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Unformatted text preview: Part 1  True or False (16% — 4% each)
Mark whether each of the following statements is true or false. No explanation required. 0 In a maximization LP, if all reduced costs are nonpositive, the current basis is optimal. TRUE FALSE 0 A minimum cost network ﬂow problem is a special case of a shortest path problem. TRUE FALSE v 0 Suppose we are running the simplex algorithm and our current basis is B. We want to increase
nonbasic variable raj and have a direction d for this purpose. To compute the multiplier 0
that tells us how far we can move along d, we only need to check ratios for the positive components of d3. p o If the optimal value of the Phase 1 LP is zero, the original LP is feasible. FALSE Part 2  Modeling (30%) You manage a production facility that needs to plan for the next two months’ demand, which you have forecasted to 70 units in Month 1 and 120 units in Month 2. Each month, demand must
; be met in full (no backlogging), using either inventory left over from the previous month or that month’s production. You have no inventory at the start of Month 1.
3 Each month, you have 800 hours of regulartime labor available; the'cost of this labor is sunk
and does not inﬂuence the production plan. In addition, 400 hours of overtime labor are available
each month, at $20 per hour. If additional labor is still needed, you may hire a skeleton crew to
work the graveyard shift; 200 hours per month are available at $50 per hour. Each unit produced.
requires 10 hours of regular or overtime labor. However, because the skeleton crew is smaller than
a regular crew, it takes 20 hours of graveyard shift time to produce one unit. Inventory holding costs are $40 per unit at the end of Month 1. Any units left over at the end of Month 2 have a salvage value of $30 per unit. Formulate an LP to minimize the cost of meeting
your projected demand. a) Write down your decision variables. Explain what each one represents clearly. (8%) V XE : meg {brooms/ham ,WLL t
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+ (4? ”%w++> i, — (ROAMJr) L. c) Write the inventory balance constraints. (5%) (1) Write the capacity constraints. (4%) (/0 MW»; ) h s 800 )W
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demand for Month 2 might change. You’d'like to know how a change in this demand could
affect your bottom line. Which quantity gives this marginal value? (4%) ‘ ﬂL, 514244604” fr/(a/ M 50417404. ﬂy
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a Part 3  Simplex and Sensitivity Analysis (40%)
Consider the following LP: min 2331 +0152 +4203 +0174 +3175 s.t. 2m1—w2+3x3—m4+m5=3
ml—m2+a:3+0x4+a:5=2
3:20. a) Show that the basis B = {1:1, :35} is optimal. The following may come in handy: [3 3H: ‘21] Is the corresponding BFS the unique optimal solution? (20%) p; = “2%?ij ,
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‘ increases this variable, keeps other non—basic variables at zero, and maintains equality? Show
that this direction is unbounded. (10%) c) Suppose a new variable, 276, is introduced into the problem. This variable has a coefﬁcient of
1 in the ﬁrst constraint and —1 in the second constraint. The objective coefﬁcient for :26 is
g 06. What values of ca maintain the optimality of the current basis? (10%) I I v/ I i 2:; = c4, v 52 JL : C4,  [2 ll 3 ' : CE ‘ I ' Mad 2; > (O 74: 7%: cot/MM W .
WW , So we Aee/ Ce 7/ . Part 4  Shortest Paths (14%)
Consider the following network: a) Formulate the problem of ﬁnding a shortest path from node 1 to node 5. (10%) Df CU" be. We (1])? LOJ% {W MZZMCZ). Wm (age/I C(l XL} st. Y” + ﬁg L >( 1" I (In/0642 I)
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3; " >19; : *wmﬁw%[email protected]@w,wm' b) Suppose I tell you the length of the shortest path from node 1 to node 5 is 15. I also tell you
the length of the shortest path from node 4 to node 5 is 4. Why are these two statements I contradictory? (4%) ‘ L4! : 0“"! gt; 5 a Vl‘a/q/é ML M , (3'.— qu g C“, = IO
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 Spring '08
 JuanPabloVielma
 Optimization, Shortest path problem, 2%, $0, » Hum

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