The capacitated multi-echelon inventory system withserial structure. 2. An average cost approximationmethodSpeck, C.J.; van der Wal, J.

Published: 01/01/1991

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Citation for published version (APA):Speck, C. J., & Wal, van der, J. (1991). The capacitated multi-echelon inventory system with serial structure. 2.An average cost approximation method. (Memorandum COSOR; Vol. 9140). Eindhoven: Technische UniversiteitEindhoven.

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TECHNISCHE UNIVERSITEIT EINDHOVENFaculteit Wiskunde en Informatica

Memorandum COSOR 91-40

The capacitated multi-echelon inventorysystem with serial structure:

2. An average cost approximation method

C.J. SpeckJ. van der Wal

Eindhoven University of TechnologyDepartment of Mathematics and Computing ScienceP.O. Box 5135600 MB EindhovenThe Netherlands

Eindhoven, December 1991The NetherlandsISSN 0926-4493

THE CAPACITATED MULTI-ECHELON INVENTORYSYSTEM WITH SERIAL STRUCTURE:

2. AN AVERAGE COST APPROXIMATION METHOD

C.J. Speck and J. van der Wal

Eindhoven University of Technology

This paper considers a multi-echelon, periodic review inventory model with

discrete demand. We assume finite capacities on various production/order

sizes and backordering of excess demand. We have already seen that modi­

fied base-stock policies work quite well under an average cost criterion. Here

a method will be presented which provides an approximation of the aver­

age costs corresponding to a modified base-stock policy in a certain class

of multi-echelon serial systems. In this the moment-iteration method devel­

oped by De Kok plays a central role.

1. Introduction. In this paper we consider a basic production (or inven­tory) model, in which the stock of a single item must be controlled underperiodic review. We assume demands in each period to be independent,nonnegative, and integer-valued. Further, all stockouts are backordered andproduction, holding, and shortage costs are linear. There are no fixed ordercosts.

For the long-run-average cost criterion Federgruen and Zipkin[1986] haveshown that in the capacitated I-echelon inventory system with serial struc­ture a modified base-stock policy is the optimal periodic review strategy:If the echelon stock has dropped below a certain level, enough should beproduced to raise total stock to that level if attainable; otherwise, the maxi­mum feasible amount should be produced. On the other hand, if the echelonstock is above that level, nothing should be produced.

Analyzing the capacitated N -echelon serial system with N ~ 2 we nu­merically proved in a preceding manuscript (Speck and Van der Wal[1991Dthe possible appearance of a so-called 'push ahead'-effect within the optimalpolicy: Due to certain combinations of already known yet to arrive ship­ments, magnitude of demand, and finite capacities it might sometimes be

I

profitable to exploit the maximum capacity in order to possibly avoid futureordering limitations.

Neither of the aforementioned papers gives an exact calculation of the av­erage cost for the optimal policy. But already a few approximation methodshave been developed, that successfully cope with the difficulties arising from .the finite capacities, such as an application of the moment-iteration methodof De Kok[1989], and a related method of Zijm (unpublished manuscript).Both methods however are as yet only applicable to the capacitated 1­echelon serial system.

This paper focuses on the development of an average cost approximationmethod on behalf of the N -echelon serial system with N ~ 2. In section 2we introduce notation, definitions, and assumptions. Section 3 contains adetailed description concerning the average cost calculation when employ­ing a modified base-stock policy. The moment-iteration method of De Kokwill be brought forward, which fills an essential part in our approximationmethod. For clarity reasons we will illustrate our approximation method onthe basis of the capacitated 2-echelon serial system.

2. Definitions and assumptions. As already mentioned we will re­strict ourselves to the capacitated 2-echelon serial system. Extension of themethod given in section 3 to the N -echelon serial system with N > 2 ispossible, but very laborious.

First of all we recapitulate the model definitions as given in Speck and Vander Wal[1991]:

1. We define the echelon stock of a given installation as the stock at thatinstallation plus all the stock in transit to or on hand at any installationdownstream minus the backlogs at the most downstream installation.

2. Next, the echelon inventory position of an installation denotes theechelon stock plus all items heading for that installation, that alreadyleft the preceding installation (or the external supplier).

The capacitated 2-echelon model we analyzed is characterized by the follow­ing parameters:

Uj = maximal production/order size, i = 1,2.it = leadtime of the route from installation 2 to installation 1.12 = leadtime of the route from external supplier to installation 2.p = shortage cost per unit per period at echelon 1.

2

hi = additional holding cost per unit per period at echelon i, i = 1,2.Xi = stock at echelon i at the beginning of a period, i = 1,2.Yi = inventory position at echelon i at the beginning of a period, i = 1,2.

D t = demand in period t, t E IN.qw = 1P{D = w}, w = 0,1, ... ,the demand probabilities.

F(u) = E:;'=o qw, u E IN, the demand distribution.

We assume Ui to be finite for i = 1,2, while Xi and Yi are always integer­valued and may be negative since stockouts are backordered.

•

Itt '.

\2 7 \ 11 D

V V•

UI

h2 hl +h2, P

We distinguish the following activities during every period: At the begin­ning of the period (a) each echelon inventory position is increased, next (b)the external demand is met, and at the end of the period (c) a cost deter­mination takes place.

period

!b

The expected costs at the end of a period consist of linear holding andshortage costs (linear ordering costs are not taken into account). Hence,the expected costs at the end of a period attached to an echelon stock Xi,

i = 1,2, at the beginning of that period are described by the well-knownNewsboy-formulas, here displayed in the discrete form:For Xi E 7Z, i = 1,2,

00

LI(XI) .- L: qw[(hl +h2)(XI - w)+ +P (Xl - w)-] - h2XI (1)w=o

L2(X2) .- h2X2. (2)

3

Finally, referring again to Speck and Van der WaJ[1991] we assume withoutloss of generality

(3)

3. Construction of an approximation method. De Kok[1989] hasshown that an approximation of the average costs corresponding to a modi­fied base-stock policy in the capacitated N -echelon serial system with N = 1and 11 E IN can be obtained by reformulating the problem as a DIGII queue.This is done by comparing the shortfall on the desired echelon inventory levelto the length of a waiting-queue.

Suppose we want to determine the average costs associated with themodified base-stock policy d1 , with d1 E IN. Let Wt E IN be the shortfallon the desired echelon inventory position level d1 at the beginning of periodt. Then, starting from a shortfall of Wt on d1 at the beginning of period tand meeting demand D t leads to a shortfall of Wt +D t at the end of periodt. If this amount exceeds the capacity Ull then the shortfall at the beginningof period t + 1 shrinks to Wt +D t - U1 ; on the contrary, if Wt +D t ~ U1 ,

then the shortfall is neutralized and the ideal level, d1 , again is achieved.Hence

(4)

This is the well-known relation for the waiting-time of the (t +l)st customerin the standard DIGll queue, for which De Kok has developed a moment­iteration method. The algorithm will be shown later on; it enables us toderive approximations for the waiting-time distribution for instance.

So if we take the collection of echelon inventory positions associated withthe modified base-stock policy d1,

Z [ d1 ] = { db d1 - l, d1 - 2, ... },

or, equivalently, the set of shortfall positions,

W={O,l,2, ... },

(5)

(6)

as state space we are able to form the enclosed transition matrix, say Ql by

4

means of (4):

F(Ud qUI+! qUI+2

F(Ul -1) qUI qUI+!

Ql := F(Ul - 2) qUI-l qUI (7)

Notice its fine diagonal structure, apart from the first column. Then, theexact average costs going with the modified base-stock policy dl is given by

9 =: Qfr[ dl ], (8)

in which Qr denotes the equilibrium matrix, and r[ d l ] the cost vectorconsisting of the expected costs per state from Z[ dl ]. The equilibriummatrix Qr, here consisting of identical rows of equilibrium probabilities11"0' 1I"j, 11"2' .••, can be constructed by solving the system

(9)

in which 11" denotes a vector of probabilities. The fact of having a non-finitetransition matrix Ql however, causes the system (8) to be infinite and there­fore does not permit an exact calculation of the equilibrium probabilities 11";.But we can derive approximations *; by using De Kok's moment-iterationmethod: Every 11";, denoting the equilibrium probability on a gap of i itemson level db in fact specifies the probability on a customer's waiting-time ofi units in a DIGl1 equilibrium system!

Algorithm 1 (Moment-iteration)

Initialisation: Choose lE{Wo} = 0 and 1E{W6} = o.Iteration: Compute

lE{Wn +Dn } =: lE{Wn } + lE{Dn }jlE{(Wn + Dn )2} =: lE{W~}+ 21E{Wn }1E{Dn } + lE{D~}.

5

Fit a distribution function FWn+Dn to the first two moments of Wn + D n •

Then calculate

lE{Wn+I} = (oo(y_ Ul)dFwn+Dn(Y),JUt

lE{W~+I} = roo (y _ Ud2 dFWn+Dn(Y).JUt

Termination: Stop whenever

IIE{Wn+I} -IE{Wn }1 < Cl and IIE{W~+I} -IE{W~}I< C2,

for some prespecified Cl > 0 and C2 > o. Then

Fw(Y) ~ FWn+D(Ul + y) , Y > o.

(10)

(11)

(12)

o

This method has been grafted upon fitting distributions to the first two mo­ments of an arbitrary distribution. E.g. Tijms[1986] provides us a methodin which mixtures of Erlang distributions yield excellent results.

Thus, applying of the moment-iteration method of De Kok provides anaccurate approximation for the waiting-time distribution of the equilibriumsystem, or translated to the inventory system, an approximation for theequilibrium distribution of the shortfall. Then, discretizing the generatedwaiting-time, or shortfall-amount, distribution we obtain the estimated equi­librium probabilities if;. Truncating (8) finally results in an adequate ap­proximation of the average cost associated with a modified base-stock policy

dl ·

So far the recapitulation of an application of De Kok's moment-iterationmethod to the capacitated I-echelon serial system.

We have been able to extend the idea of De Kok to the N -echelon inven­tory serial system with N ~ 2, but in which Ii =1 for all 1 ~ i ~ N. Supposewe want to determine the average cost corresponding to the modified base­stock policy characterized by echelon inventory position levels (d2, dl ). Weexplicitely assume the system to start from the ideal echelon inventory stocklevels di, i = 1,2. This leads to the definition of the state space Z[ (d2 ,dl )]

belonging to the modified base-stock policy (d2, dl ) as the collection of pos­sible echelon inventory positions (Z2, Zl) at the beginning of a period beforemeeting demand:

Z[ (d2,dl )]:= { (z2,zd E yz2 I Zl ~ dl , Zl ~ Z2 ~ d2 }. (13)

6

Thus, at the beginning of the first period the system finds itself in thesituation as depicted underneath.

d2

dl

\27 W D•

U2 V UI Vh2 hI + h2 , P

Consequently, when demand equals D in the first period the stock at echelon1 and 2 amounts to dl - D respectively d2 - D (remember 12 = It = 1), andthus there is a fysical stock of d2 - d l on hand at installation 2!

d2 - D

d2 - dl dI - D

\ 11• \ 2 7U2 V UI V

h2 hI +h2 , P

Then, by looking at the amount d2 - dl available at installation 2 there arethree separate cases distinguishable:

1. 0 ~ d2 - d l ~ U2

2. U2 < d2 - dl ~ UI

3. UI < d2 - dl •

First of all we can skip the case UI < d2 - dl • Namely, every modifiedbase-stock policy (d2 , dt} aims at a stock of d2 - dl on hand at installation 2

7

at the end of an arbitrary period. In the case UI < d2 - d l this means goingfor a strictly positive stock at installation 2 during the next period, for atmost UI can be shipped. For this stock amount holding costs are charged,while no advantage is yielded by this stock, see the analoguous argumentthat established the preliminary result in Speck and Van der Wal[1991].

In the other two cases the state space Z[ (d2 , dI ) ] reduces to an irre­ducible state space in each case. Suppose we practise a modified base-stockpolicy (d2, dI ) in which 0 $ d2 - dl $ U2. Then we are able to construct apartial flow scheme containing the possible transitions from the ideal state(d2,dI):

By considering the possible transitions from the other states depicted abovewe finally attain an irreducible state set Ztl (d2,dI )],

Ztl (d2,dI)] := {(d2,dI),(d2,dI-l), .•. ,(d2,d2 - U2),

(d2 - 1, d2 - U2 - 1), (d2 - 2, d2 - U2 - 2), ... }.

8

The corresponding transition matrix has the following structure:

F(d2 - dl ) qd2-dl+I qU2 qU2+I qU2+2

F(d2 - dl ) qd2-dl+I qU2 qU2+I qU2+2

F(d2 - dl ) qd2-dl+I qU2 qU2+I qU2+2

F(d2 - dl ) qd2-dl+I qU2 qU2+I qU2+2

F(d2 - dl - 1) qd2- dl qU2-I qU2 qU2+I

F(d2 -dl -2) qd2-dl-1 qU2-2 qU2-I QU2

For the remaining case (U2 < d2 - dl ::; UI ) a similar flow scheme canbe drawn, which shows the reduction of Z[ (d2 , dd ] to an irreducible statespace Z2[ (d2,dl )],

Z2[ (d2, dl )] := {(d2, dl ), (d2 - 1, dl ), ... , (di +U2, dl ),

(di +U2 -I,dl-I),(dl +U2 - 2,dl - 2), ... }

In this case we obtain a nicely structured transition matrix, reminiscent ofthe one in the I-echelon system.

F(U2) QU2+I Qd2- dl Qd2-dl+1 Qd2-dl+2

F(U2 -1) QU2 Qd2-dl-1 Qd2-d1 Qd2 -d1 +I

F(U2 - 2) QU2- I Qd2-dl-2 Qd2-dl-1 Qd2-dl

F(a) Qa+1 Qa+2 QU2+I QU2+2

F(a - 1) Qa Qa+I QU2 QU2+I

F(a - 2) Qa-I Qa QU2-I QU2

in which a := 2U2 - d2+di .

9

(d2 , *), and carrying out a corresponding clustering within the transitionmatrix associated with ztr (d2 , d1 ) ], we achieve both transition matricesbeing the same:

F(U2 ) qU2+I qU2+2

F(U2 -1) QU2 QU2+1

Q2 := F(U2 - 2) QU2-1 QU2 (14)

Inherently, Zl [ (d2 , d1 ) ] changes to

Zl[ (d2 ,dd] ={ (d2, *), (d2 - 1, d2 - U2 - 1), (d2 - 2, d2 - U2 - 2), ... }

The exact average costs associated with the modified base-stock policy(d2 ,dd is given by

9 = Q2'Tl[ (d2 ,dd] (15)

or

(16)

depending on the value of d2 - d1• In both cases Q2' is the equilibriummatrix of the system, and Ti[ (d2,dd] for i = 1,2 the cost vector consistingof the expected I-period costs per state out of the state space Zi[ (d2 , d1 ) ]

associated with a modified base-stock policy (d2 , dd. The costs are definedas

T(Z2,Zl):= 100

{L2(Z2 - w) + L1(Zl - w)} dF(w). (17)

Evidently, when at the beginning of a period t the inventory position ofechelon i, i = 1,2, is raised to level Zi, then the stock of echelon i equalsZl - Dt at the beginning of period t +1. Then the costs at the end of periodt +1 associated with echelon i amount to Li(Zl - Dt).

The equilibrium probabilities 7ro, 7ri, 7r2' ... again can be derived by sol­ving the system

(18)

10

Notice that in case of 0 ~ d2 - dl ~ U2 the equilibrium probabilities of thestates (d2 ,dt}, (d2 ,dI - 1), ... , (d2 ,d2 - U2 ) are obtained by multiplyingthe equilibrium probability 11'0 by the relative probabilities within the junc­tion state (d2 , *).Though in general it is not possible to determine the equilibrium probabili­ties 11'; exactly we can provide estimations *;. Namely, as Q2 has the verysame structure as the matrix QI in the capacitated I-echelon serial system,we again invoke the moment-iteration method of De Kok in order to achievean approximation for the waiting-time (or 'shortfall') distribution Fw of theequilibrium system. Discretisation of this generated distribution Fw thenyields us an approximation for the probabilities 11';:

*0 .- FwU),*; .- Fw(i +!) - Fw(i - !), i 2: l.

Truncating (15) or (16) yields an estimation of the average costs associatedwith the modified base-stock policy (d2 , dI ), provided that the demand dis­tribution F is explicitly given in order to calculate exactly the expectedI-period costs r(z2, ZI) per state (Z2, ZI) in (17). If only the first two mo­ments of the demand D are given, we can only approximate the expectedI-period costs: As in the first iteration of algorithm 1 a distribution is fittedto the moments of D, we have FWo+Do, or FD, at our disposal. By meansof this fitted distribution FD we can easily determine the (approximated)expected I-period costs.

We have executed the average cost approximation on a class of 2-echelonserial systems, in which, besides 12 = 11 = 1,

.lE{D} = 100u{D} = 70

hI = 2h2 =2p = 200

CI = 10-3

C2 = 10-3 •

Notice that due to the values of .lE{D} and u{D} the demand distribu­tion is approximated by an Erlang mixture with the same s,cale parameter,

11

according to Tijms[1986].

First we generated the average cost of an optimal (Le. base-stock) pol­icy for the uncapacitated system by means of the program LIJNSYSTEEM3given in Van Houtum and Zijm[1990]. LIJNSYSTEEM3 provides an esti­mation of the optimal (base-stock) policy by a recursive calculation of in­complete convolutions in which the fit-method described in Tijms[1986] isused, plus an exact calculation of the average costs associated with the gen­erated base-stock policy. At the chosen parameter values this happens tobe the base-stock policy (d'f,d'f') = (614.1,498.9). Assuming the demanddistribution to be exactly an Erlang mixture with the same scale parameterwe see that the average cost gOO associated with (614.1,498.9) amounts to1669.03.Next, for increasing U2 we have determined the average costs associatedwith the modified base-stock policy (ld'fJ, ld'f'J) = (614,498), see table 1.Thereoff successively can be read the capacity U2 , the average shortfall

(d'f,d'f') = (614.1,498.9)

gOO = 1669.03

U2 lE{W} n 9 ctime

110 215.42 656 14302.1 6.6 s

125 72.09 121 3110.9 3.7 s

150 26.99 38 1922.1 3.0 s

200 7.45 14 1713.5 2.7 s

250 2.60 9 1681.0 3.0 s

300 0.98 6 1672.6 3.0 s

400 0.15 4 1669.5 3.2 s

500 0.02 3 1669.3 3.4 s

Table 1: Increasing U2

lE{W} on the desired echelon inventory levels in the equilibrium system, the

12

number of moment-iterations n carried out, the generated average cost ap­proximation 9 attached to (LdrJ, LdfJ), and an indication ofthe consumedcalculation time in seconds ctime on an Olivetti M240 PC with numericalcoprocessor.Probabilities smaller than 10-8 were set equal to O.We find that both average shortfall and average costs decrease if U2 in­creases, which is according to our expectations. Further we see that theaverage cost 9 converges towards 1669 (U2 = 500 almost corresponds withthe infinite capacity case). But we have to keep in mind, that 1669.03 is theexact average cost belonging to the modified base-stock policy (614.1,498.9),not (614,498)! Further research however pointed out that the exact averagecost associated with the base-stock policy (614,498) in the uncapacitatedmodel amounts to 1669.04, thus indicating the average cost function to bevery flat near the minimum.

4. Finding a nearly optimal modified base-stock policy. The costcalculation method presented above enables us, referring to our conclusionsin Speck and Van der Wal[1991], to derive a nearly optimal modified base­stock policy serving in its turn as an approximation for the optimal periodicreview policy.

If for the capacitated 2-echelon serial system an initial average cost valueis given associated with an optimal base-stock policy (dr,df) from the un­capacitated model, then the search for a nearly optimal modified base-stockpolicy can be restricted to the grid points of the upper right quadrant inthe 2-dimensional space with (LdfJ, LdfJ) as origin. The extensive scala ofnumerical procedures then enables us to achieve a further efficient reductionof the searching process.

References

[1] De Kok, A.G. (1989), A moment-iteration method for approximating the waiting­time characteristics of the GIIGl1 queue, Probability in the Engineering and Infor­mational Sciences 3 : 273-287.

[2] Federgruen, A., and Zipkin, P. (1986), An inventory model with limited produc­tion capacity and uncertain demands: the average-cost criterion, Mathematics ofOperations Research 11 : 193-207.

[3] Speck, C.J., and Van der Wal, J. (1991), The capacitated multi-echelon inventorysystem with serial structure: 1. The 'push ahead'-efJect, Dept. of Maths. andCompo Science, Memorandum COSOR 91-39, Eindhoven University of Technology,submitted for publication.

13

[4] Tijms, H.C. (1986), Stochastic modelling and analysis: a computational approach,Wiley, New York: 397-400.

[5] Van Houtum, G.J., and Zijm, W.H.M. (1990), Computationalproceduresjor stochas­

tic multi-echelon production systems, International Journal of Production Eco­

nomics 23 : 223-237.

14

EINDHOVEN UNIVERSITY OF TECHNOLOGYDepartment of Mathematics and Computing SciencePROBABILITY THEORY, STATISTICS, OPERATIONS RESEARCHAND SYSTEMS THEORYP.O. Box 5135600 MB Eindhoven, The Netherlands

Secretariate: Dommelbuilding 0.03Telephone : 040-473130

-List of COSOR-memoranda - 1991

Number

91-01

91-02

91-03

91-04

91-05

91-06

91-07

91-08

91-09

91-10

91-11

Month

January

January

January

January

February

March

March

April

May

May

May

Author

M.W.I. van KraaijW.Z. VenemaJ. Wessels

M.W.I. van KraaijW.Z. VenemaJ. Wessels

M.W.P. Savelsbergh

M.W.I. van Kraaij

G.L. NemhauserM.W.P. Savelsbergh

R.J.G. Wilms

F. CoolenR. DekkerA. Smit

P.J. ZwieteringE.H.L. AartsJ. Wessels

P.J. ZwieteringE.H.L. AartsJ. Wessels

P.J. ZwieteringE.H.L. AartsJ. Wessels

F. Coolen

The construction of astrategy for manpowerplanning problems.

Support for problem formu­lation and evaluation inmanpower planning problems.

The vehicle routing problemwith time windows: minimi­zing route duration.

Some considerationsconcerning the probleminterpreter of the newmanpower planning systemformasy.

A cutting plane algorithmfor the single machinescheduling problem withrelease times.

Properties of Fourier­Stieltjes sequences ofdistribution with supportin [0,1).

Analysis of a two-phaseinspection model withcompeting risks.

The Design and Complexityof Exact Multi-LayeredPerceptrons.

The Classification Capabi­lities of ExactTwo-Layered Peceptrons.

Sorting With A Neural Net.

On some misconceptionsabout subjective probabili­ty and Bayesian inference.

COSOR-MEMORANDA (2)

91-12

91-13

91-14

91-15

91-16

91-17

91-18

91-19

91-20

91-21

91-22

91-23

May

May

June

July

July

August

August

August

September

September

September

September

P. van der Laan

I. J. B. F. AdanG.J. van HoutumJ. WesselsW.H.M. Zijm

J. KorstE. AartsJ.K. LenstraJ. Wessels

P.J. ZwieteringM.J.A.L. van KraaijE.H.L. AartsJ. Wessels

P. DeheuvelsJ.H.J. Einmahl

M.W.P. SavelsberghG.C. SigismondiG.L. Nemhauser

M.W.P. SavelsberghG.C. SigismondiG.L. Nemhauser

P. van der Laan

P. van der Laan

E. LevnerA.S. Nemirovsky

R.J.M. VaessensE.H.L. AartsJ.H. van Lint

P. van der Laan

Two-stage selectionprocedures with attentionto screening.

A compensation procedurefor multiprogrammingqueues.

Periodic assignment andgraph colouring.

Neural Networks andProduction Planning.

Approximations and Two­Sample Tests Based onP - P and Q - Q Plots ofthe Kaplan-Meier Estima­tors of Lifetime Distri­butions.

Functional description ofMINTO, a Mixed INTegerOptimizer.

MINTO, a Mixed INTegerOptimizer.

The efficiency of subsetselection of an almostbest treatment.

Subset selection for an-best population:

efficiency results.

A network flow algorithmfor just-in-time projectscheduling.

Genetic Algorithms inCoding Theory - A Tablefor A3 (n, d) .

Distribution theory forselection from logisticpopulations.

COSOR-MEMORANDA (3)

91-24

91-25

91-26

91-27

91-28

91-29

October

October

October

October

October

November

I.J.B.F. AdanJ. WesselsW.H.M. Zijm

I. J. B. F. AdanJ. WesselsW.H.M. Zijm

E.E.M. van BerkurnP.M. Upperman

R.P. GillesP.H.M. RuysS. Jilin

I.J.B.F. AdanJ. WesselsW.H.M. Zijm

J. Wessels

Matrix-geometric analysisof the shortest queueproblem with thresholdjockeying.

Analysing MUltiprogrammingQueues by GeneratingFunctions.

D-optimal designs for anincomplete quadratic model.

Quasi-Networks in SocialRelational Systems.

A Compensation Approach forTwo-dimensional MarkovProcesses.

Tools for the InterfacingBetween Dynamical Problemsand Models withing DecisionSupport Systems.

91-30 November G.L. NernhauserM.W.P. SavelsberghG.C. Sigismondi

91-31 November J.Th.M. Wijnen

91-32 November F.P.A. Coolen

91-33 December S. van HoeselA. Wagelmans

91-34 December S. van HoeselA. Wagelmans

Constraint Classificationfor Mixed Integer Program­ming Formulations.

Taguchi Methods.

The Theory of ImpreciseProbabilities: Some Resultsfor Distribution Functions,Densities, Hazard Rates andHazard Functions.

On the P-coverage problemon the real line.

On setup cost reduction inthe economic lot-sizingmodel without speculativemotives.

91-35

91-36

December

December

S. van HoeselA. Wagelmans

F.P.A. Coolen

On the complexity of post­optimality analysis of0/1 programs.

Imprecise Conjugate PriorDensities for the One­Parameter ExponentialFamily of Distributions.

CaSaR-MEMORANDA (4)

91-37

91-38

91-39

91-40

December

December

December

December

J. Wessels

J.J.A.M. BrandsR. J .G. Wilms

C.J. SpeckJ. van der Wal

C.J. SpeckJ. van der Wal

Decision systems; therelation between problemspecification and mathema­tical analysis.

On the asymptoticallyuniform distribution modulo1 of extreme order statis­tics.

The capacitated multi­echelon inventory systemwith serial structure:1. The 'push ahead'-effect

The capacitated multi­echelon inventory systemwith serial structure:2. An average cost approxi­mation method