The stochastic dynamic productioninventory lot-sizing problem by taoyni


									                                                 ARTICLE IN PRESS

                                      Int. J. Production Economics 88 (2004) 105–119

       The stochastic dynamic production/inventory lot-sizing
               problem with service-level constraints
                                S. Armagan Tarima, Brian G. Kingsmanb,*
                               Department of Computer Science, University of York, York YO10 5DD, UK
               Department of Management Science, The Management School, Lancaster University, Lancaster LA1 4YX, UK

                                          Received 17 April 2003; accepted 26 May 2003


   This paper addresses the multi-period single-item inventory lot-sizing problem with stochastic demands under the
‘‘static–dynamic uncertainty’’ strategy of Bookbinder and Tan (Manage. Sci. 34 (1988) 1096). In the static-dynamic
uncertainty strategy, the replenishment periods are fixed at the beginning of the planning horizon, but the actual orders
are determined only at those replenishment periods and will depend upon the demand that is realised. Their solution
heuristic was a two-stage process of firstly fixing the replenishment periods and then secondly determining what
adjustments should be made to the planned orders as demand was realised. We present a mixed integer programming
formulation that determines both in a single step giving the optimal solution for the ‘‘static–dynamic uncertainty’’
strategy. The total expected inventory holding, ordering and direct item costs during the planning horizon are
minimised under the constraint that the probability that the closing inventory in each time period will not be negative is
set to at least a certain value. This formulation includes the effect of a unit variable purchase/production cost, which
was excluded by the two-stage Bookbinder–Tan heuristic. An evaluation of the accuracy of the heuristic against the
optimal solution for the case of a zero unit purchase/production cost is made for a wide variety of demand patterns,
coefficients of demand variability and relative holding cost to ordering cost ratios. The practical constraint of non-
negative orders and the existence of the unit variable cost mean that the replenishment cycles cannot be treated
independently and so the problem cannot be solved as a stochastic form of the Wagner–Whitin problem, applying the
shortest route algorithm.
r 2003 Elsevier B.V. All rights reserved.

Keywords: Inventory; Chance-constrained programming; Integer programming; Modelling; Optimisation

                                                                      1. Introduction

                                                                         The study of lot-sizing began with Wagner and
                                                                      Whitin (1958), and there is now a sizeable
  *Corresponding author. Tel.: 44-1524-593-848; fax: 44-1524-         literature in this area extending the basic model
   E-mail addresses: (S.A. Tarim),
                                                                      to consider capacity constraints, multiple items,,                multiple stages, etc. However, most previous work
(B.G. Kingsman).                                                      on lot-sizing has been directed towards the

0925-5273/03/$ - see front matter r 2003 Elsevier B.V. All rights reserved.
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106                   S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119

deterministic case. The reader is directed to De              optimality under the ‘‘static–dynamic uncertainty’’
Bodt et al. (1984), Potts and Van Wassenhove                  strategy of Bookbinder and Tan.
(1992), Kuik et al. (1994) and Kimms (1997) for a                The optimal solution to the problem is the (s; S)
review of lot-sizing techniques.                              policy with different values for each period in the
   The practical problem is that in general much, if          time varying demand situation. Where the demand
not all, of the future demands have to be forecast.           level changes slowly, it is usually satisfactory to
Point forecasts are typically treated as determinis-          use a steady-state analysis with constant S and s
tic demands. However, the existence of forecast               values updated once per year. However, this is
errors radically affects the behaviour of the lot-            inappropriate if the average demand can change
sizing procedures based on assuming the determi-              significantly from period to period. This presents a
nistic demand situation. Forecasting errors lead              non-stationary problem where the two control
both to stockouts occurring with unsatisfied                   parameters change from period to period. The
demands and to larger inventories being carried               uncertainty in the timing of future replenishments
than planned. The introduction of safety stocks in            caused by an (s; S) policy may be unattractive
turn generates even larger inventories and also               from an operational standpoint.
more orders. It is reported by Davis (1993) that a               Although the inventory problems with station-
study at Hewlett-Packard revealed the fact that               ary demand assumption are well known and
60% of the inventory investment in their manu-                extensively studied, very little has appeared on
facturing and distribution system is due to demand            the non-stationary stochastic demand case. Re-
uncertainty.                                                  cently, Sox (1997) and Martel et al. (1995) have
   There has been increasing recognition as illu-             described static control policies under the non-
strated by Wemmerlov (1989) that future lot-sizing            stationary stochastic demand assumption in a
studies need to be undertaken on stochastic and               rolling horizon framework. Sox (1997) presents a
dynamic environments that have at least a                     mixed integer non-linear formulation of the
modicum of resemblance to reality. Inevitably,                dynamic lot-sizing problem with dynamic costs,
the forecast errors have to be taken into account in          and develops a solution algorithm that resembles
planning the future lot-sizes. Similar concerns have          the Wagner–Whitin algorithm but with some
been expressed by Silver: ‘‘One should not                    additional feasibility constraints. Martel et al.
necessarily use a deterministic lot-sizing rule when          (1995) transform the multiple item procurement
significant uncertainty exists. A more appropriate             problem into a multi-period static decision pro-
strategy might be some form of probabilistic                  blem under risk. Other notable works on non-
modelling.’’                                                  stationary stochastic demand adopt (s; S) or
   Silver (1978) suggested a heuristic procedure for          base-stock policies and are due to Iida (1999),
the stochastic lot-sizing problem assuming that the           Sobel and Zhang (2001), and Gavirneni and Tayur
forecast errors are normally distributed. A similar           (2001). In Iida (1999), the periodic review dynamic
heuristic, having a different objective function, was         inventory problem is considered and it is shown
presented by Askin (1981). Bookbinder and Tan                 that near myopic policies are sufficiently close to
(1988) proposed another heuristic, under the                  optimal decisions for the infinite horizon inventory
‘‘static–dynamic uncertainty’’ strategy. In this              problem. In Sobel and Zhang (2001), it is assumed
strategy, the replenishment periods are fixed at               that demand arrives simultaneously from a deter-
the beginning of the planning horizon and the                 ministic source and a random source, and proven
actual orders at future replenishment periods are             that a modified (s; S) policy is optimal under
determined only at those replenishment periods                general conditions. Gavirneni and Tayur (2001)
depending upon the realised demand. The total                 use the derivative of the cost function that can
expected cost is minimised under the minimal                  handle a much wider variety of fluctuations in the
service-level constraint. In this paper, we propose a         problem parameters. The above studies have
mixed integer programming formulation to solve                adopted either static control policies in a rolling
the stochastic dynamic lot-sizing problem to                  horizon framework or dynamic control policies
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like (s; S), although the static-dynamic uncertainty           vestigation of the minimal service-level criterion is
model is a more accurate representation of                     given by Chen and Krass (2001). For a discussion
industrial practice as pointed out by Sox (1997).              of different service constraints see Van Houtum
   Most companies use MRP in some form for                     and Zijm (2000) and Rosling (1999). Remember
their production planning and thus ordering on                 that demand is not deterministic with different
suppliers. They typically issue advance schedules              values in each period. Thus, the service-level
of requirements. In talking to companies in the                criterion in this case is defined as specifying a
supply chain who are in this situation a common                minimum probability, say a; that at the end of
complaint is that their customers continually                  every period the net inventory will not be negative.
change their schedules, the timing of orders as                In general, finding the solution to this problem can
well as the size of orders. It is the changing of the          be formulated as solving a chance-constrained
timing that they find worse. This issue of system               programming model (see for example Birge and
nervousness is an active current research area. If             Louveaux, 1997). It can be expressed as the
there is to be more co-operation and co-ordination             minimisation of the total expected cost, EfTCg;
in supply chains, then a model that attempts to                over the N-period planning horizon subject to the
determine a schedule for the timing of orders in               service-level constraints, as given below:
advance taking account of the stochastic demand,
                                                               minimise EfTCg
which remains fixed, is a contribution of practical                  Z Z     Z X
interest. This need to fix the deliveries in advance,
                                                                 ¼        ?     ðadt þ hIt þvXt g
whilst allowing reasonable flexibility in the order                       d1    d2    dN t¼1
size has been at the heart of the problem of buying
                                                                     Â g1 ðd1 Þg2 ðd2 ÞygN ðdN Þdðd1 Þdðd2 ÞydðdN Þ ð1Þ
raw materials on fluctuating price markets. You
have to determine which future months or half                  subject to
months in which to have your delivery of wheat or                   (
cocoa, etc. Once this is fixed, suppliers do not                       1 if Xt > 0;
                                                               dt ¼                           t ¼ 1; y; N;         ð2Þ
allow it to be changed, although the quantities                       0 otherwise;
ordered can be varied to some extent. Waiting
until the month or half month of delivery and                            X

using the standard (s; S) model to decide whether              It ¼ I0     ðXi À di Þ;    t ¼ 1; y; N;             ð3Þ
or not to place an order and then its size is not a
good option, as the price you pay tends to be much             PrfIt X0gXa;         t ¼ 1; y; N;                   ð4Þ
higher as sellers know you are desperate for the
material. Thus, investigating models and solution              Xt ; It X0;      t ¼ 1; y; N;
procedures for the static-dynamic uncertainty
strategy is potentially important from the practical           where N is the number of periods in the planning
application perspective.                                       horizon, Xt the replenishment order placed and
                                                               received (i.e., no lead-time) in period t; dt a {0,1}
                                                               variable that takes the value of 1 if a replenishment
2. Problem statement                                           order is placed in period t and 0 otherwise, It the
                                                               inventory level at the end of period t; I0 the stock
   The stochastic dynamic lot-sizing problem can               on hand at the beginning of period 1, dt the
be formulated either by assuming a penalty cost                demand in period t; a the fixed procurement cost, n
for each stockout and unsatisfied demand or by                  the marginal cost of purchasing an item, h the
minimising the ordering and inventory costs                    linear holding cost incurred on any unit carried in
subject to satisfying some customer service-level              inventory over from one period to the next.
criterion. In this paper we consider meeting a                    The demand dt in period t is considered as a
specified customer service case, where we use a                 random variable with known probability density
‘‘minimal service-level’’ criterion. A detailed in-            function, gt ðdt Þ: The distribution of demand may
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108                    S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119

vary from period to period. Demands in different               nistic lot-sizing model given below:
time periods are assumed independent. A fixed                                        XN
procurement (ordering or set-up) cost, a; is                   minimise EfTCg ¼         ðadt þ hEfIt g þ vXt Þ              ð5Þ
incurred each time a replenishment order is placed,                                      t¼1
whatever the size of the order. In addition to the             subject to
fixed ordering cost, a variable purchasing cost is                   (
incurred depending on the size of the order, vXt : A                  1 if Xt > 0;
                                                               dt ¼                            t ¼ 1; y; N;                 ð6Þ
replenishment order is assumed to arrive instanta-                    0 otherwise;
neously at the beginning of each period, before the
demand in that period occurs. A linear holding                                  X
                                                                                t            X
                                                               EfIt g ¼ I0 þ          Xi À         Efdi g;   t ¼ 1; y; N;
cost h is incurred on any unit carried in inventory                             i¼1          i¼1
over from one period to the next. For simplicity in
the analysis, inventory holding costs are assumed
to be incurred only on the inventory at the end of             X
each period. The extension to taking the average                     Xi XGDðtÞ ðaÞ À I0 ;      t ¼ 1; y; N;                 ð8Þ
inventory over a period is straightforward. The                i¼1

probability that at the end of each and every time             Xt ; EfIt gX0;     t ¼ 1; y; N:
period the net inventory will not be negative is set
to be at least a: Hence, it is implicitly assumed that,        The above formulation is solved by the Wagner–
since normally the desired service level is quite              Whitin algorithm. Bookbinder and Tan give the
                                                                                                 Pt           À1
high, the value a incorporates the perception of the           equivalent linear constraint,        i¼1 Xi XGDðtÞ ðaÞ À
cost of backorders, so that shortage cost can be               I0 ; for the service-level constraint PrfIt Z0gXa;
ignored in the model. The above assumptions are                where GDðtÞ ¼ Gd1 þd2 þ?þdt ð:Þ is the cumulative
valid for the rest of the paper.                               probability distribution function of d1 þ d2 þ ? þ
                                                               dt : It is assumed that G is strictly increasing,
                                                               therefore G À1 is uniquely defined. They also
                                                               examine the analogy between the deterministic
3. The Bookbinder–Tan strategy for the                         model and the ‘‘static uncertainty’’ model, and
probabilistic lot-sizing problem                               conclude that there is correspondence between the
                                                               respective terms of the two models.
   One of the few attempts to produce more                                  À1         À1                À1
                                                                   Since Gd1 ðaÞpGd1 þd2 ðaÞp?pGDðNÞ ðaÞ; the
implementable near optimal solutions to the                                Pt            À1
                                                               constraint i¼1 Xi XGDðtÞ ðaÞ À I0 is binding for t ¼
stochastic dynamic lot-size problem is given by                                PN            À1
Bookbinder and Tan (1988). In Bookbinder and                   N and gives        t¼1 Xt ¼ GDðNÞ ðaÞ À I0 ; which im-
Tan’s paper three strategies were analysed, named              plies that the total direct item cost,       t¼1 vXt ; is
‘‘static uncertainty’’, ‘‘dynamic uncertainty’’ and            constant. Therefore, Bookbinder and Tan ignore
‘‘static–dynamic uncertainty’’. In all cases, their            the total direct item cost term, which has no effect
analysis was based on a service-level constraint on            on determination of the best schedule, as in the
the probability of a stockout at the end of each               deterministic problem.
period.                                                            Another approach for determining the timing and
   The ‘‘static uncertainty’’ decision rule assumes            size of the replenishment orders is the ‘‘dynamic
that the values of all the decision variables, the             uncertainty’’ strategy. In the ‘‘dynamic uncertainty’’
timing and size of replenishment orders, are                   strategy decisions are made every period on the basis
determined at the beginning of the planning                    of the demands that have become known as you
horizon. All the Xt can be considered as determi-              move forward through time. This approach ignores
nistic decision variables. The values and timing of            the ordering cost a; and may require a replenishment
the replenishment orders is determined by initially            order almost every period. Obviously, for large
reducing the problem to the equivalent determi-                ratios of a=h this result is clearly undesirable.
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   Bookbinder and Tan then combined the features                   to period Ti in the desired buffer stock level and is
of both strategies to give a two-stage heuristic,                  given as follows:
which they called the ‘‘static–dynamic uncer-
tainty’’ approach. The first stage was to fix the                    XT1 ¼ YT1 ;
timing of the replenishment orders using the                       XT2 ¼ YT2 þ dT1 þ ? þ dT2 À1 ;
‘‘static uncertainty’’ model. Exploiting the analogy                                                                                   ð10Þ
between the deterministic model and the ‘‘static
                                                                   XTm ¼ YTm þ dTmÀ1 þ ? þ dTm À1 :
uncertainty’’ model, the replenishment periods can
be determined by using dynamic programming
                                                                   Hence, the replenishment size XTi can be deter-
(Wagner and Whitin, 1958), network flows (Zang-
                                                                   mined by means of the deterministic decision
                                                                                                         PTi À1
will, 1968) or integer programming (Karni, 1981).
                                                                   variable YTi after the demand           k¼TiÀ1 dk is
The second stage was then to calculate the
                                                                   realised. It follows from Eq. (10) that
adjustments made to those orders at the times
they were scheduled as the demand was realised                     X
                                                                   i               X
                                                                                   i              X
                                                                                                  Ti À1
over the planning horizon. These adjustments were                         X Tj ¼         YTj þ            dk ;     i ¼ 1; y; m;        ð11Þ
expressed as margins to add to the total demand                    j¼1             j¼1            k¼1
received over all the periods since the immediately
preceding order was received.                                      and that the inventory level It at the end of period
   The Bookbinder and Tan analysis for the second                  t; given by Eq. (9), can be expressed as
stage is briefly as follows. Consider a review
                                                                                   i              X
schedule, which has m reviews over the N period                    It ¼ I0 þ             Y Tj À          dk ;
planning horizon with orders arriving at                                           j¼1            k¼Ti
fT1 ; T2 ; y; Tm g; whereTj > TjÀ1 ; Tm pN: Remem-
                                                                      Ti ptoTiþ1 ;             i ¼ 1; y; m:                            ð12Þ
ber that the Ti have been determined at the first
stage. For convenience T1 ¼ 1 is defined as the
                                                                   Therefore, the random components of the closing
start of the planning horizon and Tmþ1 ¼ N þ 1
                                                                   inventory level, It ; depends only on the demand
the period immediately after the end of the
                                                                   since the most recent stock review in period Ti ;
horizon. The review schedule may be generalised
                                                                   i ¼ 1; y; m: Note that YTi is not a random
to consider the case where T1 > 1; if the opening
                                                                   variable; rather, it is a deterministic decision
stock, I0 ; is sufficient to cover the immediate needs
                                                                   variable whose value is determined using a linear
at the start of the planning horizon. The associated
                                                                   programming model.
stock reviews will take place at the beginning of
                                                                      As in the static model, Bookbinder and Tan say
periods Ti ; i ¼ 1; y; m: In the considered dynamic
                                                                   that the probabilistic constraints,
review and replenishment policy, clearly the orders
Xt are all equal to zero except at the replenishment                X
periods T1 ; T2 ; y; Tm : The inventory level It                          Yti XGdt þdt þ1 þdt þ2 þ?þdt                    ðaÞ À I0 ;
                                                                                     j     j      j              jþ1 À1
carried from period t to period t þ 1 is the opening                i¼1
stock plus any orders that have arrived up to and                     j ¼ 1; 2; y; m;
including period t less the total demand to date.
Hence it is given as                                               hold as an equality at optimality. By substituting
            i             X
                          t                                        Gdtm þdtm þ1 þdtm þ2 þ?þdt À1 ðaÞ À I0 for  i¼1 Yti ; we
It ¼ I0 þ         XTj À         dk ;                               observe that their total unit variable cost expres-
            j¼1           k¼1
                                                                           nP         Ptm À1       o
                                                                   sion, v   i¼1 Yti þ i¼1 Efdi g ; takes a constant
  Ti ptoTiþ1 ;       i ¼ 1; y; m:                        ð9Þ
                                                                   value. In other words, we can conclude that the
Bookbinder and Tan transform the decision                          unit variable cost has no role in replenishment
variable XTi to a new variable YTi AR; which                       policies determined according to the Bookbinder–
may be interpreted as the change from period TiÀ1                  Tan heuristic.
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4. An optimisation model for the probabilistic                reviews are still variables to be determined and not
lot-sizing problem                                            preset values.            P
                                                                Defining RTi ¼ I0 þ ij¼1 YTj and substituting
   The Bookbinder and Tan static-dynamic proce-               for It ; Eq. (12) becomes
dure splits the solution into two independent                               X
stages, ignoring the interactions between them. It            It ¼ RTi À           dk ;
is therefore a heuristic solution procedure. The                            k¼Ti
question then arises as to its accuracy and the                  Ti ptoTiþ1 ;         i ¼ 1; y; m:             ð13Þ
likely level of cost penalty that it incurs.
   To evaluate the accuracy of the heuristic, the             Note that RTi may be interpreted as an order-up-
problem must be formulated in a way that gives                to-level to which stock should be raised after
the optimal solution under the ‘‘static–dynamic               receiving an order at the ith review period Ti ; and
uncertainty’’ strategy. This will allow the simulta-          RTi À t i dk is the end of period inventory.
neous determination of the number and timing of               Thus, instead of working in terms of decision
the replenishments and the information necessary              variables YTi ; as in the model proposed by
to determine the size of the replenishment orders             Bookbinder and Tan, the problem can be ex-
that will minimise the expected costs of meeting              pressed in terms of these new decision variables
demand over some finite planning horizon, given a              RTi : The problem is to determine the number of
set of forecasts of the demands and a service-level           reviews, m; the Ti ; and the associated RTi for i ¼
constraint on the probability of a stockout. This is          1; y; m:
done in this section. It basically models the                    If there is no replenishment scheduled for period
Bookbinder and Tan transformation process in a                t; then Rt equals the opening inventory level in
way that allows the simultaneous determination of             period t: Now, Eq. (13) can be expressed more
the timing and size of the orders as a single step.           simply as
The initial decisions depend upon what informa-               I t ¼ R t À dt ;     t ¼ 1; y; N:                ð14Þ
tion can be expected to become available in the
future and how best to react to this information.             It follows that the variable Rt must be equal to ItÀ1
Therefore, the replenishment times to use are                 if no order is received in period t and equal to the
determined at the beginning of the planning                   order-up-to-level if there is a review and the receipt
horizon considering the interdependency between               of an order. The first case applies if no stock
the stock levels to have available at the start of            review takes place in period t; which is indicated by
those periods and their timing. The actual order              the integer variable dt ¼ 0: If dt ¼ 0 then Rt must
quantities at future replenishment periods are                equal ItÀ1 : This is achieved by the two linear
determined only at those replenishment periods                inequalities given by Eq. (15), since the constraints
and will depend upon the demand that is realised              become Rt pItÀ1 and Rt XItÀ1 ; respectively, for
period by period over the planning horizon. It is             dt ¼ 0:
assumed that negative orders are not allowed, so              Rt À ItÀ1 pMdt ;
that if the actual stock exceeds the order-up-to-                                         t ¼ 1; y; N;         ð15Þ
                                                              Rt XItÀ1 ;
level for that review, this excess stock will be
carried forward and not returned to the supply                where M is some very large positive number.
source.                                                          Whilst if dt ¼ 1 then the constraints require Rt
                                                              to lie between infinity and ItÀ1 satisfying the other
                                                              condition on Rt : The values for the order-up-to-
4.1. Reformulating the constraints of the                     level variables, Rt ; when dt ¼ 1 are then those that
chance-constrained model                                      give the minimum expected costs EfTCg: The
                                                              desired opening stock levels, as required for the
   We start with Eq. (12) of Section 3, but with the          solution to the problem, will then be those values
difference that the number and timing of the                  of Rt for which dt ¼ 1:
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   It is, therefore, clear from the above explana-                                            À1
                                                                        zero elsewhere, then GdT þdT þ1 þ?þdt ðaÞ can be
                                                                                                           i    i
tions that constraints (6) and (7) can be replaced                      expressed as
with Eqs. (15) and (14), respectively.                                                            X
   As mentioned before, a is the desired minimum                         À1                               À1
                                                                        GdT þdT þ1 þ?þdt ðaÞ ¼           GdtÀjþ1 þdtÀjþ2 þ?þdt ðaÞPtj ;
                                                                              i     i
probability that the net inventory level in any time                                              j¼1
period will actually be non-negative and deter-                            t ¼ 1; y; N;
mined subjectively. With this regard the chance
constraint                                                              and similarly Eq. (19) can be expressed as!
                                                                             Xt                              X
PrfIt X0gXa;           t ¼ 1; y; N                           ð16Þ       It X       À1
                                                                                 GdtÀjþ1 þdtÀjþ2 þ?þdt ðaÞ À   dk Ptj ;
                                                                                  j¼1                               k¼tÀjþ1
can, using Eq. (13), be written alternatively as
  (              )                                                         t ¼ 1; y; N:                                             ð20Þ
          X t
Pr RTi X      dk Xa; t ¼ 1; y; N;                ð17Þ                   The result Ptt ¼ 1 means that the stock review was
                                                                        in period 1, the start of the planning horizon,
which implies                                                           whilst Pt1 ¼ 1 means that the stock review was at
                                                                        the start of period t itself. There can at most be only
GdTi þdTi þ1 þ?þdt ðRTi ÞXa;
                                                                        one most recent order received prior to period t:
  Ti ptoTiþ1 ;          i ¼ 1; y; m;                         ð18Þ       Thus, the Ptj must satisfy
and                                                                     X t
                                                                            Ptj ¼ 1; t ¼ 1; y; N:                          ð21Þ
                                    t                                   j¼1
It XGdT þdT þ1 þ:::þdt ðaÞ À               dk ;
        i   i
                                    k¼Ti                                Three other conditions as given below are neces-
                                                                        sary to identify uniquely the period in which the
  Ti ptoTiþ1 ;          i ¼ 1; y; m:                         ð19Þ
                                                                        most recent review prior to any period t took
The right-hand side of Eq. (19) can be calculated                       place.
or possibly read from a table, once the form of gt ð:Þ                                        Pt
                                                                                tÀjþ1 ¼ 1 and    k¼tÀjþ2 dk ¼ 0; so all sub-
                                                                        * If d
is selected.
                    À1                                                     sequent dk for k ¼ t À j þ 2; t À j þ 3; y; t are
   In Eq. (19), GdT þdT þ1 þ?þdt ð:Þ can only be
                     i   i                                                 0, then we must have Ptj ¼ 1; as in these
determined after the replenishment periods Ti
                                                                           circumstances period t À j þ 1 had the most
have been determined. But, as these are chosen
                                                                           recent stock review prior to period t:
to minimise the expected costs, the stock replen-                       * If d
                                                                               tÀjþ1 ¼ 0 and    k¼tÀjþ2 dk ¼ 0; then Ptj ¼ 0
ishment periods cannot be determined until the
               À1                                                          since the most recent review prior to period t
appropriate GdT þdT þ1 þ?þdt ð:Þ values to use in the
                   i        i                                              must have been earlier than t À j þ 1:
model are known. There is an obvious circularity                               tÀjþ1 ¼ 1 and   k¼tÀjþ2 dk X1; then Ptj ¼ 0;
                                                                        * If d
here in trying to solve the problem. Since Book-                           since other reviews prior to period t occur after
binder and Tan separate the determination of                               period t þ j À 1:
the timing of the replenishment orders and the
adjustments to those orders, they avoid the                                All of these three conditions can be satisfied by
circularity by sacrificing optimality. One way to                        the equality condition given in Eq. (21) and the
overcome this problem, by not sacrificing optim-                         single constraint given below, which are designed
ality, is to formulate it as a mixed integer linear                     to identify uniquely the periods in which the most
programming model.                                                      recent order prior to t took place:
   Since the problem has a finite planning horizon
of N periods, GdT þdT þ1 þ?þdt ð:Þ can be calculated                                      X
                        i       i
for all relevant cases. If the 0/1 integer variable Ptj                 Ptj XdtÀjþ1 À             dk ;
is defined as taking a value of 1 if the most recent
order prior to period t was in period t À j þ 1 and                        t ¼ 1; y; N;      j ¼ 1; y; t:                           ð22Þ
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4.2. The mixed integer programming model to                         X
                                                                          Ptj ¼ 1;   t ¼ 1; y; N;                           ð28Þ
determine the optimal policy under the static
dynamic uncertainty strategy
   The chance-constrained programming model                         Ptj XdtÀjþ1 À               dk ;
can be expressed as minimising the objective                                         k¼tÀjþ2
function given earlier as Eq. (1) subject to the                        t ¼ 1; y; N;      j ¼ 1; y; t;                      ð29Þ
constraints given by Eqs. (14), (15), (20)–(22) and
the non-negativity conditions and the 0/1 integer                   EfIt g; EfRt gX0; dt ; Ptj Af0; 1g;
values for dt and Ptj : Since the decision rule for the
                                                                      t ¼ 1; y; N; j ¼ 1; y; t:
above stochastic optimisation problem is chosen to
be the ‘‘static–dynamic uncertainty’’ strategy, the                 This model thus determines the optimum number
timing of the replenishments, obtained from dt and                  of replenishments as well as the optimum replen-
Ptj ; must be decided once and for all, before any of               ishment schedule, the timing of the replenish-
the demands, dt ; become known. Therefore, the                      ments, together with the optimum values to use for
expectation operator must be applied to the                         dynamically determining the sizes of the replen-
stochastic variables It ; Rt and dt in the constraint               ishment orders as demand is realised, that give the
equations and objective function. In the chance-                    minimum expected total costs. The problem is to
constrained programming model given below, the                      determine the values of the 0/1 integer variables,
expected value of It and Rt are denoted by EfIt g                   dt for t ¼ 1; y; N and Ptj for j ¼ 1; y; t; t ¼
and EfRt g; respectively.                                           1; y; N; and the non-negative continuous vari-
   Such an analysis is completed at the beginning                   ables EfIt g and EfRt g for t ¼ 1; y; N; that
of the horizon by taking expectations. Hence the                    minimise the objective function. To comply with
deterministic equivalent model for the chance-                      the non-negativity constraint on the lot-sizes of the
constrained programming model under the static-                     original model, we must have EfRt g À EfItÀ1 gX0;
dynamic uncertainty strategy may be obtained by                     t ¼ 1; y; N which is already given in Eq. (15) and
taking expectations (see Bookbinder and Tan).                       incorporated into the model. The times of the
The model then is                                                   stock reviews are given by the values of i such that
                      X N                                           di ¼ 1: The associated order-up-to-levels, for each
minimise EfTCg ¼          ðadt þ hEfIt g                            i; are given as EfRi g:
                                                                       In Section 3, it was shown that the Bookbinder–
                          þ vEfRt g À vEfItÀ1 gÞ;       ð23Þ        Tan heuristic ignores the unit variable cost in
where the total expected direct item cost                           determination of the best schedule as in the
                         PN                         t¼1
vEfXt g is written as                                               deterministic problem. However, in contrast to
                            t¼1 ðvEfRt g À vEfItÀ1 gÞ
following the definition of order-up-to-levels, Rt ;                 the Bookbinder–Tan heuristic, the above formula-
   subject to                                                       tion does not ignore the unit variable cost. A
                                                                    rearrangement of the unit variable cost terms in
EfIt g ¼ EfRt g À Efdt g;           t ¼ 1; y; N;         ð24Þ       the objective function gives
EfRt gXEfItÀ1 g;         t ¼ 1; y; N;                    ð25Þ

EfRt g À EfItÀ1 gpMdt ;             t ¼ 1; y; N;         ð26Þ       v     ðEfRt g À EfItÀ1 gÞ
EfIt gX         GdtÀjþ1 þdtÀjþ2 þ?þdt ðaÞ                                 ¼ ÀvI0 þ v          ðEfRt g À EfIt gÞ þ vEfRN g
          j¼1                                                                          t¼1
                                                                                       N À1
          À             Efdk g Ptj ;     t ¼ 1; y; N; ð27Þ                ¼ ÀvI0 þ v          ðEfdt gÞ þ vEfRN g
              k¼tÀjþ1                                                                  t¼1
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using Eq. (24) for t ¼ 1; y; N À 1: Thus, one can                             Table 2, the calculated values of the coefficient
write the total unit variable cost component as f þ                              À1
                                                                              GdtÀjþ1 þdtÀjþ2 þ?þdt ðaÞ are given. For instance, ele-
vEfRN g; where f is a constant with a value equal                             ment (t ¼ 7; j ¼ 3) gives the value of
to v multiplied by the given opening stock plus the                              À1
                                                                              Gd5 þd6 þd7 ð0:95Þ ¼ 2833 which corresponds to the
expected demand over the first N À 1 periods. It is
                                                                              opening inventory level in period t À j þ 1 ¼ 5 to
clear that the total unit variable cost is a function
                                                                              satisfy the demands of periods 5–7 with prob-
of the model variable EfRN g; which is a variable
                                                                              ability of at least a ¼ 95%: These values can be
rather than a constant. Thus the solution will be
                                                                              calculated using any spreadsheet program. The
affected by the variable production cost.
                                                                              solution to the problem is calculated using both
                                                                              BT heuristic and the optimisation model derived in
                                                                              the previous section, denoted by (TK).
5. A numerical example                                                           The first step in the two-step BT heuristic is the
                                                                              determination of the replenishment times. The
   To illustrate the technique we shall use the                               replenishment times are fixed by means of the
demand forecasts presented in Table 1. The initial                            ‘‘static uncertainty’’ model, applying the Wagner–
inventory level is taken as zero. It is assumed that                          Whitin algorithm to the problem formulated as a
the demand in each period is normally distributed                             deterministic one using the analogy of Bookbinder
about the forecast value with a constant coefficient                           and Tan. This gives periods 1, 5 and 7 as the
of variation, C ¼ st =mt ¼ 0:333: The other para-                             replenishment periods. Following that the ‘‘static–
meters of the problem are a ¼ $2500 per order,                                dynamic uncertainty’’ model is used to determine
h ¼ $1 per unit per period, and a ¼ 0:95                                      the order-up-to-levels for these replenishment
(za¼0:95 ¼ 1:645). Since BT heuristic ignores the                             periods. The results are given in Table 3 and Fig. 1.
unit variable cost, the example takes into account                               The same problem is also solved by means
only the holding and ordering costs (i.e., v ¼ 0). In                         of the deterministic equivalent mixed integer

Table 1
Forecasts of period demands

Period (k)          1            2            3             4             5            6             7         8         9       10

Efdk g              800          850          700           200           800          700           650       600       500     200

Table 2                               P                           P               1=2
Calculated GdtÀjþ1 þdtÀjþ2 þ?þdt ðaÞ ¼ t
                                        k¼tÀjþ1 E fdk g þ z0:95 C
                                                                    t        2
                                                                    k¼tÀjþ1 E fdk g     values

t            j

             1            2            3             4             5               6             7         8         9          10

 1           1239
 2           1316         2290
 3           1084         2154         3096
 4            310         1299         2364          3304
 5           1239         1452         2293          3304          4223
 6           1084         2083         2293          3106          4096            5003
 7           1006         1874         2833          3042          3841            4818          5718
 8            929         1735         2568          3508          3716            4507          5475      6370
 9            774         1528         2307          3127          4056            4264          5050      6013      6904
10            310          995         1742          2518          3335            4264          4471      5256      6219       7110
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Table 3
Replenishment policy of BT

Period (t)                                         1              2            3        4         5        6       7        8        9        10

Order-up-to-level (EfRt gdt ¼1 )                       3304       —            —        —         2083     —       2518     —        —        —
Exp. opening inv. (EfRt g)                             3304       2504         1654     954       2083     1283    2518     1868     1268     768
Exp. closing inv. (EfIt g)                             2504       1654          954     754       1283      583    1868     1268      768     568

Total expected cost                                19,704

                  3500                                                                      arrive in periods 1, 3, 5 and 8. As stated the
                  3000                                                                      opening stock at the time of a replenishment is
Inventory Level

                                                                                            known and set by the policy. However, the sizes of
                                                                                            the replenishment orders are determined dynami-
                                                                                            cally as demand is realised. A given period’s lot-
                   500                                                                      size, Rt À ItÀ1 ; cannot be found until the realised
                     0                                                                      demand is known. In other words one waits for the
                         1   2    3    4    5     6      7    8       9   10       11       demands to become known, having decided in
                                                                                            advance how this knowledge will be used. Simi-
                             Fig. 1. Replenishment policy of BT.                            larly the actual opening stocks realised in the
                                                                                            intermediate periods would depend on the realised
                                                                                            demands. On the assumption of independence of
programming model presented in Section 4.2. The                                             successive period demands, Figs. 1 and 2 show
results are presented in Table 4 and Fig. 2. It is                                          what the average opening stock would be if the
seen that this optimal solution has four rather than                                        actual 10 period situation were repeated many
three replenishments which occur in periods 1, 3, 5                                         times.
and 8.                                                                                         In the above example, the unit variable cost has
   Figs. 1 and 2 show the differences between the                                           been set to zero. At this point, the effect of having
near optimal BT and the optimal TK policies                                                 unit variable cost in the problem on the inventory
under the ‘‘static–dynamic uncertainty’’ strategy.                                          policy should be investigated. To do so, the above
The probability of shortage occurring in each                                               problem is solved again for v ¼ 4 using both BT
time period is given for both policies in Table 5.                                          and TK. Since BT heuristic ignores the unit
Since the probability of a shortage occurring                                               variable cost in both the ‘‘static uncertainty’’ and
never exceeds 5% in any period, it is clear from                                            ‘‘static–dynamic uncertainty’’ models, the replen-
Table 5 that both policies satisfy the service-level                                        ishment periods, which were {1,5,7}, and the
constraints that at the end of each period the                                              calculated expected lot-sizes remain the same.
probability that the net inventory will not be                                              Hence, Fig. 1 still depicts the replenishment policy
negative is at least a ¼ 95%: The heuristic BT                                              for BT. On the other hand, TK approach replaces
method costs 1.55% more than the expected cost                                              the replenishment periods f1; 3; 5; 8gv¼0 with
of the optimal policy.                                                                      f1; 3; 5; 7; 9gv¼4 : Fig. 3 shows the revised replen-
   The opening stocks in Fig. 1 (the BT heuristic                                           ishment policy. A comparison of the statistics for
policy) for periods 1, 5 and 7 when replenishments                                          both approaches is given in Table 6.
arrive are values given by the policy. The opening                                             From Table 6, the difference between the
stocks for intermediate periods are expected                                                expected total inventory costs of BT and TK
values, assuming that the period demands occur                                              solutions for v ¼ 0 is $300 ($19,704–$19,404). The
at their expected value. It is the same for Fig. 2                                          introduction of unit variable cost, v ¼ 4; increases
(the optimal TK solution) where replenishments                                              this difference to $939 ($45,975–$45,036).
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Table 4
Replenishment policy of TK

Period (t)                                                          1              2            3        4          5       6        7       8       9        10

Order-up-to-level (EfRt gdt ¼1 )                                        2290       —            1299     —          2833    —        —       1742    —        —
Exp. opening inv. (EfRt g)                                              2290       1490         1299     599        2833    2033     1333    1742    1142     642
Exp. closing inv. (EfIt g)                                              1490        640          599     399        2033    1333      683    1142     642     442

Total expected cost                                                 19,404

                  3500                                                                                       optimal solution for v ¼ 0 (v ¼ 4). It is useful to
                  3000                                                                                       gain some idea of how well it would perform over
Inventory Level

                                                                                                             a wider set of examples. This section presents the
                                                                                                             results of a cost comparison of the BT heuristic
                  1000                                                                                       with the optimal solution, the TK model, for a
                   500                                                                                       wide range of problems. The planning horizon is
                     0                                                                                       set to 20 periods with no initial inventory in all
                         1        2       3       4        5     6        7    8       9   10       11       the experiments. The service level is set at a
                                                                                                             constant a ¼ 0:95 for all periods and the ordering
                              Fig. 2. Replenishment policy of TK.                                            cost a ¼ $1000 per order. The unit variable cost is
                                                                                                             ignored. The problems selected are defined in
                                                                                                             terms of:
Table 5
Probability of shortage (%)
                                                                                                             *   The holding cost, which will affect the average
                                                                                                                 number of periods covered by an order, the
Period                        1       2       3        4        5        6     7       8   9        10           values being $1, $2, $3, $4, $5, $7.5 or $15 per
BT policy 0.0 0.0 1.8 5.0 0.0 5.0 0.0 0.0 1.7 5.0                                                                unit per period.
TK policy 0.0 5.0 0.5 5.0 0.0 0.0 5.0 0.0 0.7 5.0                                                            *   The coefficient of variation, showing the effect
                                                                                                                 of the size of random variation in demand
                                                                                                                 about the mean. The values selected were 1/3,
                                                                                                                 1/4, 1/5 and 1/10 and were the same for each
                                                                                                                 period’s demand.
                                                                                                             *   The pattern of the mean demands over time.
Inventory Level

                  2000                                                                                           There are four different demand patterns taken
                  1500                                                                                           from Berry (1972), as given in Figs. 4a–d. These
                  1000                                                                                           range from a constant level, through a contin-
                  500                                                                                            uous sinusoidal changes to a very erratic
                         1    2       3       4        5         6        7    8       9   10       11           pattern. Demand is normally distributed about
                                                               Period                                            the forecast value under the non-stationarity
                             Fig. 3. Replenishment policy of TKv¼4 :
                                                                                                                The number of test problems, generated for
                                                                                                             seven different holding costs, four different coeffi-
6. Numerical comparisons                                                                                     cient of variations and four different mean data
                                                                                                             sets, therefore totals to 112. For each test problem
  In the illustrative example in the previous                                                                the percentage cost difference between the BT
section, it is seen that the Bookbinder and Tan                                                              heuristic and the optimal solution (TK method) is
heuristic costs 1.55% (2.09%) more than the                                                                  calculated and the results are listed in Table 7.
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Table 6
A comparison of v ¼ 0 and 4 cases

                                             BTv=0                        BTv=4                TKv=0             TKv=4

Replenishment periods                        {1,5,7}                      {1,5,7}              {1,3,5,8}         {1,3,5,7,9}
E(average inventory level)                   1520.4                       1520.4               1240.3            1035.4
E(average buffer stock level)                64.5                         64.5                 54.5              48.0
Total E(order quantity)                      6568                         6568                 6442              6295
E(total cost)                                19,704                       45,975               19,404            45,036

                                 Mean Demand Set 1                                  Mean Demand Set 2

                   120                                              200
                    60                                              100
                     0                                                0

                                Mean Demand Set 3                                   Mean Demand Set 4
                   400                                              500

                   300                                              400
                   100                                              100
                     0                                                0


                                                       Fig. 4. Demand patterns.

   The results indicate that as the demand becomes                    gives the optimal solution in 36% of cases. The
more erratic, i.e. more non-stationary, either in                     average cost penalty is 1.24%, similarly to the first
terms of the mean demand pattern or the size of                       two data sets. However, the worst cost penalty is
the coefficient of variation, the cost penalty above                   significantly higher at 9.5%. In the highly erratic
the optimal solution from using the BT heuristic                      demand data set 4, the BT heuristic is only optimal
tends to become larger. For the stationary case,                      in 11% of cases. The average penalty cost incurred
data set 1, there are 78% of cases where the BT                       is 2.4%, so much higher than the other three cases.
heuristic gives the optimal solution. The average                     The worst penalty is 9.3%, much the same as for
penalty cost above the optimal solution incurred is                   data set 3.
0.5%, whilst the worst penalty cost is 3.3%. For                         The percentage cost difference between BT and
data set 2, the sinusoidal pattern to demand over                     the optimal solution tends to become smaller as
time, the BT heuristic gives the optimal solution                     the coefficient of variation decreases. This is easy
for 14% of cases. The average cost penalty is again                   to explain. The suboptimality of the BT heuristic
only 1.3%, whilst the worst penalty is 3.8%, very                     arises from the first step of the method. Since the
similar to case 1. For the more erratic almost                        replenishment periods are fixed under the ‘‘static
sinusoidal pattern of data set 3 the BT heuristic                     uncertainty’’ strategy and the dynamic nature of
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Table 7
Percentage cost increases of BT above TK solutions (for v ¼ 0)

              s=m            h¼1            h¼2            h¼3           h¼4            h¼5            h ¼ 7:5     h ¼ 15

Set 1         1/3            0.0            0.0            2.6           0.0            0.0            3.3         0.0
              1/4            0.0            2.5            2.6           0.0            0.0            0.2         0.0
              1/5            0.0            0.0            1.8           0.0            0.0            0.0         0.0
              1/10           0.0            0.0            0.0           0.0            0.0            0.0         0.0

Set 2         1/3            1.9            3.5            3.8           2.2            3.3            1.6         0.1
              1/4            1.0            2.5            2.6           1.7            2.3            0.8         0.0
              1/5            0.5            2.0            2.0           1.3            1.4            0.3         0.0
              1/10           0.0            0.4            0.8           0.4            0.0            0.1         0.0

Set 3         1/3            0.8            3.3            9.5           0.1            1.1            0.7         0.0
              1/4            0.1            2.0            6.8           0.0            0.5            0.5         0.0
              1/5            0.0            1.1            5.1           0.0            0.1            0.4         0.0
              1/10           0.0            0.0            2.3           0.0            0.1            0.1         0.0

Set 4         1/3            5.6            1.0            1.9           5.9            9.3            2.1         0.2
              1/4            4.3            0.3            0.7           3.8            7.5            1.6         1.0
              1/5            3.5            0.0            1.4           2.4            6.1            1.4         0.9
              1/10           1.5            0.0            0.6           0.0            2.6            0.4         0.4

the problem is ignored, in general the replenish-                 cost performance of the heuristic changes drama-
ment schedule is not optimal. Therefore, a                        tically.
decrease in the coefficient of variation improves                     If the unit variable cost is ignored then the
the performance of the ‘‘static uncertainty’’                     stochastic lot-sizing problem can be modelled as a
approach and yields a better replenishment sche-                  stochastic form of the Wagner–Whitin problem
dule. It should be noted that if the replenishment                and solved by a shortest route algorithm, where
schedule is optimal, then the second step of the                  the arc cost (i; j) corresponds to the minimum cost
method produces the optimal solution for the                      of placing an order in period i to cover the next
adjustments to be made in the lot-sizes.                          j À i þ 1 periods and satisfy the service-level
   In all four demand data sets it can be seen                    constraint. However, the shortest route approach
clearly that the cost penalty increases as the                    considers the replenishment cycles independent of
coefficient of variation increases. There is no such               each other, and as a consequence of this the
simple pattern as the holding cost increases. For                 closing stock of one cycle may be above the
example, in the deterministic demand data set 1,                  opening stock for the next cycle. It is clear that this
the BT heuristic gives the optimal solution for the               gives a negative replenishment level, and therefore,
smallest, middle and largest holding costs but not                is an infeasible solution. Such infeasible solutions
for all of the others. This is mainly due to the                  are observed particularly in the erratic demand
discrete nature of the model used. The replenish-                 case, demand data set 4. In this case, the shortest
ment schedule obtained by the ‘‘static uncertainty’’              route method gives infeasible solution for all
model will remain unaltered over some range                       problems when s=m is in {1/3,1/4,1/5} and hX14:
for h: As a result of this, for a specific replenish-              This does not happen with the Bookbinder and
ment schedule the cost performance deteriorates as                Tan heuristic nor the optimal mixed integer
h approaches to the limits of its range. One                      programming model presented in this paper.
observes that at the limits the ‘‘static uncertainty’’            Moreover, if the unit variable cost cannot be ig-
model gives new replenishment policies and the                    nored, then it is not possible to treat replenishment
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118                   S.A. Tarim, B.G. Kingsman / Int. J. Production Economics 88 (2004) 105–119

cycles independently and therefore to apply the               treating the replenishment cycles independently
shortest route algorithm.                                     and applying the shortest route algorithm.
   All experiments were done on a 1.2 GHz                        Although, it has been assumed that the replen-
Pentium III, 512 MB RAM machine. The gener-                   ishment lead-time is zero, it is possible to extend
al-purpose solver CPLEX 8.0 (2002) is used with               the model for the non-zero replenishment lead-
the default settings and in all cases the optimal             time situation without any loss of generality. A
solution to the mixed integer programming model               similar model, incorporating the shortage cost, in
is found in less than a quarter of a minute.                  place of service-level constraints is currently being
Although the BT heuristic provides an almost                  developed by the authors. Further work could be
immediate solution, since the computation time                invested in evaluating the performance of the
required by the TK model is not excessive, it may             optimal ‘‘static–dynamic uncertainty’’ strategy in
be worth having the optimal solution.                         the rolling horizon environment compared to
                                                              Bookbinder and Tan’s heuristic.

7. Conclusions

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