assignment3 - V 1 Problem 6 page 261(BHM The caterer’s...

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Unformatted text preview: V, 1. Problem 6, page 261 (BHM) The caterer’s problem can be solved using a min—cost flow model (you should draw the graph): Restaurant nodes (for clean napkins): 0,1,2,3,* Laundry nodes (for dirty napkins): A,B,C,D Arcs for overnight storage of clean napkins: 01,12,23,3* unit cost = 0.0 Arcs to pass dirty napkins to laundry: 0A,1B,2C,3D unit cost = 0.0 Arcs for overnight cleaning: A1,B2,C3 unit cost = 0.025 Arcs for normal cleaning: A2,B3,C*,D* unit cost 2 0.015 Decision variables for flows mm for each arc ij e.g., $012, 5603, 1131),. , etc. LP formulation: min total cost subject to: flow out of node 0 = 350; flow into node * 2 350; flow out — flow in = 0, for all other nodes; 300,; = 175,3301 :: 175, $13 = 300,.1‘20 = 325, 133D = 275; a?” 2 0 V arcs z'j. To accommodate dirty napkin holdover, we could add arcs (0.0 unit cost) AB,BC,CD, with correspond- ing decision variables. If all napkins must eventually be cleaned, will this change alter the optimal solution? 2. Problem 13, page 263 (BHM) This production problem can be modeled as a transportation problem: Demand nodes 1,2,3,4,5,6 with demand amounts 100,150,200,100,200,150 Supply nodes for regular production R1,R2,R3,R4,R5,R6, each with 100 units supply Supply nodes for overtime production Ol,O2,03,04,05,06, each with 75 units supply An additional demand node * to handle excess supply Production arcs Rij from Ri to j, Vj Z 2' unit cost is 10+(j—i)(.5) (production/ carrying) Overtime arcs Oij from Oi to j, W Z 2' unit cost is 12+(j~i)(.5) Arcs Ri* and Oi* for each 2’ , which handle fictitious production unit cost = 0.0 Can you suggest a simple way to determine an optimal solution for this model? Lin uhv y ‘3 Problem 26, page 269 (BHM) Suppose, in general, that we have m letters which are used to make n words. Consider the bipartite graph with (left) nodes 1,2,. . .,m for the letters and (right) nodes 1,2, . ..,n for the words, which has an edge 2' j if the ith letter appears in the j th word. Thinking of the letters as supply nodes, each able to supply one unit, and the words as nodes each having unit demand, we wish to know whether there is a feasible solution to the corresponding transportation problem, i.e., a flow in which each “word is supplied with a unique letter. A solution provides letters to serve as a so—called system of distinct representatives for the words. 1.} Coarse “Glegs Application 1 9.4 Flyaway Kit Problem Many companies (e.g., computer companies or telephone companies) own, leas or warrantee a wide range of equipment that they must maintain at geographically dispersed field locations. In performing a given job, the repair crew often require various types of parts (and tools). In many cases the crews carry some replaceme V parts in a kit rather than storing them at the equipment site. If all the required parts are in the kit, the crew member can repair the equipment. But if any of these ite _ is not available, the service call is incomplete and the job is a “broken job.” Broken jobs are costly for several reasons: (1) they increase equipment downtime, (2) the repair crew must make an extra trip for parts, and (3) partially repaired equipmen might be unsafe or vulnerable to damage. On the other hand, carrying more items in the kit increases handling and inventory costs. In the flyaway kit problem, W: need to obtain the optimal kit of parts (and tools) that minimizes the sum of at handling and inventory costs and the costs of broken jobs. ‘ 7 Suppose that we number the parts required for servicing the jobs as l, 2, . ~ ’- r. We assume that the repairman restocks the kit between jobs, but with a fixed and specified content. For our purposes we define a job by the set of parts (and t0015 that it requires. Making this association defines a collection of job types .11. J2, - ' ' J1, that encompasses all the known possibilities that a repairman might encounter. -, The job type J j is defined by the set B,- of the parts required by that job. Let 1,- denot the expected number of job types Jj serviced in one year, and V]- ‘denote the penalty, cost we incur whenever job j is a broken job. " A stocking policy of a kit consists of a fixed set of parts M Q {1, 2, . ' - , r} i that a crew would carry. Let H. denote the yearly handling and inventory cost for ' carrying part i in the kit. Then the total handling cost is 216M Hi- Moreover,» .2; the total expected cost of broken jobs per kit per year for policy M would be 2{ 1334M} lej. Therefore, policy M incurs a total expected yearly cost per kit of z(M) = 2 H,- + 2 L]. iEM {j:B/¢Ml In this expression, L] 2 lej. The optimal policy would, of course, be a set M g {1, 2, . . . , r} that minimizes z(M). Notice that minimizing z(M) is equivalent to max- imizing —z(M), which we can restate as ”Z(M) : 2 Lj‘” 2 HiWL, {j:B/;M} ieM by letting L = EJLI LJ, a constant. Consequently, our objective is to identify a policy M that maximizes 2{j:3j;M}Lj ~ 2,9” Hi. This problem is a special Ease of the maximum weight closure problem on the bipartite network shown in Figure 19.6. This network contains two types of nodes: those representing parts and those representing jobs. It also contains an arc from a node representing job type 11» to each part node in Bj. Notice that a node 11- can be in the maximum weight closure only if the closure also contains each part (and tool) in Bj. Figure 19.6 Network for the fly away kit problem. A.» A “n...- ~fl.----~ Q6. The general min-cost flow problem for the network G = (V, E) is: mi” 22‘ Zj Ciflz‘j 5.25. 2]. aiij - 23- 9331‘ £17- < mij bi Vi E V w, j) e E t/\ I! E The fij and uz-j are, respectively, lower and upper bounds on are flows and the bi give the net supply at each node, i.e., b1- > 0 for supply and b;- < 0 for demand. (a) Show that for feasibility of this model we must have 21. bi = 0 . (b) The transformation $2)» =2 9325 — 1213- produces an equivalent model with are flows which satisfy 0 _<_ 5% _<_ Uij —— Eij V(z‘, j) E E . Note that this transformation may also change the values of the bi . Does this affect the condition stated in part (a)? Why? (0) In view of (a) and (b), we will assume that 21- b,- = 0 and that the model has been transformed so that Zij = 0 ‘v’(z’, j) E E . Give a max-flow model Whose solution either determines or proves nonexistence of a feasible solution for the transformed model. {no} Add 0? wnsWndS :>:X;~ 23X} 35>} «av 3 d J .L “1’er L':\ swan, » once, ‘5‘?“ ‘ same 66¢“ X3 We? 5° cl “’1‘“ m “El/‘7' ” 50". 2:0 to sew,» So “5 {:ev // } kl 5H3 TmmS Ave/A 3" b hx‘é'd _ 5 c: V30 'Cé ‘\s;\a~oto\ “m“ J2A5“ Cb) NO . ‘m is) Q 3 C34 no memo: “056° as o“ €°3° (C) Rail ' L Li EV: \m at mode r‘ 5g?’3\fi “0A?" g, GOCV‘ 3“ l tar eodfi \ \ B 50K ) 3 mde‘lfl The: ~ U ‘01 ' (‘5) I &\ wAM (MC COQC’ ‘59“ C0? } \i h<0> 559:. «356 Ufa U“ A (a?) \n Nuflneady Z. (\ 6 A (Kerb 06"., “(3% C,\f\Cf\j€ %\ w Xfi anvog 5(0 ‘ ’, swig} 0‘ U 7 mox ' @ Him/W53 wow 3 {a t, \ «r ° ‘\ (3&3: §O\U€, WUQX”!§\W L M” 3133)? 3V) VthOgdfl ii», ...
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