2012 Fall Final Exam+Solution

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  • Topic: Optimization, Minimum spanning tree, Flow network
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Sample Exam # 2 solutions

ESI 6314 Deterministic Methods in Operations Research

1. Valdosta Tile Co. has two clay mining pits, P1 and P2, each of which supplies 15 tons of clay per month to three tile making kilns, K1, K2 and K3, each of which require 10 tons of clay per month. The shipping costs ($ per ton) are given in the table below: [pic]

a) Formulate a transportation linear program to minimize shipping costs b) Determine a “Northwest Corner Rule” basic feasible solution for this problem. Calculate the total transportation cost associated with this solution. c) Perform one iteration of the transportation simplex method. What do you observe?

a)
min 1.2 x11 + 0.6 x12 +1.5 x13 +3.6 x21 +2.4 x22 +1.2 x23 s.t. x11 + x12 +x13 =15
x21 + x22 +x23 =15
x11 + x21 =10
x12 + x22 =10
x13 + x23 =10
xij>=0

b)
| |1.2 |0.6 |1.5 | | | |10 |5 | |15 | | |3.6 |2.4 |1.2 | | | | |5 |10 |15 |

101010
The total cost is: 10*1.2 + 5*0.6 + 5*2.4 + 10*1.2 = 39

c) We have: u1=0, u2=1.8, v1=1.2, v2=0.6, v3= -0.6
Using these values, we have:
_
c13 = 0 + (-0.6) – 1.5 = -2.1
_
c21 = 1.8 + 1.2 – 3.6 = -0.6
So, this is the optimal solution.
2. Think of the network shown in the figure below as a highway map, and the number recorded next to each arc as the travel time for each arc. A traveler plans to drive from node 1 to node 12 on this highway. Find the best path for this traveler using Dijkstra’s algorithm. [pic]

Noded(j)pred (j)
100
211
3102
441
552
6125
7104
8105
9128
101311
11128
12139

The shortest path from 1 to 12 is: 1-2-5-8-9-12 with length 13.

3. Find minimum spanning trees for the networks shown in the following Figures. Start from any node. For each network, specify the order in which the nodes are added to the tree and the total length of the tree. [pic]

a) Starting from node 1 and using Prim’s algorithm, we get the following two possible orders of nodes: 1-4-3-2-6-5-8-7-9 or 1-4-3-6-5-8-7-9-2

The minimum spanning tree is the following with total cost of 35:

b) Again, starting with node 1 we get the following order of nodes: 1-2-3-6-9-5-7-8-4

The minimum spanning tree is the following with total cost of 101: 4. Find the Maximum Flow for the network below, assuming that node 1 is the source node, and node 9 is the sink node. Report the optimal flow in the network based on the paths you have allocated.

Also, find the cut in this network with the total capacity equal to the amount of maximum flow.

[pic]
The maximum flow is 17 with the following arc flows:
f12 = 10
f13 = 7
f24 = 7
f25 = 3
f34 = 3
f36 = 4
f46 = 3
f48 = 7
f45 = 0
f47 = 0
f57 = 3
f79 = 3
f89 = 10
f69 = 4

which can be obtained by sending:
3 units thorough 1-2-5-7-9
7 units thorough 1-2-4-8-9
3 units thorough 1-3-4-6-8-9
4 units thorough 1-3-6-9

The corresponding minimum cut is defined by the sets {1,3} and {2,4,5,6,7,8,9}

5. A company is considering opening warehouses in cities 1, 2, 3 and 4. Each warehouse can ship 100 units per week. The weekly cost of operating each warehouse is $400, $500, $300, and $150 for cities 1 - 4, respectively. Region 1 needs 80 units per week, Region 2 needs 70 per week, and Region 3 needs 40 per week. The cost of shipping (per unit) from city i to Region j is given and equal to pij. It is necessary to meet the weekly demand of the regions and to satisfy the additional conditions:

• If the warehouse...
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