A Stochastic Bottleneck Assignment Problem
Author(s): Uri Yechiali
Source: Management Science, Vol. 14, No. 11, Theory Series (Jul., 1968), pp. 732-734
Published by: INFORMS
Stable URL: http://www.jstor.org/stable/2628198
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732
COMMUNICATIONS TO THE EDITOR
the maximum of ui variables with the common distribution
Pr {Y y-
=y "t
for y E [0, 1]. Homogeneityof the expression(1) completesthe proof as before.
For irrationalvalues of the wi , the proofcan be extendedby continuity.Thus, in
any case, our scheme gives the value (1) for VR . Since expectations are additive, it
follows that this scheme coincides with b for all games.
We point out, finally, that the form of the schemeguaranteesthat, for monotonic games,the modifiedvalue 4 will be an imputation.
GuillermoOwen
Fordham University
References
1. SHAPLEY, L. S. 'A Value for n-Person Games', Annals of MathematicsStudy 28, Prince-
2.
-
ton, 1953.
, Additive and Non-Additive Set Functions. Ph.D. Thesis, Department of Mathematics, Princeton University, 1953.
A STOCHASTIC BOTTLENECK ASSIGNMENT PROBLEM* t
URI YECHIALI
Columbia University
Consider an n-job stochastic bottleneck assignment problem in which the
production rates are independent random variables rather than constants.
Assume that the production rate, Riy, (Riy L 0; i, j = 1, 2, - *, n) of man i
when assigned to job j is: 1) exponentially distributed with mean 1/Xii or 2)
Weibull distributed with scale parameter Xis and shape parameter p. It is shown
that in either case, the assignment that maximizes the expected rate of the
entire line is found by solving the deterministic assignment problem with 'cost'
matrix [Xij4.
The classical'BottleneckAssignmentProblem'[21is described[1]as follows:
n men are availableto be assignedto n operationsor jobs comprisinga productionline. Associatedwith each man i-job j combinationis a knownand con0; i, j = 1, 2, ***, n). The production rate of
stant production rate R j (Rtj
i
will
line
for
the
any given assignment be equalto the slowestratein the line, i.e.,
the rate of the line over all n!
the bottleneck.The problemthen is to maxmmize
possibleassignments.
1, 2, ... , n}, find a permutaMathematically, given the set {R1j > 0; i, j
,
n
so
to
as
achieve
on
ir*
tion
the integers1, 2, *
(1)
Max, {MiniRi,i} .
Now, supposethat for each man-jobcombination,the correspondingRij is not
* Received December 1967.
t This research was supported in part by National Science Foundation Grant NSF-GK1584.
COMMUNICATIONSTO THE EDITOR
733
a constantbut a continuousnonnegativerandomvariablewith distributionfunction Fij( - ). For any permutationir the rate of production,R, of the entireline is
also a randomvariable definedby: R = MiniRi,.) . Our objective then is to
findan assignmentthat will maximizethe expectedrate of productionof the line,
i.e., we seek a permutation&* so as to achieve
(2)
Max, E{R}.
Assumingthat the Rij's are independentrandomvariableswith finite means,
the distributionfunctionof R, FR( ), is found to be:
(3)
FR(r)
=
(r)
[1 -F
1-Il
and the expectedrate of productionis given by:
(4)
{T=1 [1 - Fi,()(r)]} dr.
E{R} -1
We considernow two cases:
CASE 1: Let Rij be exponentiallydistributedwith a knownparameterXi > 0,
i.e., with p.d.f.:
(5)
fij(r)
=
XiJexp -Xer},
=
0,
r > 0,
otherwise,
for ij = 1,2,-... ,n
Consider also the following 'Assignment Problem': Given Xis > 0,
i,j = 1, 2 ..., n, find a permutation r* so as to achieve
(6)
Mit
_1 )Ii,Tr(S
.
We show that in the exponentialcase, a solutionof (6) is a solutionof (2). To
see this we write:
(7)
E{IR}
=
=
fexp
1
/E
{=
(t1:
7
)r}
dr
Xi,7(i)X
It is clear that Max, E{R} is achievedwhen Minr Z?=1Xi,,(i) is achieved, thus
provingthe assertion.
CASE 2: Let Rij be distributedaccordingto the Weibullprobabilitylaw with
scale parameterXij andshapeparameter,3 (equalfor all man-jobcombinations).
The p.d.f. of Rij is:
(8)
fij(r) - =
{-\r',
ijr'exp
= 0,
otherwise,
Xis, # > 0, and the distributionfunction:
(9)
Fij(r) = 1
r > 0,
-
exp {-Xir').
734
COMMUNICATIONSTO THE EDITOR
For a given permutationr, the expectedrate of productionequals:
(10)
E{RJ = fexp
=
{-(EffjXr())r$}
dr
( 1/E1= xi(.) ) *r(1/i3 + 1),
wherer(.) denotes the gammafunction. Since ,3 is fixed, E{R} is again maximized when Min7 f=il Xiw(i) is achieved.
Note that the exponentialcase is seen to be a specialcase of the Weibullwith
= 1 for all i andj.
To summarize,it is shownthat, in both cases, the assignmentthat maximizes
the expected rate of productionof the entire line in a randomizedbottleneck
assignmentproblem,is found by solving a 'deterministic'assignmentproblem
with 'cost' matrix[Xij].
It shouldalso be pointedout that even if a positivelowerbound,G,-'location
parameter'-is imposedon each Rij, (i.e., Rej _ G > 0, all i, j)-the resultsstill
hold.
References
D., I. GLICKSBERGAND 0. GROSS,"A Production Line Assignment Prob1. FULKERSON,
lem," The RAND Corporation Research Memorandum RM-1102, Santa Monica,
California, May 27, 1953.
2. GROSS,O., "The Bottleneck Assignment Problem," The RAND Corporation, P-1630,
March 6, 1959.
A Clarification in LIFO vs. FIFO*
In his investigation of inventory depletionpolicies, Bomberger[1] examines
conditionson the field life function, L(S), for which either LIFO (last in, first
out) or FIFO (firstin, first out) is an optimalissue policy. His TheoremI states
"If L-'(S) is concaveincreasing,then LIFOis optimalforn 2 2." In this theorem
L-1(S) is not explicitlydefined,and althoughthe authorrefersto this functionas
the inverseof L(S), the readershouldbe cautionedthat he intendsL-1(S) to be
the reciprocalof L(S) and not the usualinversefunctionwhosecompositionwith
L(S) yields the identity function.
That this may be a point of confusionis evidenced,for example,by the fact
that Hillierand Lieberman[2]cite the aboveresultin theirrecenttextbook (page
370), rephrasingit in the form,"Finally,Bombergerhas shownthat if the inverse
of the fieldlife function,i.e., LU (S), is concaveincreasing,then LIFO is optimal
for n > 2." Again, a question arises as to the intended meaningof the phrase
"inverseof the field life function."The reasonwhy this clarificationis neededis
that the writerhas shown[3]that if, for example,L(S) is convexwith L'(S) > 1
for all S > 0, in whichcasethe usualinversefunctionLV1(S) is indeedconcaveincreasing,then not only iFO nouLI bmal,t optis e t thopposingpolicy FIFO is
* Received September 1967, and revised January 1968.