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Unformatted text preview: GE330: Operations Research Methods for Proﬁt and Value Engineering
Final Exam May 7, 2008 NAME: UIN : Please Show all the steps. Good luck and have a nice summer. Sbialiﬁvxﬁ /20
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/17 Total /100 NOTE: Formulas are provided on Page 2 and 3. Problem 1. [18 points]
Consider the following LP. max 2: 3m + 2222 + 533 5.13. $1 + 2562 i 333 S 430 (1)
3331 £ 23:3 5 460 (2)
:81 + 4332 S 420 (3) $11 3327 503 Z 0 The optimal tableau is: Basic $1 x2 1173 m4 595 336 Solution
2 4 0 0 1 2 0 1350 332 E 1 0 E Z 0 100
mg g 0 1 0 % 0 230
3:6 2 0 0 2 1 1 20 where 224, 2:5, and as are slack variables for constraints (1), (2), and (3), respectively. (a) (3 points) Write down the dual LP. (b) (3 points) What is the optimal dual solution? (c) (6 points) If the righthand side of constraint (1) changes to 370, what is the new optimal value?
what is the new optimal solution? ((1) (6 points) In the objective function, if the coefﬁcients of $1 and :33 are ﬁxed, give the range for the
coefﬁcient of 2:2 in which the current solution is still optimal. SDi/wi'imAS: (a) min w = 4750311' 44>de t 42033
3‘ + 331+ 3'5 :5
13‘ + 435 22
31+ it i5
3‘ 351, 55 z o 20] F1
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t‘l‘iﬁil 2’0 Problem 2. [17 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 are given in the following table. Notice that the problem is unbalanced, if a unit of demand
of a distribution center is not satisﬁed (by any of the plants), a penalty cost is incurred at the rate of
$3, $5 and $2 per unit for distribution centers 1, 2, and 3, respectively. Additionally, all the demand at
distribution center 3 must be satisﬁed. We want to minimize the total cost. Distribution Center (a) (4 points) Formulate the problem as a balanced standard transportation problem, provide the
transportation tableau. (b) {4 points) Find an initial basic feasible solution.
((1) (5 points) Calculate reduced costs for all nonbasic variables in the initial basic feasible solution. (d) (4 points) Choose an entering variable and ﬁnd the leaving variable. Update the current solution. 0»). Tetelc gab13?”; ge+$o+2o 1:60 60<70
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i0 (intentionally left blank) E E at ~— 0 Problem 3. [10 points]
Consider the following integer program. max z = 3931 + $62 + 3273 S.t. —331 + 2562 + $3 5 4 4332 — 3533 S 2 {1:1 — 3332 + 233 S 3 3:1, 372, 3:3 2 0 and integer
The following tableau is the optimal tableau of the LP relaxation.
Basic 331 (1)2 333 :84 $5 335 Sol 2 0 0 0 2 3 5 29
$3 33 '3
3 $2 $3
$1 1_ 5i D D
C) O—I O
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II—mII—ml
Impml ,_ (a) (5 points) Suppose we use the branch—and—bound method, which variables can be the branching
variables? Write down the two sub problems for each possible branching variable. (b) (5 points) Derive a valid cutting plane based on the optimal tableau. {00% @Y ”(5 Com iiiL “cl/UL brow/Mug, Uan‘ahle 
if 5le . “ll/LU,“ X19. 5 + Of‘lalrwo/L ’FY‘OBlQM “M I 6 Jr swat/ml IWDIDlim if (Y 5 M“ X; g 3 ‘+ erljl’fxwl ?Wbl1\m
(X534 "i aﬂjlwuL TWlDlP/e lb)" 543 +%m+ 7‘11”“ 531%,: “s 15+ m ﬁve + [0+J5’Xg T Mime: g+ % “l ‘1 ‘1 3 . Problem {18 points} Customers arrl  at a onewindow drivehin bank according to a Poisson distribution, with a v an of 15
per hour. The servr : time per customer is exponential, with a mean of 3 minutes. Ther :r. e four spaces in front of the window, 'ciuding the car being served. Other arriving cars line : outside this 4car
spacer (a) (3 points) What is the probabi‘ > that an arriving car can enter a e of the 4car space?
(13) (3 points) How long is an arriving cus .  er expected to » it before starting service? (c) (4 points) How many car spaces should be pro '  a in front of the window (including the car being
served) so that an arriving car can ﬁnd a sp  the : at least 95% of the time? (d) (4 points) If cars are not allowed to ait outside the 4—car ace, what is the probability that an
arriving car will leave without 1' ing service? (6) (4 points) Since the arr' . rate increases to 35 customers per hour, the i ;. l k decided to open a new
window with the y. e average service time. How long is an arriving custo : expected to spend
before ﬁnishi : service. NOTE: For this part, the cars are allowed to wait outsi  he 4—car space. (intentionaily left blank) [glaml
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1/ ‘ : Ci.%m:w+ EMMA: I2;%m:n. W§ZW17+ 7&— 10 ' blem 5. [10 points] Joe lov  eat out in restaurants. His favorite foods are Mexican, Italian, and Chines cm the average,
Joe pays $10 0 : exican meal, $15 for a Italian meal, and $12 for a Chines  al. Joe’s eating habits
are predictable: There » ~ I% chance that today’s meal is a repeat of _ *' erday‘s, and equal probabilities
of switching to one of the rem''ng two. (NOTE: If you I . ‘ aifﬁculty in solving a system of linear
equations, just set it up and then exp ~. the answe  e questions in terms of the variables.)  (a) (4 points) Express the situation : . arkov in.
(b) (4 points) How 111 . noes Joe pay on the average for his I ;. dinner. (c) (2 u ' 5 How Often does Joe eat Chinese food? .’ ‘Tl/m Transition W‘tvii:
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30 EKLH gitlb 4r 5 f 5 .3 ~— % .BUQY7 '5 0107/3 . 11 Problem 6. An item is consumed at te of 45 items per day. The holding cost per un' ay is $05, and the setup cost is $200. Suppose that no tage is allowed and that the asng cost per unit is $10. The
lead time is 21 days. (a) {4 points) Determine the optimal inv (b) (6 points) If the man
units, What urer now provides a discount of $3 per un' r orders more than 1000 be the optimal inventory policy. k: ‘0: K3200 .Fw“ﬁ _ é ..
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Consider the following function f(ar:) = f(m1,:cg,$3) = x? +mg +m§ — :31— 23:3 —w2w3. (a) {3 points) Calculate the stationary point(s) of f(:r). (b) (3 points) For each stationary point, specify whether it is a local minimum or a local maximum.
(Hint: Calculate the Hessian matrix)  id be the moving direction? How do . * 'e the step length? (Just
give the optimization problem for a  '' g e s  e  ; , do NOT solve it). (e) (3 points) If we want to minimize f (w) subject to the constraints 321 +302 +323 = 10 and 2331 +3532 +
433 = 20, write down the conditions for 213* being a stationary point of this constraint optimization problem. » .l.
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 Spring '09
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