1-s2.0-s0925527300000505-main

*Corresponding author. Tel.: #46-13-28-1504; fax: #46-13-28-8975.

E-mail address: [email protected] (R.W. GrubbstroK m).

Int. J. Production Economics 68 (2000) 123}135

Modelling rescheduling activities in a multi-periodproduction}inventory system

Robert W. GrubbstroK m*, Ou Tang

Department of Production Economics, Linko( ping Institute of Technology, S-581 83 Linko( ping, Sweden

Received 1 April 1998; accepted 19 April 2000

Abstract

Decisions for planning production activities for multi-period production}inventory systems have been studied ina number of papers applying input}output analysis and the Laplace transform. The decisions have concerned activitiesspread out over time without having the opportunity to adjust future decisions when the external and/or internalcircumstances change. In this paper, we extend the analysis to situations when rescheduling is possible. Firstly, di!erentclasses of causes justifying rescheduling activities are presented including periodic rescheduling and `net changea.Secondly, in terms of previously developed theory, we model the behaviour of a simple single-level production}inventorysystem for which its production plan may be modi"ed at one point in the future. ( 2000 Elsevier Science B.V. All rightsreserved.

Keywords: Rescheduling; Laplace transform; Multi-period production}inventory system; Renewal process

1. Introduction

The purpose of a production}inventory system isto meet the external market requirement by e$-ciently using the resources in the system. In multi-level production}inventory systems, material re-quirements planning (MRP) is usually employed asan information system to connect the relationshipsbetween demand and supply of di!erent items.Starting from the master production schedule(MPS), MRP explodes detailed production deci-sions from top-level items down to the lower-levelitems. In several previous papers [1}5], optimisa-

tion conditions for production planning are derivedfor such systems applying input}output analysisand the Laplace transform. The production sched-ule for end items is used as the MPS. When deter-mining the MPS, safety stock considerations havebeen included to enhance system performance. Inthe theory hitherto developed, the opportunity foradjusting the production schedule due to thechanged background circumstances has been disre-garded within the planning horizon.

In a real production}inventory system, when thesystem updates information about its operationalenvironment and the market forecast, there may bereasons to change the previously decided produc-tion schedule, although this schedule earlier wasregarded as the optimal one. Because of internaland external unforeseen developments, there canbe a need to revise the schedule considerably. In

0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 5 0 - 5

practice, this is often done either periodically, orwhen a strong need is experienced.

On the other hand, every rescheduling of "nishedtop-level items usually causes changes concerninglower-level production plans due to the e!ects frominternal demand a!ected by the parts explosion.Such adjustments of plans also have an augmentede!ect in an assembly system, and this is often refer-red to as system nervousness. Nervousness may be-come the obstacle in the implementation of MRPand even cause the collapse of the whole system.A general discussion on rescheduling and therelated nervousness topic can be found in [6}9].

In this paper, we "rst review related work con-cerning the rescheduling problem in multi-level andrelated production}inventory systems. The basicmechanism for justifying rescheduling is discussedin an analytical framework. As a few preliminarydevelopments, a "rst model for reschedulingdeparting from the net present value (NPV)approach is analysed. Based on this model, a num-ber of numerical examples are studied. The majorpart of our "ndings is summarised in a concludingsection together with some ideas for developing thistheory in new directions.

2. Motives for rescheduling

Rescheduling refers to the process of changingorder or operation due dates, usually as a result oftheir being out of phase with when they are needed[10]. Due to an increasing uncertainty of the in-formation while the previous plan is implemented,MRP frequently needs to update unplanned eventswithin or outside the production}inventory systemin order to keep the information accurate. Unplan-ned events of these kinds are usually considered toemanate from the following four sources, whichalso provoke the requirement for rescheduling,namely (i) uncertainty in external demand; (ii) un-certainty in supply conditions; (iii) e!ect of a rollingplanning horizon, and (iv) a `system e!ecta.

Changes in external demand conditions are verycommon reasons for performing rescheduling ac-tivities. Normally, the MPS is determined accord-ing to a forecasted demand and assumed supplysituation, including considerations such as capacity

limitations. As soon as a customer changes hisdemand pattern, either quantitatively or timely, theoriginal forecast needs to be adjusted. If the MPS isfrozen and such changes are ignored, di!erencesbetween demand and supply will develop resultingfor instance in a depressed service level or an in-creased cost of holding inventory due to additionalcapital tied up.

Neither the supply from outside nor the internalsupply of intermediate items is entirely predictablein the real world situation. There can be a variationin the vendor or production lead times, scrap canexceed estimated quantities, errors are discoveredin inventory data, and machine breakdowns occur[11]. All these causes re#ect that the informationon available inventory may be inaccurate. Whetheror not an original MRP plan should continue to beimplemented, depends on how to remedy the prob-lem by rescheduling and the economic conse-quences thereof.

The third motive for initiating a new schedule isthe rolling planning horizon. Blackburn et al. [12]discussed the horizon sensitivity embedded inmany dynamic lot-sizing algorithms. When thetime of planning moves forward, new demand ap-pears, which causes changes in part, if not all of theprevious MPS. A forward lot-sizing rule such as theSilver-Meal algorithm, can diminish the horizonsensitivity by accumulating the changes near to theend of the horizon. However, such an algorithmusually misses the optimal solution. Studies of ner-vousness and lot-policy relationships can also befound in [13}15]. Besides traditional lot-sizingrules, we also observe that other optimal produc-tion planning procedures, for example when usingthe net present value (NPV) approach [2], aresensitive to the planning horizon. The MPS alwaysfaces a pressure to be rescheduled when time isrolling forward.

Basically the last reason for rescheduling is a by-product from the aforementioned three sources. Ina multi-level system, whenever the production deci-sion is changed at a higher level, the MRP plan forthe related lower-level items will be out of phase.This chain reaction downwards through the prod-uct structure and upstream along the productionchain refers to excessive sensitivity and responsive-ness of the whole MRP system to minor changes.

124 R.W. Grubbstro(m, O. Tang / Int. J. Production Economics 68 (2000) 123}135

Fig. 1. Trade-o! between di!erent rescheduling decisions.

This is what is called `MRP nervousnessa. Thisnervousness leads to increased costs, it reduces pro-ductivity, it lowers the service level and it createsgeneral states of confusion on the shop #oor [16].Nervousness is probably the main obstacle againstconsidering rescheduling opportunities.

3. Methods for rescheduling

Orlicky [17] pointed out that there aretwo essential methods for updating informationand rescheduling in an MRP system. The "rst is`schedule regenerationa and the second is `netchangea.

According to the American Production andInventory Control Society, regeneration is de"nedto be a process approach in which the masterproduction schedule is totally re-exploded downthrough all bills-of-materials, to maintain validpriorities [10]. New requirements and planned or-ders are completely recalculated and regenerated atthat time. One obvious advantage of this method isthat it purges errors that have accumulated in thesystem so that the system information becomesmore accurate. Since it concerns all elements in thesystem, a good solution can be obtained. On theother hand, regeneration is limited by its frequencydue to the computational e!orts involved and theircorresponding cost.

A `net changea is an approach in which thematerial requirements plan is continually retainedin the computer. Whenever a change is needed inrequirements, open order inventory status, or bill-of-materials, a partial explosion and netting ismade for only those parts a!ected by the change.Therefore, all irrelevant items are left untouched.The calculation e!ort in this case is smaller. Orlicky[17] and Oden et al. [18] mentioned that netchange might cause more nervousness because itresults in a high frequency of many minor changes,and sometimes a rescheduling has to be made toa recently rescheduled plan.

Most MRP software packages include bothmethods. Net change is applied more frequently toupdate information on a continuous basis whileregeneration is applied in a longer time interval torefresh the whole system.

As mentioned in the previous section, thepurpose of rescheduling is to "ll the gap betweenthe forecasting information and the actual state.Therefore, the service level of the system is expectedto increase and inventory costs are reduced. Never-theless, the negative result from rescheduling is thatit creates nervousness. Fig. 1 illustrates that there isa trade-o! between the rescheduling and oppositenon-rescheduling (stay) alternatives.

Carlson et al. [13] discussed the cost of ner-vousness. Since a production plan is less sensitive tothe change of order volume or to cancel an order,they concentrated on the cost of scheduling a newsetup. The cost is then supposed to be a function oftime. If the rescheduled setup is moved backwardsbeyond the minimum lead time, the cost is in"nitelylarge. In other intervals, they proposed using a sim-ilar estimation procedure as for the usual setup costto obtain a rescheduling cost. However, we maynote that their nervousness cost is based on thelimitations of a single-level item system. The chaine!ect of nervousness was ignored because of itscomplexity. Van der Sluis [19] also stated that thenervousness cost is determined by the early replen-ishments, which cause all scheduled replenishmentsto be moved forward. This follows the same frame-work as of Carlson et al. but Van der Sluis focusedon a multi-level system.

Inderfurth [20] indicated that the consequencesfrom nervousness are di$cult to evaluate in termsof costs, since rescheduling expenses not only de-pend on the situation-speci"c availability of theplanning capacity, but they are also a!ected by the

R.W. Grubbstro(m, O. Tang / Int. J. Production Economics 68 (2000) 123}135 125

performance of the whole system and the moralreaction of the sta! involved.

Instead of investigating the trade-o! between res-cheduling decisions in monetary terms, some re-searchers have put e!ort into studying dampeningmechanisms for stabilising system nervousness. Ho[21] suggested a dampening "lter to be used to screenout insigni"cant rescheduling messages which de-pend on the decision criterion used in the "lter, suchas limitation of time changes and the cost of ner-vousness. Penlesky et al. [22] compared four di!er-ent "ltering heuristics for rescheduling. All theseheuristics perform statistically better than the "xeddue-date procedure (no rescheduling) and dynamicprocedure (frequent rescheduling). A similar studywas also carried out by Christy and Kanet [23]. Inshort, the advantage to accomplish rescheduling un-der appropriate circumstances is obvious.

From our literature review, we have noticed thatmost research carried out has been based on simu-lation methodology when studying the reschedul-ing problem. Analytical models appear only tohave been presented by Inderfurth [20], in whichhe studied the impact on the stability of productionplanning from di!erent parameters in inventorycontrol rules, such as from the (s,S) and (s, nQ)policies. The objective function thus applied mea-sured the stability of the system rather than in-cluded economic consequences.

4. Basic problem

Our basic problem treated is how to determinethe appropriate circumstance for when and whennot to reschedule. In other words, what is to triggerthe act of rescheduling. Essentially, we envisagethat there are two main approaches. The "rst isa time trigger, like regeneration and net changetaking place at regular pre-determined points intime. The frequency is determined in advance andthe production}inventory system may be inspectedperiodically to examine if there is a need or not toreschedule. The second approach is that a state ofthe system reached automatically triggers a res-cheduling activity. Information concerning sucha state may be the inventory or backlog situation orhow much the actual demand has departed from

its previous forecast, etc. Similar two classes ofpolicies are studied in basic inventory theory withperiodic-review models and reorder-point systems.

Apart from limiting ourselves to a single-levelsystem, in our treatment to follow, we also con"neourselves to a case in which there is only one pointof time in the future at which we can reconsidera previously decided production plan. Hence, weavoid complications from recurring reschedulingopportunities. These limitations will be relaxed infuture studies since analysing multi-level produc-tion}inventory systems is our main target. We alsoassume that the horizon of the original plan (thelength of the time interval covered initially) doesnot change at the time a new schedule is made, i.e.we are not considering a rolling planning process.This enables the possibility of a comparison be-tween rescheduling or not. If and when the scheduleis changed, it concerns the remaining part of theoriginal horizon. As in previously developed the-ory, demand will be assumed to follow a stochasticrenewal process.

As mentioned in Section 3, the cost from carryingout a rescheduling operation is di$cult to deter-mine in practice. For our purpose we assume thatthere is a "xed exogenously determined cost asso-ciated with deciding to reschedule.

At "rst, let us introduce the following notation.The predetermined point in time at which a pre-viously determined plan can be revised is ¹, thesequence of production decisions prior to this pointin time are denoted p and those at and beyond thispoint are denoted p@.

Let d and d@ represent dependent stochastic se-quences of a demand process, before and after thepossible rescheduling time ¹. Assume that thesehave probability distributions written dF(z) anddF@(uDz), respectively, meaning that the probabilityof an outcome d@"u might possibly depend on theoutcome of the previous sequence d"z.

The general framework for rescheduling is illus-trated in the diagram in Fig. 2. At time 0, we makedecisions for production p and p@ based on demandsequences d and d@. At the rescheduling point, wecalculate the expected net present value with andwithout rescheduling and make a comparison.

In Fig. 2, squares indicate decision points andcircles chance points at which stochastic sequences


Fig. 2. Decision-tree diagram for rescheduling.

are realised. The notation pirefers to di!erent pos-

sible choices of p, and dj

to di!erent possible out-comes of d, and similarly for p@

land d@

k. For

expository reasons, the "gure assumes that thereexist "nite sets of opportunities for choosing theproduction decisions and for demand outcomes,but in our treatment below this limitation is notrequired. The symbol in the "gure refers toa `tolla amounting to the economic consequencesfrom making a new schedule. The stay option refersto the opportunity to stick to the originally decidedplan for the period succeeding the reschedulingpoint. The decision to reschedule is immediatelyfollowed by a choice of new plan p@

l.

Let NPV(p, p@Dd, d@) be the stochastic economicoutcome, based on these four sequences. The objec-tive function to be maximised is assumed to be theexpected value

E(NPV(p,p@Dd, d@))

"PPNPV(p, p@Dz,u) dF@(uDz) dF(z). (1)

A "rst basic question addressed, is whether thereexist simple conditions concerning the functionNPV(p,p@Dd, d@) in order for the "rst part of thedecision sequence p to be optimal irrespective ofwhether there is an opportunity to choose a newsecond part of the sequence p@ at ¹ after the "rstoutcome d has been revealed, or not.

We compare the two cases after introducing theabbreviations x"p, y"p@ and

a(x, y, z)"Pu

NPV(p, p@Dz,u) dF@(uDz)

"Pu

NPV(x, yDz, u) dF@(uDz). (2)

In the "rst case, both the "rst and the second part ofthe production decision sequence x"p and y"p@are decided at the initial time t"0.

The optimisation problem for this case thenreads

NPVH"Maxx

MaxyPa(x, y, z) dF(z). (3)

Let xH and yH, respectively, denote the optimalsolution.

In the second case x"p is chosen at t"0 and y"p@ chosen at t"¹ after the outcome z"d is known.We now instead have the optimisation problem

NPVHH"MaxxPMax

y

a(x, y, z) dF(z). (4)

Let xHH denote the optimal solution for the "rstpart of the decision sequence and yHH(z) the optimalremaining decision sequence (depending on theoutcome z) in this second case.

It is obvious that the second case in generalproduces a higher value of the objective function asgiven by the statement

NPVHH*NPVH.

If there were no cost associated with changing theplan, in general, it would always pay to make thechange. You can never lose by doing this.

A major question remaining is whether or not theoptimal initial decision sequence remains the samewith and without the option to reschedule? In thefollowing theorem, we show that in a speci"c case ofindependence between the periods, the original opti-mal sequence for both parts of the horizon remainsoptimal, also after the outcome z is revealed.

Theorem 1.

IfL2a(x, y, z)

LyLz,0, then xHH"xH and yHH"yH.


Hence, yHH will be independent of the outcome z. Thisalso implies that NPVHH"NPVH.

Furthermore, assuming a second, weaker type ofindependence between the two parts of the horizon,we "nd a su$cient condition for the "rst part of thedecision sequence to remain optimal whether ornot there are rescheduling opportunities:

Theorem 2.

IfL2a(x, y, z)

LxLy,0, then xHH"xH.

Theorems 1 and 2 provide su$cient conditionsfor the optimal "rst part of the decision sequence tobe the same, irrespective of whether or not there areopportunities to revise the decisions. Their proofsare included in the appendix. Necessary conditionsremain to be investigated.

As a simple example illustrating the reschedulingproblem, we consider the following variation of theNewsboy problem. Let x be an amount producedinitially before a stochastic demand z occurs and lety be an amount which can be produced to covera possible backlog when z is known. Let r be theunit revenue, c the unit production cost, K thesetup cost and c( a unit penalty for each unit soldfrom the second batch. Let us further assume thatz is exponentially distributed with a parameterj and that r'c#c( . We distinguish three intervalsfor z creating di!erent expressions for total pro"ts(for simplicity written NPV):

a(x, y, z)"!c(x#y)!K(sgn(x)#sgn(y))

#Grz if z(x,

rz!c( (z!x) if x)z(x#y,

r(x#y)!c( y if x#y)z.

In the stay case, we have

NPVH"Max

x

Max

yP

=

0

je~jza(x, y, z) dz

"Max

x

Max

yAr(1!e~j(x`y))!c( e~jx(1!e~jy)

j

!c(x#y)!K(sgn(x)#sgn(y))B

which is maximised for

xH"ln(r/c)

j, yH"0, if K(

r!c(1#ln(r/c))

j,

and xH"0, yH"0, otherwise.When rescheduling is considered, we instead

obtain the problem

NPVHH"MaxxP

=

0

je~jz Maxy

a(x, y, z) dz.

The inner maximisation has the solution

yHH"Gz!x if z'x#K/(r!c!c( ),

0 elsewhere.

Inserting this function and developing the integralgives us

NPVHH

"Max

xAr(1!e~jx)#(r!c!c( )e~j(x`K@(r~c~c( ))

j

!cx!KB,which has the unique solution

xHH"

Gln ((r!(r!c!c( )e~jK@(r~c~c( ))/c)

jif K(

r!c

j!cxHH,

0 otherwise.

Since r!(r!c!c( )e~jK@(r~c~c( )(r, the opti-mum initial production xHH is smaller for therescheduling alternative compared to the staycase and with positive production xH (a small K).We may also note that the threshold value of K,beyond which no production is optimal, is smallerfor the stay case than for the rescheduling case,since

r!c(1#ln(r/c))

j(

r!c

j!cxHH.

Obviously, the opportunity to reschedule af-fects the initial decision even in this simpleexample.


5. Extension of NPV theory to cover non-zero initialnet inventory

5.1. Stockout and inventory functions

The previously developed theory, in whichamong other questions safety stock issues havebeen focused upon [1,5], has treated cases when noinitial inventory nor any initial backlog have beenpresent. For treating a basic rescheduling problem,the theory needs to be extended to cover cases withopportunities for a non-zero initial net inventory.The decision sequences p and p@ from Section 4 arenow interpreted as sequences of batches producedand the demand sequences d and d@ as realisationsof a renewal process, i.e. a sequence of unit demandevents separated by independent stochastic timeintervals.

The net inventory written R(t) is de"ned to beR(t)"S(t)!B(t), where S(t)*0 and B(t)*0 areinventory and stockouts, respectively, and R(t) canbe either positive or negative. The relationship be-tween expected inventory E(S(t)), expected stock-outs E(B(t)), expected cumulative demand E(DM (t))and cumulative production PM is given by

E(S(t))!E(B(t))"R(0)#PM !E(DM (t)), (5)

where R(0) is the initial net inventory. In terms ofthe Laplace transform, we have

E(SI (s))!E(BI (s))"R(0)

s#PMI (s)!E(DMI (s)), (6)

where the tildes denote transforms of the corre-sponding time functions. The symbol s is used forthe complex Laplace frequency.

In previous papers, for instance [2], the stockoutfunction has been developed in terms of the trans-form for demand following a renewal process, as-suming R(0)"0:

E(BI (s))"fI PM `1

s(1!fI )"E(DMI (s))!

1

s

PM

+j/1

fI j, (7)

which holds for time intervals during which PM isconstant. Here fI (s) is the transform of the densityfunction of the time between two consecutive de-mand events. In the case that R(0)#PM *0, weimmediately obtain the generalisation

E(BI (s))"fI R(0)`PM `1

s(1!fI )"E(DMI (s))!

1

s

R(0)`PM

+j/1

fI j. (8)

When R(0)#PM (0, we instead have

E(BI (s))"E(DMI )!R(0)#PM

s. (9)

Therefore, for the appropriate intervals duringwhich PM is given:

E(BI (s))"

GfI R(0)`PM `1

s(1!fI )"E(DMI )!

1

s

R(0)`PM+j/1

fI j, R(0)#PM*0,

E(DMI )!R(0)#PM

s, R(0)#PM(0.

(10)

Similarly, we obtain the expected inventory func-tion

E(SI (s))

"GR(0)#PM

s!

f (1!fI R(0)`PM )

s(1!fI ), R(0)#PM *0,

0, R(0)#PM (0.

(11)

5.2. Objective function and optimisation conditions

As shown in previous papers, for instance [3],the net present value of the cash #ow in this systemcan easily be presented in terms of the Laplacetransform. By ignoring a possible lost sales event atthe end of the horizon, the expected Net PresentValue for the backlogging case can be written

NPV"r[E(DI (o))!oE(BI (o))#B(0)]

!

n+k/1

(K#c(PMk!PM

k~1))e~otk , (12)

where r, c, K and o are the given parameters beingthe unit sales price, the unit production cost, the"xed setup charge and the continuous interest rate,respectively. We always assume that r'c, other-wise there would never be any chance of makinga pro"t from the production. Cumulative produc-tion is a staircase function described by the levelPMk

during the interval tk)t(t

k`1. This staircase


LE(BI (s))LPM

k

"[E(BI (w))]PM k`1

![E(BI (w))]PM k

"

1

2piPbì=

w/b~i=

([E(BI (w))]PM k`1

![E(BI (w))]PM k

)e~(s~w)tk!e~(s~w)tk`1

s!wdw

"G!

1

2pi:bì=w/b~i=

fI R(0)`PM k`1

s

e~(s~w)tk!e~(s~w)tk`1

s!wdw, R(0)#PM

k*0,

!

1

2pi:bì=w/b~i=

1

s

e~(s~w)tk!e~(s~w)tk`1

s!wdw, R(0)#PM

k(0.

(15)

constitutes the production decisions to be taken.The term oE(BI (o)) accounts for delayed paymentsfrom backlogging and B(0) is the initial stockout attime t"0. The objective function NPV is to bemaximised subject to constraints of the typesPMk`1

'PMk'0 and t

k`1't

k*0, for k"1,

2,2, n, where n is the total number of batches untilthe horizon.

Because cumulative production PMk

is a staircasefunction, the expected stockouts E(BI (o)) in theabove equation are the sum of £~1ME(BI (s))N multi-plied by impulses of unit height and of a durationrestricted to each interval (the characteristic func-tion). This multiplication in the time domain isequivalent to a convolution in the frequency do-main. Therefore,

E(BI (s))"n+k/0

[E(BI (s))]PM k *

e~stk!e~stk`1

s

"

1

2pi

n+k/0

Pbì=

w/b~i=

[E(BI (w))]PM k

]e~(s~w)tk!e~(s~w)tk`1

s!wdw, (13)

where the asterisk denotes the convolution opera-tion and where b is real and chosen so that theintegral converges. By convention, we let t

0"0

and PM0"0. The time derivative is

LE(BI (s))Lt

k

"e~stk[[E(B(tk))]

PM k~1![E(B(t

k))]

PM k] (14)

and the di!erence with respect to PMk

(being inte-ger-valued) is

Since E(DI (o)) and B(0) in the objective function areindependent of the decision variables t

kand PM

k, the

necessary optimisation conditions thus read

LNPV

Rtk

"oe~otk(!r([E(B(tk))]

PM k~1![E(B(t

k))]

PM k)

#(K#c(PMk!PM

k~1))))0, (16)

LNPV

LPMk

"!ro([E(BI (o))]PM k`1

![E(BI (o))]PM k

)

!c(e~otk!e~otk`1))0, (17)

for k"1, 2, 2, n.For the "rst batch time we have the weak in-

equality t1*0, whereas for later times t

k'0, for

k*2, which create equalities in Eq. (16). Thecomplementarity conditions for the "rst time there-fore requires

LNPV

Lt1

t1"0. (18)

5.3. Initial net inventory cases

The initial net inventory, either as an inventoryor a stockout, has an impact on the productionplan following. When there is an initial backlog,which means that R(0)(0 and B(0)'0, the fol-lowing theorem for the system applies.

Theorem 3. The optimal xrst batch PM1

is at leastB(0), i.e. PM

1*B(0), or PM

1#R(0)*0. This means

that any initial backlog is immediately covered by thexrst batch.


Table 1Parameters of the production}inventory calculations

Case 1 Case 2 Case 3 Case 4 Case 5

Planning horizon ¹K 500 500 500 500 500Interest rate o 0.01 0.01 0.01 0.01 0.01Demand rate j 1 1 1 1 1Sales price r 5000 2000 1000 1000 1000Fixed setup cost K 100 100 100 300 600Variable production cost c 100 100 100 100 100

Fig. 3. Relative gain in NPV discounted to time t"¹ vs.rescheduling time ¹ for Cases 1}3.

Regarding conditions for the "rst batch time tobe positive or zero, we have results given by thefollowing theorem.

Theorem 4. (a) If the revenue from the initial stockoutis larger than the total cost of the xrst batch,rB(0)'K#cPM

1, there is a requirement for an im-

mediate setup, t1"0. (b) If initial stockout B(0) is

less than K/(r!c), we must always wait for the xrstsetup, i.e. t

1'0. Hence, if the xrst setup is immedi-

ate, i.e. t1"0, then we must have B(0)*K/(r!c).

We also easily "nd that t1'0 implies

PM1*r(B(0)!K)/c. However, we have not yet been

able to "nd su$cient conditions for an immediatesetup expressed only in terms of the productionparameters K, c, r and o.

The results above are used for determining theoptimal future production plan when choosing therescheduling alternative. Proofs are given in theappendix.

6. Numerical examples

Based on the rescheduling problem stated inSection 4, we calculate the value from reschedulingin the following "ve examples. First we releasea production plan at t"0 according to the initialnet inventory situation. At the rescheduling point¹, we then compare the di!erence between theexpected future NPV for the rescheduling and stayalternatives calculated from t"¹ and onwards.External demand is assumed to follow a Poissonprocess. The stay alternative is based on initial

decisions assuming no rescheduling to be possible(the xH solution in Section 4). The reschedulingdecisions are determined based on prior batch deci-sions and the backlog at ¹ (the function yHH(xHH, z)in Section 4). The comparisons made thus implicitlyassume that the optimal initial decisions are equal(xHH"xH) irrespective of the opportunity to res-chedule. The parameter values chosen for the pro-duction}inventory system are listed in Table 1.

When choosing di!erent alternative points atwhich the production plan can be revised, the cal-culations show that the value from reschedulingincreases with the rescheduling time ¹ in case thetime discount factor e~oT is disregarded(Figs. 3 and 4). These curves also indicate a fairlylinear relationship. For di!erent sales prices, theslope of the curve changes dramatically (Fig. 3). Onthe other hand, di!erent setup costs generate some-what the same slopes (Fig. 4). When the discountfactor e~oT is considered (NPV is calculated withreference to time zero), we have always found


Fig. 4. Relative gain in NPV discounted to time t"¹ vs.rescheduling time ¹ for Cases 3}5.

Fig. 5. Relative gain in NPV discounted to time t"0 vs. res-cheduling time ¹ for Cases 1}3.

Fig. 6. Relative gain in NPV discounted to time t"0 vs. res-cheduling time ¹ for Cases 3}5.

Fig. 7. The NPV vs. net inventory for the rescheduling and stayalternatives. For negative net inventories below s

1and positive

net inventories above s2

it is advantageous to reschedule.

a maximum point for the NPV gain in di!erentcases. However, Figs. 5 and 6 indicate that themaximum point is insensitive with respect to theparameters r and K.

Even though the expressions for the optimisationconditions are complicated, the numerical exampleshows that its NPV is quite linear with respect tothe net inventory at ¹ (Fig. 7). This linearity mightsuggest that there is an independence of a kind thatwould justify the assumption xH"xHH.

A zero net inventory generates the lowest NPV.For a positive net inventory, the NPV of the stayalternative is almost constant (slightly increasing).This means that additional initial inventory doesnot have an obvious impact on the expected stock-out. On the other side, for a negative net inventory,when making it more and more negative, NPV will"rst increase but approaches a constant level forvery high backlogs. For small stockout levels at

t"¹, the NPV di!erence is marginal between therescheduling and stay alternatives. This is due tothe insigni"cant di!erence in the optimum res-cheduling and stay production decisions (staircasefunction) for a small stockout.

If we include a rescheduling cost in the model(c@ in Fig. 7), the size of the gap between the NPV ofthe rescheduling and stay alternatives determinesan interval, outside of which it is advantageous torevise the plan. Consequently, the boundary gapsdetermine the states that should trigger a res-cheduling.

From the numerical example, we also notice thatthe "rst batch time of the new schedule tends to-wards zero when the initial stockout is highenough. When this "rst batch time is zero, the stepsof the resulting staircase function (PM

k#R(0), t

k) are

identical, which simpli"es the analysis. Therefore,


"nding the boundary value of the initial stockoutfor the "rst batch to be zero is imperative. Never-theless, this su$cient condition still needs to befurther investigated as mentioned in Section 5.

7. Summary

This paper aims to take a "rst glance at therescheduling problem in a production}inventorysystem based on our previous research. After a lit-erature review of related topics, we designed a gen-eral model for addressing the rescheduling problemin the single-level case. The gap between the res-cheduling and stay alternatives can be calculatedfrom the model and can be compared with anexogenously determined rescheduling cost.

By relaxing the initial inventory constraints inthe previous NPV theory, we have also obtainedsome insights into the impact of the initial netinventory level on the optimal production plan.The optimal production decision for the "rst batch,especially the "rst batch time, is important in a res-cheduling study. So far as we know, this will alsoconstitute the main constraint for the reschedulingof lower-level items in a multi-level extension.However, the su$cient conditions for an immedi-ate setup still cannot be expressed only in terms ofthe production parameters such as K, c, r, o. It alsodepends on PM

1, which is a decision variable in the

system. This should be an important problem to besolved in a future study.

Our discussion in Section 4 has shown su$cientconditions for the optimal solutions xH and xHH tocoincide. However, in general, the optimal decisionsequences before the rescheduling point are di!er-ent for the cases with and without reschedulingoptions. Even in a simple case, as the example inSection 4 indicated, the optimal solutions xH andxHH are di!erent. There is therefore a strong need toinvestigate such conditions further.

Appendix A

Proof of Theorem 1. The general solution to[L2a(x, y, z)]/(Ly Lz)"0 may be written as the sumof two arbitrary functions having the arguments

according to a(x, y, z)"A(x, y)#B(x, z). The "rstmaximisation case then reads

NPVH"Maxx

MaxyP(A(x,y)#B(x, z)) dF (z)

"Maxx

Maxy

(A(x, y)#PB(x, z) dF (z))

"Maxx

(A(x, yH)#PB(x, z) dF (z))

"A(xH, yH)#PB(xH, z) dF (z),

and the second

NPVHH"MaxxPAMax

y

(A(x, y)#B(x, z)B dF (z)

"MaxxAMax

y

(A(x,y)#PB(x, z) dF (z)B"Max

xAA(x, yHH)#PB(x, z) dF (z)B.

Hence, yHH is determined by the same maximisa-tion as yH and therefore yHH"yH. Then the remain-ing maximisation with respect to x will also be thesame as before, i.e. xHH"xH.

Proof of Theorem 2. The general solution to[L2a(x, y, z)]/(Lx Ly),0 may be written as the sumof two arbitrary functions having the argumentsaccording to a(x, y, z)"M(x, z)#N(y, z).

For the "rst maximisation case we have

NPVH"Maxx

MaxyP(M(x, z)#N(y, z)) dF (z)

"MaxxPM(x, z) dF (z)#Max

yPN(y, z) dF (z)

"PM(xH, z) dF (z)#PN(yH, z) dF (z).

For the second case we instead have

NPVH"MaxxPAM(x, z)#Max

y

N(y, z)B dF (z)


"MaxxPM(x, z) dF (z)#PMax

y

N(y, z) dF (z)

"PM(xHH, z) dF (z)#PN(yHH(z), z) dF (z).

Therefore, the maximisation with respect to x in thetwo cases coincide and xHH"xH, whereas this isnot true in the general case concerning the maxi-misation with respect to y.

Proof of Theorem 3. We assume the converse,namely that PM

1#R(0)(0. Then from Eq. (15) we

have

LNPV

LPM1

"!ro([E(BI (o))]PM 1`1

![E(BI (o))]PM 1

)

!c(e~ot1!e~ot2)

"!roA!1

2piPb`i=

w/b~i=

1

o

]e~(o~w)t1!e(o~w)t2

o!wdwB

!c(e~ot1!e~ot2)

"roAReso/0

1

oe~(o~w)t1!e~(o~w)t2

o!w B!c(e~ot1!e~ot2)

"roA1

o(e~ot1!e~ot2 )B!c(e~ot1!e~ot2 )

"(r!c)(e~ot1!e~ot2 )'0.

This contradicts the optimisation condition (17).Hence, it pays to increase PM

1inde"nitely as long as

PM1#R(0)"PM

1#B(0)(0. Therefore, in the opti-

mum we must have PM1*B(0), meaning that the

"rst batch at least must cover the initial backlog.

Proof of Theorem 4. Since fI j(s)/s is the transform ofa cumulative probability distribution with nojumps, it always holds that £~1M fI j(s)/sN*0 andthat lim

t?0£~1M fI j(s)/sN"lim

s?=s fI j(s)/s , for all j.

For a system with an initial stockout, we haveR(0)(0. Then

LNP<

Lt1

"oe~ot1(!r[E(B(t1))]

PM 0

![E(B(t1))]

PM 1)#K#cPM

1)

"oe~ot1A!rA[E(DM )]#B(0)

!A[E(DM )]!£~1G1

s

PM 1~B(0)+j/1

fI jHBB#K#c(PM

1!PM

0)B

"oe~ot1A!rAB(0)#£~1G1

s

PM 1~B(0)+j/1

fI jHB#K#cPM

1B)oe~ot1(!rB(0)#K#cPM

1).

Hence, if K#cPM1(rB(0) then LNPV/Lt

1(0 and

from Eq. (18) t1"0. From Theorem 3, we have

PM1*B(0). Therefore, if K'(r!c)B(0), it must

hold that t1'0.

When instead B(0)"0, which means R(0)*0,the derivative is evaluated as

LNPV

Lt1

"oe~ot1A!rA!£~1G1

s

R(0)+j/1

fI jH#£~1G

1

s

R(0)`PM 1+j/1

fI jHB#K#cPM1B.

Then, by Theorem 3 we obtain

LNPV

Lt1Kt/0

"o(K#cPM1)*o(K#cB(0))'0,

which contradicts the optimisation conditionEq. (16). Therefore, t

1'0.

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