MSCI222 CWA 2020
In the following, you should replace X by the number between 0 and 99 formed by the 5th and 6th digits of your library card number, and Y by the number between 0 and 99 formed by the 7th and 8th digits of your library card number. For example, if your library card number is 08123456, then X = 34 and Y = 56.
Section A
Let D(x) be rounded down integer value of x. That is, D (3.5) =3.
Consider the following scenario where three buyers are bidding to buy combinations of two products (P1 and P2) that a seller owns. Seller has 10 units of P1 and 5 units of P2 for sale. The buyer bids are summarized as follows:
Buyer
P1
P2
Bid value (£)
1
10
4
11+D(X/10)
2
8
1
8
3
5
10
10 +D(Y/10)
Where Bid value represents the amount the buyer is willing to pay for his bid. For example, buyer 1 is willing to pay 11£ for 10 units of P1 and 4 units of P2 when 0 <= X <= 9. Assuming:
– Fractional bids are acceptable (for eg., ½ bid of buyer 1 will be 4 units of P1 and 2 units of P2 with a value of 5.5£, when 0 <= X <= 9),
– A bid is acceptable more than once (for e.g., 1.25 times of buyer 2 will be 10 units of P1 and 1.25 of P2 with a value of 10£ when, 0 <= X <= 9).
Q1) Formulate the seller’s revenue maximizing problem as a linear programming problem. (10 marks)
Q2) Solve it using simplex algorithm either by hand or by LINDO. (5 marks)
Q3) Analyze the sensitivity report. (10 marks)
Consider a similar scenario as above where seller owns four products P1, P2, P3, P4 of which he has 2, 4, 2, and 5 units respectively. As before, suppose that 3 buyers are interested in following combinations of products:
Buyer
P1
P2
P3
P4
Bid value (£)
1
1
1
0
1
10
1
0
0
1
1
10
2
1
1
0
0
6
2
0
0
1
0
6
3
0
0
0
1
3
Q4) Give two LP formulations that maximize seller’s revenue. The two formulations should have completely different variables. Solve them both and interpret the output including sensitivity reports. (25 marks)
Section B
IMC is currently investigating their production scheduling for one of their PC monitors. The monitor is currently produced at three different factories located in different parts of the country: in Northtown, Midtown, Southtown. Because of the system of nationwide dealerships that IMC operate, production from each of the factories is shipped to one of four distribution depots, which are located in different parts of the country: Depots A, B, C and D. Sales demand for the product occurring in one particular part of the country is then met from the nearest distribution depot.
Transportation costs in $’s are given in table below:
Depot A
Depot B
Depot C
Depot D
Northtown
20
11
12
13
Midtown
11
10
13
15
Southtown
9
12
15
20
The weekly production schedules of the three factories are known to be as follows:
Northtown
400+X
Midtown
700
Southtown
500
The sales forecast have identified the maximum demand that each distribution depot will have for this item. There are
Depot A
200
Depot B
400
Depot C
500
Depot D
500+Y
Any demand not met from the factories due to capacity limitations can be met from a third-party supplier who charges 20$ per unit to supply at any of the depots A, B, C and D.
Q1. (a) Model this problem as a Transportation Problem and solve it using the algorithm presented in the Lecture. (20)
Q1. (b) Are there more than one optimal solutions? If NO, give your reasoning clearly why this is the case. If YES, find alternative optimal solution and give a proof of its optimality. (5)
Q2. Use the information in your solution to answer the following questions:
(i) What would be the effect on the optimal solution and profit (if any), if the cost of supplying from Northtown to Depot B increases by 1$. (5)
(ii) What would be the effect, if any, if the cost of supplying from Midtown to depot D increased by 1$? (10)
(iii) Illustrate, using parametric programming tool of LINDO, the impact on optimal cost as the production from southtown varies between 500 and 600. (10)
