Tilburg University

A modified coordinated reorder procedure under aggregate investment and serviceconstraints using optimal policiy surfacesHeuts, R.M.J.; Bronckers, M.

Publication date:1988

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Citation for published version (APA):Heuts, R. M. J., & Bronckers, M. (1988). A modified coordinated reorder procedure under aggregate investmentand service constraints using optimal policiy surfaces. (Research memorandum / Tilburg University, Departmentof Economics; Vol. FEW 357). Unknown Publisher.

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A MODIFIED COORDINATED REORDERPROCEDURE UNDER AGGREGATE IN-VESTMENT AND SERVICE CONSTRAINTSUSING OPTIMAL POLICY SURFACESR.M. Heuts, M. Bronckers

FEW 357

Sept. 1988

1

A MODIFIED COORDINATED REORDER PROCEDURE UNDER AGGREGATE INVESTMENT ANDSERVICE CONSTRAINTS USING OPTIMAL POLICY SURFACES

R.M. Heuts and M. BronckersDepartment of EconometricsTilburg University

P.o. Box 901535000 LE Tilburg, The Netherlands.

Abstract

This paper presents a management oriented reorder technique based on acont~nuous (s,Q) inventory model, which does not require any marginal costinformation.The procedure - which is based on aggregate objectives and constraints -results in a point located on an optimal three dimensional response sur-face, showing optimal relationships among aggregate customer service,workload (total number of replenishments) and investment. By comparing ourcoordinated procedure with the classical EOQ-model, we will show the supe-riority of the simultaneous approach.

Especially demand which is highly erratic in nature will leave us withsignificant cost differences between the two approaches.

Ke wy ords~

Inventory, aggregate analysis, optimal poli.cy surface, workload, invest-ment budget, service constraint.

2

1. Introduction

In the authors' opinion a serious gap exists between the theoretical solu-tions on the one hand, and the real world problems on the other (see alsoGardner and Dannenbring [4]; Silver and Peterson [12]).We will try to bridge this gap in several ways.First we base our analysis on aggregate objectives and constraints. Byaggregating we overcome the most serious inconveniences of classical sin-gle-item models. In practice top management is responsible for thousandsof items and is primarily concerned with aggregate variables. In additionone has to operate under service constraints stated by the marketing de-partment and investment constraints fixed by the financial department. Inold - established literature (Brown [2]; Plossl and Wight [8]) one tacklesthis problem by solving a cost minimalization problem, with a servicerestiltant and neglecting any financial restrictions.In the second place we will try to avoid the prolem of a correct costparameter determination. In this context the carrying charge appears to bethe most complicating factor, as its value should not be based on theore-tical aspects, such as return on investment for example.Instead it ought to be an instrument variable . According to Brown [2]:"There is no "correct" value to use, other than the value that results inwhat management wants" (see also Silver and Peterson [12]). Over and abovethat the cost determination is heavily dependent on the accounting systemin use, it is not surprising that cost parameters can not uniquely beestablished. Furthermore the cost accounting output produces average fig-ur~~s in most cnses, while only marl;inril costs contein relevanL informa-tion. (see also Selen and Wood [11]; Schonberger and Schniederjans [10]).In this research, inventory decisions will be considered as a policytradeoff between aggregate customer service workload and investment, with-out any need to determine marginal cost figures.

2. A two-dimensional aggregate inventory analvsis

Besides the complexity of an explicit determination (see also the intro-duction) of the carrying charge ( r) and marginal fixed order cost (A),

3

it's even unwise to minimize total cost on an individual basis. Most prac-tioners are primarily concerned with aggregate inventory control, takinginto account aggregate objectives and constraints such as workload andinvestment.The traditional síngle-item analysis provides insufficient insights intoexisting interdependencies between items under control and conflictingintern business objectives. To gain more insight into the results on ex-changes curves as presented by Silver and Peterson [12], we will nextformally prove the optimality relation between both decision variables:total cycle stock investment (y) and workload (N), using a standard La-grange optimization technique. When an EOQ-strategy is used for each item,one can derive every individual EOQi-value (i-1,...n) from the resultingworkload associated with the total cycle stock investment fixed by top-management.Summarizing the above reasoning the following steps are in order:- the financial department fixes the total investment in cycle stocks

(y-y, a fixed value);- given this budget it's possible to determine the optimal workload

(N-N');- given this information it is possible to derive the (A~r)-value and the

individual EOQi- values (í-1,...,n).We now derive to above discussed optimality relation.

The objective function subject to the investment constraint is as follows:

n d.min N - F 1qi i-1 qi

n q.v.i is.t. Li-1 2

- Y

whereN- total annual replenishments (- workload).d. - annual sales in units for item iiq. - order quantity in units for item iiy- total annual cycle stock investmentvi - unit value of item i expressed in guilders per unit.

(2.1)

(2.2)

y

The Lagrangean function becomes:

n d. n q,v. lL(q1,...,9n,~ )- F 1 t~ f F 1 1- Y J .Y i-1 qi y Li-1 2 (2.3)

where ~Y - Lagrangean multiplier with respect to the investment budget.Differentiating with respect to qi, aY (i-1,...n) we obtain the followingfi-rst order conditions:

~L -di ~ vi~ - 2 t ~- - 0, ( i-1,...n);q.1 qi

~L n qí~i~a - E 2 -Y-O.

y i-1

After some algebraic manipulations we obtain the following results:

~ - N

Y Y

and

2 d.iq. - (i-1... ,n),i a v.'

Y 1

which after substitution leads to:

2.d..yi9. - . (i-1,.. ,n).i N.v.i

Therefore equation ( 2.1) can be rewritten:

(2-4)

(2.5)

(2.6)

(2.7)

(2.8)

n di n N.di.vi~. 1 n~N- E - ï - J ï d..v. (2.9)

i-1 2.di.y i-1 2'Y Y f i-1 i iN.v.i

or

5

~.~-~~. ~i-1

which is equivalent with

nN . Y - ~ . ï di.

i-1 1The above formula (2.11) is an hyperbola.

(2.10)

(2.11)

Using an EQO-stategy for each item we will obtain two more relations withwhich we can demonstrate the equivalence with the above Lagrangean result.

n q.v, n v..r ' iE 1 1- ï 1Y -

i-1 2 i-1 2

v

. J d..v.i i

- J-. 1. ï J d..v.r f i-1 i i

n d. n d.N- L 1- ï 1

i-1 qi i-1 2A.div..ri

n- ~ r 1 ~i-1 A f i i

- J - . 1 . n ~ï d..v.A ~ i-1 i i

(2.12)

(2.13)

So we can verify:

6

2r ny. N-~. j L di.vi ~.

11-1

At the same time we obtain:

y AN - r

(2.14)

(2.15)

which is equivalent to formula ( 5.29) and ( 5.30) from Silver and Peterson[12].Note the equivalence between formula (2.6) and (2.15):

N ry - y - A (2.16)

From (2.16) it's clear that a(r~A)-value is implied by fixing a multi-plier value. .4 graphical representation of formula (2.14) is to be foundin fikure (2.1):

-~- - - -- - -- ---~j

Figure 2.1. Example of the exchange curve

N

From relation (2.15) it's easily seen that any point on the hyperboliccurve implies an unique value of r.

Alternative approaches can be found in Plossl and Wight [8], Eaton C3],Prichard and Eagle [9], Groff and Muth [5], Hadley and Whitin [6].

Silver and Peterson [12] give an excellent overview of the exchange curveconcept, of which a short exposition will be given next.

Figure 2.2. Aggregate performance of different r values.

Not.ation used:

A: estimated cur.rent operating point in terms of total number of re-plenishments per year (N) and total average cycle stock (y).

B: point achieved by cutting total average cycle stock (y) at the sametotal annual number of replenishments (N).

C: point achieved by cutting the total number of replenishments per year(N) at the same total average cycle stock (y).

D: optimal point achieved by the total average cycle stock (y) fixed bythe financial department.

Given the feasible policy alternatives, management should try to improvetheir current policy by moving closer in a left downward direction to theexchange curve with regard to the current operating point A.Quoting Silver and Peterson [12]:"From a top management perspective this is far more appealing than thefact that the EOQ minimizes total costs on an individual item basis.(Particularly when the individual cost factors are so difficult to esti-mate)".

8

The solution procedure is presented in short to clarify the general con-cept:

1. Management fixes the total annual cycle stock investment (y).2. Via equation (2.11) the minimal number of replenishments (Nw) can be

determined because there exists an optimal relationship between aggre-gate average cycle stock investment and workload, when demand is deter-ministic.The point on the exchange curve which is found in this manner, deter-mines implicitly an unique A~r - value, which can be obtained withFormula (2.15).

3. Via equation (2.8) the order quantities (qi,i-1,...,n) can be deter-mined without explicitly having to estimate the marginal order cost (A)and carrying charge (r).

4. An improvement on the actual inventory performance can be obtained bymoving in figure 2.2 from the estimate current operating point A to-wards the optimal poinY. D on the so cnlled exchange curve.

Implicit.ly we have assumed a direct splitting up of the total availableinventory investment budget (Y) into a total cycle stock investment (y)and a total safety stock budget. The latter being equal to:

n n nY- y- E S. - i k. . 6 . v

i-1 1 i-1 1 Li i

where:Si : safety stock in guilders for item i;ki : safety factor for item i;

(2.17)

o~ : estimated standard deviation of errors of forecasts over a replenish-menL lead time, in units.

Most practioners will find it difficult to explicitly decompose the totalinventory budget into the above mentioned components. Second, such anexplicit decomposition of the total budget will lead to inferior resultscompared to the simultaneous approach elaborated in the next section.

9

Finally we should note that demand is considered known and more or lessconstant in time. However, with stochastic demand things are more compli-cated. To treat these complexities appropriately we will introduce athree-dimensional aggregate analysis in the next section.

3. Three-dimensional aggregate inventory analysis:a modification of the Gardner and Dannenbring approach

Stochastic demand complicates matters considerably. First, cycle and safe-ty stock investment decisions are interdependent for each item. In addi-tion interactions also exist between the products as the two budgets forcycle and safety stocks have to be allocated to the individual items.[Jntil now we have assumed a predetermined total safety stock budget (seeformula (2.1~)). Furthermore the order quantities were calculated neglec-ting the standard deviation of forecast errors, as demand was determini-stic and constant in time. With stochastic demand the standard deviationsplay an essential role in the determination of safety stocks.Therefore the concept as presented in section 2, has to be extended withanother aggregate policy variable: customer service in terms of the per-centage of annual customer requisitions which are backordered (short),which will be projected on the vertical axis, denoted as (1-PZ).To allow for a simultaneous approach we will use an extended investmentconcept. Instead of total average cycle stock (y) we now introduce totalstock investment (Y).Just as in the two-dimensional analysis we will now present a graphicalillustration of the relationships between total stock investment, workloadand customer service.

10

N

YFigure 3.1. The optimal policy surface for stochastic demand

Most of the tradeoffs which are presented by the optimal policy surfaceare straightforward. The interested reader is referred to Gardner andDannenbring [4] for more details. However, the effect of an increase inworkload is more complex. In Gardner and Dannenbring's words: "With afixed investment constraint, increases in workload are equivalent to in-creases in the number of exposures to risk of stockout".On the other hand, the increased workload leads to increased total safetystock at the cost of total cycle stock.For small and moderate values of N the marginal increase of safety stockdominates the risk of going short. But eventually the decrease in servicelevel will overwhelm the marginal increase in safety stock. We defineNlimit- for a fixed value of Y- as the value of N where a maximum servicelevel is attained.For determining the Nlimit- value, Gardner and Dannenbring [4] formulate aLagrangean model which minimizes the total number of annual shortagessubject only to an investment constraint. Next they add a workload con-strairit to locate interior points on the optimal policy surface.For more details on this subject we refer to their article [4].

11

We will now present an alternative approach which we think will be farmore appealing to management. Instead of minimizing the service level interms of the percentage of annual customer requisitions which are backor-dered (short) subject to a fixed workload constraint, we now try to mini-mize the total workload subject to a predetermined aggregate service con-straint.Before turning to the essential modified Lagrangean model formulation, wefirst present the overall aggregate simultaneous solution procedure.

Phase 1:

Locate the edge of optimality. In other words: determine the maximal work-load Nlimit given the i~ivestment constraint. Following Gardner and Dannen-bring a single point on the edge of optimality is found by minimizing thefollowing objective function:

n di ~ (xi-si)1L1 ql sf ml f(xi) dxi -(1-P2)

subject to the investment constraint:

n qi.vi~ 2 t Si~ - Y .i-1

(3-1)

(3-2)

where the symbols used will be defined at the end of phase 3. For detailson the solution procedure we refer to Gardner and Dannenbring [4].

Phase 2.

Via a trial-and-error procedure one has to determine a feasible (Y,(1-P2))combination, where (1-P2) may vary. Following Gardner and Dannenbring aninterior point associated with the (Y,(1-P2)) combination is found byadding to (3.1) and (3.2) the workload constraint N as was determinedlimitin Phase 1:

n d.ii~l qi - Nlimit (3-3)

12

Again we refer to Gardner and Dannenbring [4], for further details on thesolution procedure of this model.

Phase j:

Determine the interior point on the optimal policy surface, according tothe modified Lagrange model formulation. To locate any interior point onthe surface, to the left of the edge of optimality, the objective functionis:

n d.min z - F 1qi i-1 qi

subject to the investment and service constraints:

i-1

n q.v. lF lZl tsi J - Y

n di m (xi-si)1F1 ql sf ml f(xi) dxi -(1-P2)

(3.3)

(3.4)

(3.5)

wherez : total annual replenishments per yeardi : annual sales in units for item iqi : order quantity in units for item ivi : unit value of item i expressed in guilders per unitSi : safety stock in guilders per item iY : total investment constraint in guildersxi : leadtime demand in units for item im. : customer requisition size in units for item iif(x.) : probability density function for leadtime demand for item ii(1-P2): service constraint in terms of the percentage of annual customer

requisitions which are backordered ( short).si : reorder points in units (sum of safety stock plus leadtime demand

stock) for item í

The next step is to form the Lagrangean function, L:

n d. n q.v. lL(q1,...qn,S1,...Sn,~Y,~ )- ï 1 t a ï 1 1 t S. J - Y t

p i-1 qi Y i-1 2 1

n d.. a [ F 1 fm ( x.-s.) f(x.) dx. - (1-P )] (3.6)P 9.m. s. i i i i 2i-1 i i i

Differentiating with respect to qi, Si, ~Y and ~p, we obtain the firstorder conditions:

~L -di ~ vi ~ di m} ~ ~- f (x.-s.) f(x.) dx. -~qi - q? 2 - q2.m. si i i i i

i i i0

, (i-1,...n.) (3.7)

DL - ~ } aY.di m~Si Y qi.mi ' sf (-1) . f(xi) . dxi - 0. ( i-1,....n) (3-8)

~L n qi.vi lDa - L 2 'Si J -Y-0

Y i-1

~1 n di~~ - L q m. sfm (xi-si) f(xi) dxi -(1-P2) - 0

p i-1 i i i

We introduce some simplifying notation:

Pi - sfm f(xi) dxi~

Ei - sfm ( xi-si) f(xi) dxi~

d.i

where

mi

(3.9)

(3.10)

(3.11)

(3.i2)

(3.13)

Pi: probability of a stockout during one order cycle

14

E.: partial expectation of demand or the expected number of units shortiper order cycle

Fi: annual frequency of demand for each item.

So (3.7). (3.8). (3,9) and ( 3.10) are respectively equivalent with:

-di aY.vi ap.Fi.Ei2 } 2 - 2 -

qi qi

i-1

0 , (i-1, . ,n) (3.14)

~ -~p.Fi.Pi - 0

Y ql . (i-1....,n) (3.15)

n q,v.E 121 t Si - Y

n F.E.F 1 1 - (1-P )

i-1 qi Z

From (3.15) it follows that:

p i i~Y - ql or qi - ~Y .

(3.16)

(3.17)

(3.18)

Relation (3.16) can be rewritten as follows:

na . L F..P..v.n a.F..P..v. l p 1 1 1

1~1 p 2~Yi i t Sil - Y or ~Y - 1-n . (3.19)

J 2(Y- i Si)i-1

In addition, it follows from (3.15) that:

~Y'qiPi - a .F. ' (i-1' ' 'n)'

P 1

a .F..P. a .F..P.p i i

(3.20)

Rearrange formula (3.14) into:

15

qi

or even further into:

~ .v. d. ~ .F..E.Y2 1- 2. P 2 1 ,(i-1,...,n)

qi

~ .q..v. d. ~ .F..E.Y 2i i- 1 t P 1 1,(i-1,...,n).qi qi

Summing formula (3.22) with respect to i:

n q..v. n d. n F.E.i i 1.~ F i i

~Y iEl 2 - i~l qi P i-1 qi '

which can be rearranged into:

i

nl ~Flil l .

F Jq.

Substitution of (3.1~) into (3.24) leads to:

n qivi n di~Y (i~l ? - i~l qi

~ E qi~i - ~ diY i-1 2 i-1 qi

(1-P2)

We summarize the derived relationships:

n~ ï Fi.Pi.viY i-1

~ - nP 2 (Y- L Si)

i-1

~Y qiPi - ~ ' F.P 1

~ F ql Vi - E aiY i-1 2 i-1 qi

~p - (1-P2) ~

(3.2~)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

(3.20)

(3.25)

]G

Finally we derive an equation for qi, ( i-1,...,n) from equation (3.21):

2 2 (ditap.Fi.Ei)

qi - ~Y ~i, (i-1, . ,n) (3.27)

which can be reduced to:

2 (d.t~ .F..E.)1 p 1 iqi - ~Y ~i

, (i-1,...,n). (3.28)

A detailed flow-chart of the solution procedure can be found in the appen-dix, where the following assumptions are in order:- the lenght of the leadtime is constant;

- the customer requisition sizes for each items are constants and areindependent of the level of demand;

- the leadtime demand is normally distributed, where the distribution mo-ments are given and may differ per item.

Brown [2], Pantumsinchai, Hassan and Gupta [7] among others have alsoproposed aggregate inventory decision rules, for a stochastic demand pro-cess. However, their suggested procedures are inferior to the one above,because:- reorder quantities and reorder points are determined sequentially, i.e.not simultaneously;

- reorder quantities and reorder points are determined neglecting anyinfluences of standard deviations of forecast errors or service levels;

- total average cycle stock investment is considered as a resultant, astotal safety stock investment is determined first and independent oftotal average cycle stock investment.

4. Conclusions and suggestions for further research

As mentioned earlier, aggregate concepts are more appealing to managementthan any analysis on índividual basis.In addition we avoid all estimation problems of an explicit marginal costdetermination by working with aggregate variables.

The service measure as used in this paper is based on an average value fora group of items under consideration. A measure which takes into accountthe same service level for every item in the group could be an alterna-tive. This concept would imply the use of n service constraints which canbe attacked with existing software packages on non-linear programming.However, the computational effort is more complicated. In our searchproce-dure optimal reorder quantities and safety stocks are determined simulta-neously. The standard deviations of forecasts errors during the leadtimemay now influence the optimal results. In this way items which can beforecasted accurately consume less safety stock than those which are high-ly erratic in nature.

Furthermore, in comparison with the sequential two-dimensional analysis,

w}iere safety stocks are predetermined, we now obtain a more efficient

balance between total average cycle stock - and total safety stock

investment.

Gardner and Dannenbring (see [4]) have already compared a simultaneousreorder procedure with one where reorder quantities were determinedindependently from each other, and they obtained substantial improvements.Further research is planned to test our suggested procedure empirically.

18

References~

[1] Abramowitz, M. and Stegun, J.A., 1968,

Handbook of mathematical functions,

Dover Publications, New York.

[2] Brown, R.G., 1967,Decision rules for inventory management,Arthur D. Little, New York.

[3] Eaton, J.A., 1964,New - the limit technique,Modern Materials Handling, 19: 38-43.

[4] Gardner, E.S. and Dannenbring, D.G., 1979.Using optimal policy surfaces to analyze aggregate inventorytradeoffs,Management Science, 25: 709-720.

[5] Groff, G.K. and Muth, J.F., 1972,Operations management: analysis for decisions,Irwin, Homewood.

[6] Hadley, G. and Whitin, T.M., 1963,Analysis of inventory systems,Prentice-Hall, Englewood Cliffs.

[7J

[8]

Pantumsinchai, P., Hassan, M.Z. and Gupta, I.D., 1983.Basic programs for production and operations management,Prentice-Hall, Englewood Cliffs.

Plossl, G.W. and Wight, O.W., 1967Production and inventory control,Prentice-Hall, Englewood Cliffs.

19

[9] Prichard, J.W. and Eagle, R.H., i965,

Modern inventory management,New York.

[10] Schonberger, R.J. and Schniederjans, M.J., 1984,Re-inventing inventory control,Inter)eaces, 14: 76-83.

[11] Selen, W.J. and Wood, W.R., 1987,Inventory cost definition in an EOQ model application,Production and Znventory Management Journal, 44-47.

[12] Silver, E.A. and Peterson, R., 1985.Decision systems for inventory management and production planning,John Wiley ~ Sons, New York.

zo

A~~endix: Flowchart of the solution procedure

STARTReading of all starting values:

Y; n; x o d.~ v.; m.- i-1,.. ,nL.' L.' i' i i'i i

Phase 1:

Having determined Nlimit' using the solution procedure of Gardner and

Dannenbring, we will only use workload values which are smaller than

it~Nlim

Phase 2:We check the feasibility of the service level (1-P2)` as fixed by top

management, by determining the minimal service level in terms of thepercentage of annual customer requisitions which are backordered(short), (1-P2) associated with the ÍY.Nli ) combination from Phase 1im t

Yes (1-P2) ~ (1-P2)N

21

aDetermine ~ using (3.26):

Pn

~ ~ i Fi Pi viY- i-1 where S. - 0a n ip 2(Y- F S ) P - k, i-1,...,ni i

i-nn

~ ï ~ Fi . viY i-1So: ~ - 2YP

Calculate qi, i-1,...,n by means of formula (3.20)

. F . P~ l ipql - ~

Ywhere Pi - }, i-1,...,n,

. F~ ~p iso q - ~

iY

Determine aY and a with the help of equation (3.25)pn q.v.

1 1n d.~aY ( ï 2 ) -

1F q1 iii-

~ --

p (1-P2) '

22

Calculate qi - i-1,...,n using (3.28)2.(dit~p.Fi.Ei)

qi - ~ .v. ~ i-1,...,nY i

where:

Ei - sfm (xi-si) f(xi) dxii

~ 6L. 1 exp (-~ ki) -1 2R

d- t - 1an ' i - 1t0,2316419.k.ibl - 0,319381530b2 - -0.356563782b3 - 1,781477937b~ - -1,821255978

b5 - 1.3302784429

1 exp (-~ ki) Lbltit...tb5ti5]2n

, i-1, . ,n

Evaluate Pi, i-1,...,n with the help of (3.20)

~Y qiPi - a F.' i-1,...,n

P 1

~

23

1Now Pi, i-1,...,n is known where also

Pi - f0 f(x) dxi - kf~ fx(z) dz with z -- N(0,1)si i

Using a rational function approzimation (see Abramowitz andStegun [1]):

t t2ttc~ c2.cl.k. x t -

1 i a 2 a 3awhere

. l.tt 2.t t 3.t

kZ h P i 1t- 1J n wit i nown,

( ~

,...,n-

1

where co - 2,51551~ dl - 1,432~88

- 0,802853c d - 0,189269l 2

- 0,010328c d - 0,0013082 3

Calculate the S values, i-1, ...,n associated with the currenti

k values, i-1,...,n:i

Si - ki . 6L . vi with 6L , vi knowni i

Calculate:n q.v, n F.

1ï 2 t Si and ï 1. E.i-1 i-1 qi 1

24

n (q.v. ~E I 2 1 t Si - Y

i-1 `

andn F. ~L 1 . Ei - (i-P2)~

i-1 qi

STOP AND PRINT ALL THE RELEVANT

n d.INFORMATION INCLUDING E 1- N AND

i-1 qiS., i-1,...,n.i

WITH FORMULA (2.15) WE CAN CALCULATETHE INDIVIDUAL EOQi, i-1,...,n WHERE

nY - Y - ï Si

i-1NOTE THE DIFFERENCES BETWEEN qi ANDEOQi, i-1,...,n.

aRecalculate ~Y using (3.26):

Pn

~ E Fi.Pi.viY i-1

a np 2.(Y-ï Si)

i-1

Recalculate aY and ap with the help of (3.25)

a F qivi F aiY i-1 2 i-1 qi

(1-P2)~

where ~Y, qi, vi, di, (1-P2) are knownP

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249 Peter M. KortThe Influence of a Stochastic Environment on the Firm's Optimal Dyna-mic Investment Policy

250 R.H.J.M. GradusPreliminary versionThe reaction of the firm on governmental policy: a game-theoreticalapproach

251 J.G. de Gooijer, R.M.J. HeutsHigher order moments of bilinear time series processes with symmetri-cally distributed errors

252 P.H. Stevers, P.A.M. VersteijneEvaluatie van marketing-activiteiten

253 H.P.A. Mulders, A.J. van ReekenDATAAL - een hulpmiddel voor onderhoud van gegevensverzamelingen

254 P. Kooreman, A. KapteynOn the identifiability of household production functions with jointproducts: A comment

255 B. van RielWas er een profit-squeeze in de Nederlandse industrie?

256 R.P. GillesEconomies with coalitional structures and core-like equilibrium con-cepts

11

257 f.k1.M. Ruys, G. vrín der LaanCompuLat,ion of nn industrial equilibrium

258 W.H. Haemers, r~.E. BrouwerAssociation schemes

259 G.J.M. van den BoomSome modifications and applications of Rubinstein's perfect equili-brium model of bargaining

260 A.W.A. Boot, A.V. Thakor, G.F. UdellCompetition, Risk Neutrality and Loan Commitments

261 A.W.A. Boot, A.V. Thakor, G.F. UdellCollateral and Borrower Risk

262 A. Kapteyn, I. WoittiezPreference Interdependence and Habit Formation in Family Labor Supply

263 B. BettonvilA formal description of discrete event dynamic systems includingperturbation analysis

264 Sylvester C.W. EijffingerA monthly model for the monetary policy in the Netherlands

265 F. van der Ploeg, A.J. de ZeeuwConflict over arms accumulation in market and command economies

266 F, van der Ploeg, A.J, de ZeeuwPerfect equilibrium in a model of competitive arms accumulation

267 Aart de ZeeuwInflation and reputation: comment

268 A.J. de Zeeuw, F. van der PloegDifference games and policy evaluation: a conceptusl framework

269 Frederick van der PloegRationing in open economy and dynamic macroeconomics: a survey

270 G. van der Laan and A.J.J. TalmanComputing economic equilibria by variable dimension algorithms: stateof the art

271 C.A.J.M. Dirven and A.J.J. TalmanA simplicial algorithm for finding equilibria in economies withlinear production technologies

272 Th.E. Nijman and F.C. PalmConsistent estimation of regression models with incompletely observedexogenous variables

273 Th.E. Nijman and F.C. PalmPredictive accuracy gain from disaggregate sampl.ing in arima - models

111

2~4 Raymond H.J.M. GradusThe net present value of governmental policy: a possible way to findthe Stackelberg solutions

2~5 Jack P.C. KleijnenA DSS for production planníng: a case study including simulation andoptimization

2~6 A.M.H. GerardsA short proof of Tutte's characterization of totally unimodularmatrices

27~ Th. van de Klundert and F. van der PloegWage rigidity and capital mobility in an optimizing model of a smallopen economy

278 Peter M. KortThe net present value in dynamic models of the firm

2~9 Th, van de KlundertA Macroeconomic Two-Country Model with Price-Discriminating Monopo-lists

280 Arnoud Boot and Anjan V. ThakorDynamic equilibrium in a competitive credit market: intertemporalcontracting as insurance against rationing

281 Arnoud Boot and Anjan V. ThakorAppendix: "Dynamic equilibrium in a competitive credit market:intertemporal contracting as insurance against rationing

282 Arnoud Boot, Anjan V. Thakor and Gregory F. UdellCredible commitments, contract enforcement problems and banks:intermediation as credibilíty assurance

283 Eduard PondsWage bargaining and business cycles a Goodwin-Nash model

284 Prof.Dr, hab. Stefan MynarskiThe mechanism of restoring equilibrium and stability in polish market

285 P. MeulendijksAn exercise in welfare economics (II)

286 S. JeJrgensen, P.M. Kort, G.J.C.Th. van SchijndelOptimal investment, financing and dividends: a Stackelberg differen-tiel game

287 E. Nijssen, W. ReijndersPrivatisering en commercialisering; een oriëntatie ten aanzien vanverzelfstandiging

288 C.B. MulderInefficiency of automatically linking unemployment benefits to priva-te sector wage rates

1V

289 M.H.C. PaardekooperA Quadratically convergent parallel Jacobi process for almost diago-nal matrices with distinct eigenvalues

290 Pieter H.M. RuysIndustries with private and public enterprises

291 J.J.A. Moors ~ J.C. van HouwelingenEstimation of linear models with inequality restrictions

292 Arthur van Soest, Peter KooremanVakantiebestemming en -bestedingen

293 Rob Alessie, Raymond Gradus, Bertrand MelenbergThe problem of not observing small expenditures in a consumerexpenditure survey

294 F. Boekema, L. Oerlemans, A.J. HendriksKansrijkheid en economische potentie: Top-down en bottom-up analyses

295 Rob Alessie, Bertrand Melenberg, Guglielmo WeberConsumption, Leisure and Earnings-Related Liquidity Constraints: ANote

296 Arthur van Soest, Peter KooremanEstimation of the indirect translog demand system with binding non-negativity constraints

v

IN 1988 REEDS VERSCHENEN

297 Bert BettonvilFactor screening by sequential bifurcation

298 Robert P. GillesOn perfect competition in an economy with a coalitional structure

299 Willem Selen, Ruud M. HeutsCapacitated Lot-Size Production Planning in Process Industry

300 J. Kriens, J.Th. van LieshoutNotes on the Markowitz portfolio selection method

301 Bert Bettonvil, Jack P.C. KleijnenMeasurement scales and resolution IV designs: a note

302 Theo Nijman, Marno VerbeekEstimation of time dependent parameters in lineair modelsusing cross sections, panels or both

303 Raymond H.J.M. GradusA differentisl game between government and firms: a non-cooperativeapproach

304 Leo W.G. Strijbosch, Ronald J.M.M. DoesComparison of bias-reducing methods for estimating the parameter indilution series

305 Drs. W.J. Reijnders, Drs. W.F. VerstappenStrategische bespiegelingen betreffende het Nederlandse kwaliteits-concept

306 J.P.C. Kleijnen, J. Kriens, H. Timmermans and H. Van den WildenbergRegression sampling in statistical auditing

307 Isolde Woittiez, Arie KapteynA Model of Job Choice, Labour Supply and Wages

308 Jack P.C. KleijnenSimulation and optimization in production planning: A case study

309 Robert P. Gilles and Pieter H.M. RuysRelational constraints in coalition formation

310 Drs. H. Leo TheunsDeterminanten van de vraag naar vakantiereizen: een verkenning vanmateriële en immateriële factoren

311 Peter M. KortDynamic Firm Behaviour within an Uncertain Environment

312 J.P.C. BlancA numerical approach to cyclic-service queueing models

V1

313 Drs. N.J. de Beer, Drs. A.M. van Nunen, Drs. M.O. NijkampDoes Morkmon Matter?

314 Th. van de KlundertWage differentials and employment in a two-sector model with a duallabour market

315 Aart de Zeeuw, Fons Groot, Cees WithagenOn Credible Optimal Tax Rate Policies

316 Christian B. MulderWage moderating effects of corporatismDecentralized versus centralized wage setting in a union, firm,government context

31~ Jdrg Glombowski, Michael KrtigerA short-period Goodwin growth cycle

318 Theo Nijman, Marno Verbeek, Arthur van SoestThe optimal design of rotating panels in a simple analysis ofvariance model

319 Drs. S.V. Hannema, Drs. P.A.M. VersteijneDe toepassing en toekomst van public private partnership's bij degrote en middelgrote Nederlandse gemeenten

320 Th. van de KlundertWage Rigidity, Capital Accumulation and Unemployment in a Small OpenEconomy

321 M.H.C. PaardekooperAn upper and a lower bound for the distance of a manifold to a nearbypoint

322 Th. ten Raa, F. van der PloegA statistical approach to the problem of negatives in input-outputanalysis

323 P. KooremanHousehold Labor Force Participation as a Cooperative Game; an Empiri-cal Model

324 A.B.T.M. van SchaikPersistent Unemployment and Long Run Growth

325 Dr. F.W.M. Boekema, Drs. L.A.G. OerlemansDe lokale produktiestructuur doorgelicht.Bedrijfstakverkenningen ten behoeve van regionaal-economisch onder-zoek

326 J.P.C. Kleijnen, J. Kriens, M.C.H.M. Lafleur, J.H.F. PardoelSampling for quality inspection and correction: AOQL performancecriteria

vii

327 Theo E. Nijman, Mark F.J. SteelExclusion restrictions in instrumental variables equations

32~3 13.8. van der GenugtenLar.imation in linear regression under the presence of heteroskedas-t.icity of a completely unknown form

329 Raymond H.J.M. GradusThe employment policy of government: to create jobs or to let themcreate?

330 Hans Kremers, Dolf TalmanSolving the nonlinear complementarity problem with lower and upperbounds

331 Antoon van den ElzenInterpretation and generalization of the Lemke-Howson algorithm

332 Jack P.C. KleijnenAnalyzing simulation experiments with common random numbers, part II:Rao's approach

3j3 Jacek OsiewalskiPosterior end Predictive Densities for Nonlinear Regression.A Partly Linear Model Case

334 A.H. van den Elzen, A.J.J. TalmanA procedure for finding Nash equilibria in bi-matrix games

335 Arthur van SoestMinimum wage rates and unemployment in The Netherlands

336 Arthur van Soest, Peter Kooreman, Arie KapteynCoherent specification of demand systems with corner solutions andendogenous regimes

337 Dr. F.W.M. Boekema, Drs. L.A.G. OerlemansDe lokale produktiestruktuur doorgelicht II. Bedrijfstakverkenningenten behoeve van regionaal-economisch onderzoek. De zeescheepsnieuw-bouwindustrie

338 Gerard J. van den BergSearch behaviour, transitions to nonparticipation and the duration ofunemployment

339 W.J.H. Groenendaal and J.W.A. VingerhoetsThe new cocoa-agreement analysed

340 Drs. F.G. van den Heuvel, Drs. M.P.H. de VorKwantificering van ombuigen en bezuinigen op collectieve uitgaven1977-1990

341 Pieter J.F.G. MeulendijksAn exercise in welfare economics (III)

V111

342 W.J. Selen and R.M. HeutsA modified priority index for GQnther's lot-sizing heuristic undercapacitated single stage production

343 Linda J. Mittermaier, Willem J. Selen, Jeri B. Waggoner,Wallace R. WoodAccounting estimates as cost inputs to logistics models

344 Remy L. de Jong, Rashid I. A1 Layla, Willem J. SelenAlternative water management scenarios for Saudi Arabia

345 W.J. Selen and R.M. HeutsCapacitated Single Stage Production Planning with Storage Constraintsand Sequence-Dependent Setup Times

346 Peter KortThe Flexible Accelerator Mechanism in a Financial Adjustment CostModel

34~ W.J. Reijnders en W.F. VerstappenDe toenemende importantie van het verticale marketing systeem

348 P.C. van Batenburg en J. KriensE.O.Q.L. - A revised and improved version of A.O.Q.L.

349 Drs. W.P.C. van den NieuwenhofMultinationalisatie en codrdinatieDe internationale strategie van Nederlandse ondernemingen naderbeschouwd

350 K.A. Bubshait, W.J. SelenEstimation of the relationship between project attributes and theimplementation of engineering management tools

351 M.P. Tummers, I. WoittiezA simultaneous wage and labour supply model with hours restrictions

352 Marco VersteijneMeasuring the effectiveness of advertising in a positioning contextwith multi dimensional scaling techniques

353 Dr. F. Boekema, Drs. L. OerlemansInnovatie en stedelijke economische ontwikkeling

354 J.M. SchumacherDiscrete events: perspectives from system theory

355 F.C. Bussemaker, W.H. Haemers, R. Mathon and H.A. WilbrinkA(49,16,3,6) strongly regular graph does not exist

356 Drs. J.C. CaanenTien jaar inflatieneutrale belastingheffing door middel van vermo-gensaftrek en voorraadaftrek: een kwantitatieve benadering

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