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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 mincost 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 uzj 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 maxflow model Whose solution either determines or proves
nonexistence of a feasible solution for the transformed model. {no} Add 0? wnsWndS :>:X;~ 23X} 35>} «av
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 Summer '07
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