Operations Research Midterm#2 Dec 19, 2007 15:10-18:00
1. (10 points) A network with costs, supplies, and demands is described by Figure 1. Node a supplies 1 unit of goods, node b demands one unit, while node c is an intermediate node. Numbers along each arc (a, b), (a, c) and (c, b) are the shipping costs. The goal is to minimize the cost of sending flow.
a=1
b=-1 c=0
-1 1
-3 Figure 1
(a) (3 points) Find the minimum cost flow for the network in Figure 1.
(b) (3 points) Increase the shipping cost on each arc by 3 to obtain a new network described by Figure 2. Find the minimum cost flow in Figure 2.
a=1
b=-1 c=0
2 4
0 Figure 2
(c) (4 points) Explain why we can not solve the minimum cost flow problem of negative cost (Figure 1) by transforming to a new network whose cost on each arc is shifted by a common constant K (Figure 2)?
2. (15 points) This array describes an assignment problem with five people (A, B, C, D, E) and five jobs (1, 2, 3, 4, 5). This first person has a cost 81 if assigned to the first job;
a cost 14 if assigned to the second job; etc. The goal is to assign people to jobs in a way that minimizes total cost.
1 2 3 4 5
A 81 14 36 40 31 B 20 31 25 26 81 C 30 87 19 70 65 D 23 56 60 18 45 E 12 15 18 21 100
(a) (5 points) Show that subtracting a constant K from one row or column will not change the optimal way to assign jobs, but will lower(lift) the total assignment cost by K.
(b) (10 points) Find the optimal assignment by Hungarian method. Show step- by-step procedure.
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3. (15 points) Consider the following special linear programming problem:
min −x1− 2x2− 3x3· · · − nxn
s.t. ∆ = {x ≥ 0|
Xn i=1
xi = 1}.
(a) (5 points) Let us start the primal simplex method from a basic feasible solution.
How many iterations AT MOST the algorithm will take? Why?
(b) (5 points) Write down all simplex multipliers that are dual feasible.
(c) (5 points) Solve the dual problem.
4. (25 points) There is a furniture company that makes tables and chairs. A table requires 40 board feet of wood and a chair requires 30 board feet of wood. Wood costs $1 per board foot and 40,000 board feet of wood are available. It takes 2 hours of skilled labor to make an unfinished table or an unfinished chair. Three more hours of labor will turn an unfinished table in to a finished table; two more hours of skilled labor will turn an unfinished chair into a finished chair. There are 6000 hours of skilled labor available. (Assume that you do not need to pay for this labor.) The prices of output are given in the table below:
Product Price Unfinished Table $ 70
Finished Table $ 140 Unfinished Chair $ 60
Finished Chair $ 110
The answer to the problem is to construct only finished chairs 1333.333. Let’s assume that a sell of 13 chair is possible. The profit is $106,666.67. Use sensitivity analysis to answer the following questions.
(a) (5 points) Formulate an LP that describes the production plans that the firm can use to maximize its profits.
(b) (5 points) What would happen if the price of unfinished tables went up by 70 dollars more?
(c) (5 points) How much decrease in the price of finished chairs will force the firm to give up specializing in finished chairs? If so, what is the alternative optimal plan?
(d) (5 points) Since the firm has extra labors, how much can it increase the profit by increasing the supply of lumber so that the current production pattern is not affected? (That is, the firm will keep specializing in finished chairs.) (e) (5 points) The owner of the firm comes up with a design for a beautiful hand-
crafted cabinet. Each cabinet requires 250 hours of labor (this is 6 weeks of full time work) and uses 50 board feet of lumber. Suppose that the company can sell a cabinet for $200, would it be worthwhile?
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5. (20 points) Hollmann Cooperation owns three factories (Node A, B, C), two ware- houses (Node D, E), and 4 outlets (Node F , G, H, I). The production levels of A, B, and C and the demands from G, H, I, and J are specified, respectively, in the numbers along the nodes on the following network. All shipments are made to two warehouses before they are distributed to four of their outlets.
A 25
B 60
C 35
D 0
E 0
F 30
I 30 G 30
H 30
Figure 3 The cost matrices are
D E
A 4 7
B 8 5
C 5 6
F G H I
D 6 4 8 4
E 3 6 7 7
(a) (7 points) Solve the minimum cost flow problem using the Network Simplex method.
(b) (7 points) Formulate the problem into a transportation problem.
(c) (6 points) Solve the transportation problem in (b) to find the best way of shipping the product from factories to outlets.
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6. (15 points) A town has 7 residents R1, R2, · · · , R7; 4 clubs C1, C2, C3, C4; and 3 political parties P1, P2, P3. Their relations can be described by the following network.
P2
R2
P1
R3
R4
R5
R6
C2
C3
C4
C1
P3
R1
R7
Figure 4
where arc (Ci, Rj) means resident Rj is a member of club Ci and are (Rj, Pk) means resident Rj belongs to party Pk. Each club must nominate one of its members to represent it on the town’s governing council but one person can not represent two different clubs. For the reason of political balance, party P1 and P2 each can have at most one delegate while party P3 can have at most 2. A city council satisfying the above conditions is thus called“balanced”.
(a) (5 points) Add a source node s and a sink node t to the network. Connect (s, C1), (s, C2), (s, C3), (s, C4); also connect (P1, t), (P2, t), (P3, t). Please assign appropriate capacities to each arc so that solving the maximum flow in this network gives a balanced council for the town. Explain why this maximum flow network model is suitable for solving the balanced council problem.
(b) (5 points) Suppose R1 is sick so that he can not be a delegate and club C4
has decided to nominate R4. Find a feasible flow on this network to represent the situation and use the “f-augmenting path”to show that it is impossible to form a balanced council given this condition.
(c) (5 points) Since R1 can not be a delegate in any case, to resolve the political deadlock requires a renomination by club C4. Show that the “renomination”
can be achieved when you try to grow a f-augmenting path out of the feasible flow in (b) and find out all other resolutions(renominations) for club C4.
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