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Unformatted text preview: (313330: Operations Research Methods for Proﬁt and Value Engineering
Midterm H Apr. 3, 2008 NAME: UIN: Please Show all the steps. 9% mats Q1 /30
Q2 /15
Q3 /15
Q4 /15
Q5 /17
Q6 /8 Total  /100 Problem 1. [30 points] A company has three plants and three distribution centers. The unit cost for shipping products from
each plant to each distribution center, the supply of each plant, and the demand of each distribution
center is given in the following table. Notice that the problem is unbalanced, if a unit from a plant is
not shipped out (to any of the distribution centers), a storage cost is incurred at the rate of $2, $5 and $3 per unit for plants 1, 2, and 3, respectively. Additionally, all the supply at plant 3 must be shipped
out completely to make room for a new product. Distribution Center (a) (7 points) Formulate the problem as a balanced standard transportation problem, provide the
transportation tableau. (b) {6 points) Find an initial basic feasible solution. (c) (8 points) Calculate reduced costs for all nonbasic variables in the initial basic feasible solution.
(d) (5 points) Choose an entering variable and ﬁnd the leaving variable. (e) (4 points) Write down the new basic feasible solution. [0+30 +40 :80
.10 +30 +20 =70 ‘ 30 80>70_ Mai To add a dummy demand.
40 {NH—1A dxmnnoi (0 WWW tiwt all “rim ”PM at Plant 3 Must Be
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““13 1 U2+VQ ‘Cag ’— ‘I‘FS‘S :‘3 ‘qu L uﬁuq C24 =., “HM"; = M“ 6 (in Problem 2. [15 points] The local government is trying to construct roads to connect six villages. It is required that for each pair
of the villages, there must be a route (may through some other intermediate villages) connecting them. The following ﬁgure depicts candidate road linkage among those villages with the cost shown on each
arc. (a) {10 points) Determine the most economical road network. (b) (5 points) Because of heavy trafﬁc, the government decided that Villages 2 and 6 have to he linked directly. Other than this, other villages can be linked directly or indirectly. Determine the most
economical road network. (intentionally left blank) Problem 3. [15 points]
Given a directed graph as follows: Find the shortest path from node I to all other nodes. Fwd—l" (awﬁ‘t [A ‘ #922 l>3_>4"">1 6
1"” 1 I~—=:B 2
3641 jggﬁbﬁ} 3 I.» 5’ l."—"95‘"94 96—93 8
[A 6 [—93’5‘} “>5 7 [97 [—53ﬁ>4“§6—3‘7 ’3 (intentionally left blank) Problem 4. [20 points}
Given a directed graph as follows. We want to ﬁnd the maximum ﬂow from 3 to t on this graph. Suppose after sending ﬂow on several augmenting paths, we get the following graph with the the two numbers on each are indicating the current
ﬂow and the residual capacity on it, respectively. (15,35) (a) (3 points) Given a cut with S : {s,1,2,3} and T = {4, 5,6, 7, 8, t}, what is the capacity of this cut. 30+5+40+30 H§ 2:20 (13) (3 points) Is the path 3 —¢ 1 a 4 —> 7 —) i an augmenting path? why? No 4—57 olives M't' have regiolual afﬁnity  l (c) (3 points) Is the path 3 —> 1 "a 4 a 6 we 2 —> 5 —) 8 —a t an augmenting path? why? Yes , (d) (7 points) Given an augmenting path 3 —> 2 —> 5 —> 8 —> t, determine the maximum amount of ﬂow
we can send aiong this path. Update the ﬂows and residual capacity in the following ﬁgure. 0 ° 0,20) (50,10) ' 0
(0, 00 i 0) (40,10) o o a 7) o 0’40) 0
(20.13) (02) (L 6 (15,15) (15,5)
(15’0) (15, (435) o (L,“
 o (e) (5 points) Do part (d) again with the augmenting path 3 +> 2 —> 5 ——> 8 a 6 —+ 7 —> t. (10,1 (15,35) Problem 5. {17 points} Six projects are being evaluated over a 3—year planning horizon. The following table give the expected
returns for each project and the associated yearly expenditures. Expenditures (million $)/year Project 1 2 3 Returns (miliion $)
1 6 3 7 30
2 4 7 10 40
3 4 8 2 20
4 7 4 15
5 7 6 8 25
6 10 8 45
Avaiable Funds (million $) 30 30 30 (a) (9 points) Formulate an integer program to maximize the return. (13) (4 points) If we require that at most one of projects 1 ,3, and 5 can be picked, what constraint
should we add? (c) (4 points) If we require "that project 4 can be picked only if project 2 is picked, what constraint
should we add? (a) X 1 waﬁd‘ 3L \s chpsevx
1” Ci 0 on).
Max 2 : 50M1+4ONL+1013 TISMaizmgwrxé
Sttr M. +412. +415 +714 +71: Mom {350 3% + M; +33%. +414 +6‘xg +816 $230
7% +1012 +113 4 “X4 +81;— “”6450 mm; .Xsaank: , m, e—{on} b7 mwﬁ’xggi 0 X491; 10 Problem 6. [8 points] A company is planning to produce at least 1500 products on three machines. Each machine has a set—up
cost which only occurs if a. machine is used (if a machine is not used then there is no set—up cost for it)
The following tahie gives the data of the situation. Machine Setup Cost Production unit cost Capacity (units) 1 300 4 800
2 100 10 900
3 500 3 1000 Formulate an integer program to minimize the total cost. (Please explain your variables clearly.)
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 Spring '09
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