Each wrapper can wrap an average of 35 cakes per hour. The cakes are brought to the wrappers at the rate of 160 per hour. Assuming it is a multiple-server waiting line model; determine the average number of cakes waiting for a wrapper and the average time a cake must wait for the wrapper. What is the probability that there will be more than 5 cakes in the system?