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midterm2_3133_sol(2)

# midterm2_3133_sol(2) - Part 1 True or False(16 — 4 each...

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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 non-positive, 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 non-basic 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 regular-time 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 ‘je'. 0.1:. t u m g l? Zt : arctVZAjoU/j gwﬂ‘" w w. t ct : MA 91 amt—rt»- I'vaZ/w‘l-Wj, M e b) Write the LP’s objective. (5%) m... az°/~><%>m my.» (Wm .2.» + (4? ”%w++> i, — (ROAM-Jr) L. c) Write the inventory balance constraints. (5%) (1) Write the capacity constraints. (4%) (/0 MW»; ) h s 800 )W 06.5%?) yeé Va? é:/2 (20 ’W/Wz) Z; é 209 I” e) Write the variables’ domains. (4%) Xk,ﬂe/Zel,‘;t 20, LL"/Z i l g f) Suppose you solve this LP to obtain an optimal production plan, and then discover that your 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 m 54% [nth -&WZ/wt(4. l l ; E l I 5 i . l I i s , l l ‘r 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 , \$0 BFS h H \t l (\.\ N L4 II \" l U0 H \ i b) Suppose I want to increase the non—basic variable 352. What is the direction vector (1 that ‘ 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) x23+ x25 ~ x,2—>< :0 (M49 2) X32+X3LI1LX ‘*X*‘>< —)( :0 13 2'3 93 XH3 * X45 " XI»: 'X3L/ : O (M41 q) 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 lnl‘vl‘l’ivcw , If M1 jaw/'1le 4/;5 M H/ W HA /’S CMM7L [01,» Hum TWWM%C””W3%%v%f5W3W ...
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