W e develop an inventory placement model in the context of general multi-echelon supply chains where the deliverylead time promised to the customer must be respected. The delivery lead time is calculated based on the available
stocks of the different input and output products in the different facilities and takes into account the purchasing leadtimes, the manufacturing lead times, and the transportation lead times. We assume finite manufacturing capacities andconsider the interactions of manufacturing orders between time periods. Each facility manages the stocks of its input andoutput products. The size of customer orders and their arrival dates and due dates are assumed to be known as in manyB2B situations. We perform extensive computational experiments to derive managerial insights. We also derive analyticalinsights regarding the manufacturing capacities to be installed and the impacts of the frequency of orders on the systemcost.
Key words: inventory placement; generic multi-echelon supply chain; delivery lead time; finite capacityHistory: Received: October 2011; Accepted: January 2013, by Jayashankar M. Swaminathan, after 1 revision.
1. Introduction
In the current industrial environment, many real-world supply chains have complex network structures,which consist of multiple layers of geographically dis-persed manufacturing and distribution facilities andan international network of suppliers. Componentsare therefore purchased worldwide, and intermedi-ate and final products are manufactured in differentfacilities that are generally located in different coun-tries (Hammami et al. 2009). Such a globalization ofproduction activities increases the lead time of pur-chasing, manufacturing, and transportation through-out the supply chain. This leads to increasing thetotal production cycle time.Simultaneously, the market is being very competi-
tive and companies are more than ever obliged (bytheir customers) to meet short customer lead times(CLTs). The CLT is defined as the time intervalbetween the release of an order and the due date ofthat order (as required by the customer). When thetotal production cycle time is larger than the CLT, thesupply chain needs to hold inventories to shorten thedelivery lead time and, consequently, meet customerrequirements. The delivery lead time is commonlydefined as the real elapsed time between releasing anorder and receiving it by the customer.
Thus, companies must cope with a situation charac-terized by a large total production cycle time thatleads to the necessity of holding inventories toshorten the delivery lead time and make it smallerthan (or equal to) the CLT. According to Kaminskyand Kaya (2008), in complex supply chains whereproducts are manufactured in many different facili-ties, inventory costs make up a significant proportionof total network costs.Thus, one key and challenging question that arises
is how to best manage and coordinate inventories inthe supply chain so as to meet CLT with minimumsystem-wide inventory holding cost. The central issuein this problem is to determine the location and sizeof inventories in the different periods. The inventoryplacement problem has been dealt with in differentcontexts and with different assumptions in the litera-ture. Typically, it consists in positioning stocks of out-put products throughout a supply chain with specificstructure to cope with demand and/or lead timeuncertainties and meet the service requirements whileminimizing the total incurred inventory cost.In this study, we address this relevant question
from a new perspective. Indeed, we consider theinventory placement problem in the context ofgeneral multi-echelon supply chains without anyrestriction on the supply chain structure. Each facility
in the supply chain manages the stocks of both inputand output products. Hence, all purchased, interme-diate, and final products are involved, and bill ofmaterial constraints are considered. This implies thatwe have to coordinate the stocks of all interrelatedinput and output products over the planning horizonin the different echelons of the supply chain. This isone of the novelties of this work. Unlike most pub-lished studies, our model considers finite capacities inthe manufacturing facilities and takes into accountthe interactions of manufacturing orders betweentime periods. Thus, the lead time required to finish amanufacturing order depends, on the one hand, onthe size of the order and the availability of input itemsand, on the other hand, on the lead times of previousorders. According to Simchi-Levi and Zhao (2005), ifproduction capacities are the major concern, thencoordinating inventory policies in multi-stage supplychains poses a substantial challenge.Consider the following industrial setting that moti-
vates our modeling choices of demand. A manufac-turer of electrical harnesses (located in Tunisia,Morocco, and France) supplies an automaker (inFrance). The manufacturer purchases componentsworldwide to fill the requirements of its differentsites. Some intermediate products are exclusivelymanufactured in France (for technological reasons) orin Morocco (for economical reasons). Electrical har-nesses are finally assembled in Tunisia and Moroccoand delivered to the customer. The customer placesorders once a week (every Friday) and imposes thatthey be delivered within 7 days. The automakeradopts a just-in-time policy with the supplier. Nostocks of electrical harnesses are held in the customersite. Thus, failure to deliver within the CLT (7 days) isvery costly for the harness manufacturer (a penalty ofseveral thousand euros is charged for each hour ofdelay). Consequently, the CLT must be respected.Every Friday, the automaker provides the supplierwith the real amount of demand over the next4 weeks and an estimate of the demand of 4 weeksafter. At the beginning of a given period t, the manu-facturer knows the demand of t, t + 1, t + 2, and t + 3and has an estimate of the demand of periods t + 4,t + 5, t + 6, and t + 7 with an uncertainty level of20%. Hence, the demand is deterministic over a plan-ning horizon of 4 weeks and may be predicted with agood level of confidence over 8 weeks. This is a com-mon situation in the automotive industry. We alsoexperimented a similar situation in the electronicsindustry between a manufacturer of TV decoders andits suppliers of hard disks.Motivated by such industrial situations, we assume
that the arrival times, the due dates, and the sizes ofcustomer orders are known at the beginning of theplanning horizon. We consider the general case where
the time intervals between successive orders mayhave different lengths and the CLT may vary fromone order to another. For every order, we impose thatall demand must be satisfied and that the CLT mustbe respected (due date cannot be missed; i.e., thedelivery lead time must be smaller than the CLT). Webelieve that these constraints are reasonable andappropriately model the reality of many firms in theB2B context as underlined by Kapuscinski and Tayur(2007). To calculate the delivery lead time, we con-sider the lead times of purchasing, manufacturing,and transportation throughout the supply chain.The model determines the stock level to be kept for
each product in each facility in each period. This stocklevel may vary from one period to another. The objec-tive is to minimize the total inventory cost. We formu-late the problem as a non-linear mixed integerprogramming model. We then derive managerialinsights based on extensive computational experi-ments and/or analytical approaches. These insightsare grouped into three main categories: the determi-nation of the amount of capacity to install in a givenmanufacturing facility, the negotiation of quoted leadtime and frequency of orders with customers, and thecomparison between different inventory policies. Thepresent study makes two major contributions: (i)modeling the inventory placement problem in a com-plex context that has not been considered in thearchive literature, and (ii) deriving insights that canbe valuable to production and inventory managers.The article is organized as follows. We dedicate sec-
tion 2 to a literature review on the inventory place-ment problem. In section 3, we present the frameworkand detail the assumptions of the proposed model.The mathematical formulation of the problem isaddressed in section 4. In section 5, we discuss how toextend the model to consider situations with backor-ders and lost sales. We also show how the model canconsider some particular manufacturing and inven-tory policies under specific settings. We then performcomputational experiments in section 6 and derivemanagerial insights. Finally, we give some conclud-ing remarks and new research perspectives.
2. Literature Review
The inventory placement problem has been dealt withfrom different perspectives in the literature. The firstcategory of works focus on positioning stocksthroughout the supply chain to cope with uncertain-ties and meet the service requirements, such as theworks by Graves and Willems (2000, 2008), Simchi-Levi and Zhao (2005), Ettl et al. (2000), Glassermanand Tayur (1995), and Lee and Billington (1993). Thesecond group of studies integrates the inventoryplacement decisions with facility location (e.g., Eski-
gun et al. 2005, Sourirajan et al. 2007) and manage-ment and planning of production/distribution sys-tems (e.g., Barnes-Schuster et al. 2006, Benjaafar andElHafsi 2006, Kaminsky and Kaya 2008, Spitter et al.2005, You and Grossman 2008). These works aremotivated by the correlations that exist between theinventory decisions and the other supply chaindecisions.The assumptions adopted in the studies that deal
with the inventory placement problem are mainlyconcerned with the inventory control policy, the nat-ure of demand, the structure of the supply chain, thetypes of considered products, the capacity issues, andthe method of calculating lead times. Usually, thebase-stock method is used as the inventory controlpolicy in inventory placement models. It is wellknown that a base-stock policy is widely used in prac-tice (Simchi-Levi and Zhao 2005). Depending on themodeling approach, the main decision variables areoften the base-stock levels (e.g., Simchi-Levi and Zhao2005) or the services levels/times (e.g., Ettl et al. 2000,Graves and Willems 2000, 2008, Inderfurth andMinner 1998). In most studies, the objective is to mini-mize the total system cost.Many of inventory placement models deal with
specific supply chain structures such as serial systems(Arslan et al. 2007), assembly systems (Benjaafar andElHafsi 2006), or distribution systems (Gurbuz et al.2007). This observation was also underlined byGraves and Willems (2003). Many models consider asingle product even for multi-echelon supply chainssuch as the works by Gurbuz et al. (2007) and Moin-zadeh (2002). The majority of studies consider a con-tinuous stationary demand, which is generally a non-realistic assumption. However, there is a growingbody of work to handle non-stationary demand (e.g.,Ettl et al. 2000, Graves and Willems 2008).Most inventory placement studies assume that the
system being considered has an infinite capacity (e.g.,Arslan et al. 2007, Benjaafar and ElHafsi 2006, Ettlet al. 2000, Graves and Willems 2008, Lee and Billing-ton 1993, Moinzadeh 2002, Simchi-Levi and Zhao2005). This is a big limitation of such works comparedwith real-world situations. Regarding the lead times,most works consider the processing lead times at thesupply chain nodes and the transportation lead times.As the capacity constraints are often ignored, it is typ-ically assumed that the processing lead time, whichmainly refers to the production lead time, does notdepend on the quantity of processed products.In addition, many of inventory placement models
manage only the stocks of the output product in everyfacility and ignore the stocks of input products. How-ever, companies usually have to manage inventoriesof both input and output products, especially in gen-eral multi-echelon supply chains where an output
product is made from different input products thatare acquired from different sites.Overall, the literature seems to be rather scanty on
analytical inventory placement models that simulta-neously consider general multi-echelon supplychains, input and output products in each facility,finite manufacturing capacities, and interactions ofmanufacturing orders between time periods. Ourwork tries to fill this gap.
3. Model Description and Assumptions
In this section, we present the problem characteristicsand the modeling assumptions.
3.1. Multi-echelon Supply Chain NetworkWe model the multi-echelon supply chain as a net-work with a set of nodes N and a set of arcs A. Thenodes represent the set of facilities F, the set of suppli-ers S (nodes without upstream stages), and the set ofcustomers K (nodes without downstream stages). Theset of facilities F includes the set of manufacturingfacilitiesM and the set of distribution facilities D.An arc denotes that an upstream node provides a
downstream node with a given product. Each node inthe network may be connected to several upstreamand downstream nodes. For each facility j (j 2 F), wedenote by Up(j) the set of upstream facilities thatsupply facility j and Down(j) the set of downstreamfacilities that are supplied by facility j.For simplification reasons, we consider only one
final product (product sold to customers). We con-sider the different purchased and intermediate prod-ucts that are required to obtain this final product.Purchased products are obtained from external sup-pliers. An intermediate product is manufactured in agiven facility and used in other facilities to manufac-ture other products. We denote by P the set of allproducts. As in Simchi-Levi and Zhao (2005), weassume that each facility j (j 2 F) manages a uniqueoutput item (which may be an intermediate or a finalproduct), one unit of which is possibly obtained frommultiple units of multiple input items (which may bepurchased or/and intermediate products).Thus, to each facility j (j 2 F) correspond to a set of
input products and a unique output product. Obvi-ously, a facility that has many output items may bemodeled as different facilities in the same location.However, we need to assume that these differentfacilities are independent. We denote by In(j) the setof input products of facility j, whereas Out(j) repre-sents the output product of j. For distribution facili-ties, In(j) = Out(j). The output product of distributionfacilities is the final product delivered to customers.We take into account the bill of materials through thescalar Uqp, which indicates how many units of input
item q are required per output product unit p. Theallocation of the different products to the differentfacilities is known. Finally, we assume that each facil-ity j (j 2 F) has only one upstream node for each of itsinput items. This upstream node may be an externalsupplier s (s 2 S) or an other facility j (j 2 M).
3.2. Demand ProcessThe planning horizon is finite and divided into sev-eral time periods indexed by t = 1,2,…,T. At thebeginning of every period t, demands arrive from oneor several customers. We let Lt denote the length ofperiod t, which is the time interval between the begin-ning of period t and the beginning of period t + 1.Time periods may have different lengths. To each cus-tomer order correspond an order size and a due date.We assume that the size, the arrival time, and the
due date of every order are known at the beginning ofthe planning horizon. As the allocation of the differ-ent products to the different facilities is known, wecan obtain, for every period t, the demand of finalproduct triggered in every distribution facility j. Wedenote it by Dt
j . An illustration of the demand processis given in Figure 1. A particular case of such ademand process corresponds to the realistic industrialsituation described earlier. We believe that thisdemand process appropriately models the reality atmany firms in B2B.
3.3. Guaranteed Service LevelWe let Ct
k denote the CLT imposed by customer k forits order of period t. Recall that the CLT is the timeinterval between the due date of an order and thetime when it is placed by the customer as shown inFigure 1. The supply chain must guarantee a 100%service level. We do not allow backorders and lostsales. All demands must be satisfied within the CLT.Thus, for every order, the delivery lead time must besmaller than the CLT. Clearly, keeping inventories ofthe final product in the distribution centers may be asolution to satisfy the CLT constraint, but this is not
the only available option. One may also store invento-ries of intermediate and/or purchased items in differ-ent facilities of the network.
3.4. Lead Times and Finite CapacityWe consider three types of lead times to calculate thedelivery lead time associated with each order: (i) thepurchasing lead times associated with the externalsuppliers, (ii) the transportation lead times betweenthe different facilities and toward the customers, and(iii) the manufacturing lead times.We assume that purchasing and transportation lead
times do not depend on the involved quantity.Depending on the roles of nodes i and j, we let �p;i! j
denote the purchasing or transportation lead time ofproduct p from node i (i 2 S ∪ F) to node j(j 2 F ∪ K). The processing lead time in distributionfacilities (i.e., the elapsed time between receivingproducts from upstream facilities and making themavailable for delivery) is also assumed to be constant.Therefore, it is not explicitly considered in our model,but is included in the CLT Ct
k.The manufacturing lead time is the time from when
the manufacturing process begins until production iscompleted and available to serve demand. As we con-sider a finite manufacturing capacity, the manufactur-ing lead time depends on the manufactured quantity.According to Cachon and Terwiesch (2009), the timeto manufacture x units (x > 0) = time to obtain the first
unit + x� 1 unitflow rate
. The time required to obtain the first
unit is denoted by TFU. The flow rate, denoted by FR,measures the number of flow units that move throughthe process in a given unit of time (e.g., 10 units/hour). We denote by d(x) the manufacturing lead timeof x units.It is important to note that the manufacturing pro-
cess of a given product can begin only if both of thefollowing conditions are satisfied: (i) all requiredinput products are available and, (ii) all previousorders are completed. The second condition accounts
CLT for order 1
Length of period 1 PD : Placement date
DD : Due dateCLT: Customer Lead mePD of Order 1
=Beginning of
period 1
DD of Order 1 DD of Order 2 DD of Order 3 DD of Order 4
for the possibility that a previous manufacturingorder is still in process and therefore the current orderneeds to wait until previous ones are completedbefore it can start being processed.
3.5. Control PolicyTo calculate the delivery lead time, we need to makesome assumptions on the supply chain control policy.Graves and Willems (2003) argue that some assump-tions on the links between upstream and downstreamstages should be considered to make progress oncomplex supply chain systems.In each period, each (manufacturing or distribu-
tion) facility j (j 2 F) can place at most one procure-ment order of each input product to its upstreamfacility. The orders are placed at the beginning of theperiod. Then, the order is received in facility j afterthe required lead time. In each period, at most oneorder of each input product can be received in eachfacility.We assume that facilities have to wait until all
required input items become available before startingprocessing. In the case of a facility supplying multipledownstream facilities, we assume that the facilityreleases the requirements of all adjacent downstreamfacilities at the same time. Then, the products arereceived in each downstream facility according to thetransportation lead time.Each facility j (j 2 F) is a potential location for hold-
ing stocks not only for output products but also forinput items. The model determines the stock level thatmust be held for each product in each facility in eachperiod to satisfy the CLT constraints with the lowesttotal inventory cost. The supply chain control policyis described hereafter.
• Outputs/inputs of distributions facilities. Eachdistribution facility j must be able to:
– satisfy its demand Dtj of every period t
within the CLT. If the available stock in j is notsufficient, then the net required quantity mustbe obtained from the upstream facility.– replenish its stock level between successiveperiods.
• Outputs of manufacturing facilities. Each man-ufacturing facility j must be able to:
– satisfy the requirements of its downstreamfacilities regarding its output product p. If theavailable stock of p is not sufficient, then thenet required quantity must be manufactured infacility j.– replenish its stock of output product pbetween successive periods. This can start onlywhen manufacturing the net requirement iscompleted.
• Inputs of manufacturing facilities. Each manu-facturing facility j must be able to obtain therequired quantity of each of its input items.An input item q may be required in j in thefollowing cases:
– to manufacture the net requirement of outputproduct p, or/and– to manufacture the quantity of p required toreplenish the stock (of p), or/and– to replenish the stock of item q.
4. Model Formulation
We formulate the problem as a non-linear mixed inte-ger programming model. We consider the followingdecision variables:
• htpj: stock level of product p (p 2 P) that mustbe held in facility j (j 2 F) at the beginning ofperiod t (t = 1,2,…T).
• whtq;i! j: quantity of input product q ordered by
facility j (j 2 F) from node i (i 2 S ∪ M) at thebeginning of period h and received in j in per-iod t (between the beginning and the end of t).
• xtpj: net requirement of output product p infacility j (j 2 M) triggered by the downstreamdemand in period t. It does not include thequantity required to replenish the stock of p inj for the next period.
• stpj: quantity of output product p that must bemanufactured in facility j (j 2 M) in period t toreplenish the stock of p for the next period. Itdoes not include the net requirement.
• ztqj: net requirement of input product q in facil-ity j (j 2 F) triggered by the downstreamdemand in period t. If j 2 D, then ztqj repre-sents the quantity of final product that is stillrequired (after considering the available stockat the beginning of t) to fill the downstreamdemand Dt
j (here, q is the input and the outputproduct at the same time). If j 2 M, then ztqjrepresents the quantity of q that is stillrequired (after considering the available stockof q at the beginning of t) to manufacture thenet requirement xtpj of output product p. Notethat ztqj does not include the quantity requiredto replenish the stock levels in j either forinput product q or for output product p.
• ytqj: net requirement of input product q in facil-ity j (j 2 M) triggered by both the downstreamdemand and the stock replenishment of outputproduct in period t (not only downstreamdemand as in the case of ztqjÞ. It represents thequantity of q that is still required (after consid-ering the available stock of q at the beginning
of t) to manufacture the quantity ðxtpj þ stpjÞ ofoutput product p.
Note that additional decision variables will berequired to model the different lead times. For clarityof the presentation, they will be introduced later.
4.1. Illustrative ExampleTo help the reader understand the above decisionvariables, we consider a simple illustrative case of aserial supply chain with one supplier s, one manufac-turing facility m, one distribution facility d, and onecustomer k as shown in Figure 2. There are two prod-ucts: q (purchased product) and p (final product). Weassume that Uqp ¼ 1. We consider only one period. In
Table 1, we consider some values of inventory levels
at the beginning of the first period ðh1pjÞ and at the end
of the planning horizon ðh2pjÞ. We then deduce the
values of variables xtpj, stpj, w
htp;i!j, y
tpj, and ztpj. The cus-
tomer demand associated with the first period isassumed to be 100. In this example, as we consideronly one period and do not deal with lead timesissues, the orders placed at the beginning of period 1are assumed to be received in this same period.For instance, in the case of the last row, facility d
has only 80 units in stock at the beginning of period 1and must fill the customer demand of 100 units.Hence, it has a net requirement of 20 units. Facility ddoes not need to keep stock for the next period (giventhat h2pd ¼ 0). Therefore, it places an order of 20 unitsto facility m at the beginning of period 1ðw1;1
p;m!d ¼ 20Þ.Facility m does not have stock. Therefore, the net
requirement of output product p in m is equal to 20.Facility m must manufacture the net requirement
x1pm ¼ 20. It must also manufacture the quantity s1pmafterwards to replenish its stock of p. Here, s1pm ¼ 50
as h2pm ¼ 50. Consequently, facility m needs 20 units
of input product q to manufacture x1pm, and 50 units to
manufacture s1pm. As h1qm ¼ 0, then z1qm ¼ 20 and
y1qm ¼ 70. Facility m also needs 10 units to replenish
its stock of q (given that h2qm ¼ 10Þ. As a result, facility
m places an order of 80 units to its supplier
sðw1;1q;s!m ¼ 80Þ.
To clarify the difference between the variables, ztqjand ytqj (j 2 M), we give below the values of thesevariables in all possible situations:
• There is enough stock of input item q to manu-
facture xtpj þ stpj (p is the output product) )ytqj ¼ ztqj ¼ 0.
• There is enough stock of input item q to manu-facture xtpj, but the stock is not sufficient tomanufacture xtpj þ stpj ) ytqj [ 0 and ztqj ¼ 0.
• The available stock of q is smaller than the
quantity required to manufacture xtpj; and
stpj ¼ 0 ) ytqj ¼ ztqj [ 0.
• The available stock of q is smaller than thequantity required to manufacture xtpj; andstpj [ 0 ) ytqj [ ztqj [ 0.
4.2. Objective Function and Physical FlowConstraintsThe objective function of the model minimizes thetotal inventory cost over the planning horizon as
given by expression (1), where ICtpj represents the unit
inventory holding cost of product p in facility j in per-
iod t. Note that hTþ1pj refers to the inventory level at the
end of the planning horizon.
MinX
1� t�T
Xj2F
Xp2P
ICtpj
htpj þ htþ1pj
2
!: ð1Þ
In each facility, the flow conservation constraints ina given period ensure that, for each product, theinbound quantity is equal to the outbound quantity.In the case of a distribution facility j, the inboundquantity of product p in period t is the sum of thequantity received from the upstream facilityi ðPh;h� t w
htp;i! jÞ and the available stock in period
t ðhtpjÞ. The outbound quantity is the sum of thedemand of period t allocated to facility j ðDt
jÞ and thestock level that must be kept for the next periodðhtþ1
pj Þ. This is guaranteed by constraint (2).Xh;h� t
whtp;i!j þ htpj ¼ htþ1
pj þDtj
j 2 D; i 2 UpðjÞ; p 2 InðjÞ \OutðiÞ; 1� t�T:ð2Þ
Regarding manufacturing facilities, we distinguishbetween the case of output products and the case of
Table 1. Example of Calculation of Decision Variables
h1qm h2qm h1pm h2pm h1pd h2pd z1pd w1;1p;m!d x 1pm s1pm z1qm y 1qm w1;1
input products. For an output product p, the inboundquantity is the sum of the total manufactured quantity
ðxtpj þ stpjÞ and the available stock in period t ðhtpjÞ. Theoutbound quantity is given by the downstream
demand in period t ðPi=ðj;iÞ 2A
Ph;h� t w
thp;j!iÞ and the
stock that must be kept for the next period ðhtþ1pj Þ.
For an input product q, the expression of inboundquantity is similar to that used for distribution facili-ties. The outbound quantity is the sum of the quantityof q required by the manufacturing activities in jðUqpðxtpj þ stpjÞÞ and the stock that must be kept for thenext period ðhtþ1
qj Þ. Constraints (3) and (4) are relativeto the flow conservation conditions in the manu-facturing facilities for output and input products,respectively.
In every t, each manufacturing facility imust satisfy
its downstream demand ðPj=ði;jÞ 2A
Ph;h� t w
thp;i! jÞ. If
htpi \P
j=ði;jÞ 2A
Ph;h � t w
thp;i!j (i.e., the available stock
htpi is not sufficient), then we must manufacture the
net requirement xtpi ðxtpi [ 0Þ. If htpi �P
j=ði;jÞ2APh;h� t w
thp;i!j, then xtpi ¼ 0. Hence, we add the follow-
ing constraints to define xtpi:
xtpi ¼ maxX
j=ði;jÞ2A
Xh;h� t
wthp;i!j � htpi; 0
0@
1A
i 2 M; p 2 OutðiÞ; 1� t�T:
ð5Þ
The net requirements of input products ztqj areobtained according to constraints (6) and (7) for distri-bution and manufacturing facilities, respectively.Indeed, if there is enough stock of input product q infacility j regarding the requirements of period t, thenztqj takes the null value. Otherwise, ztqj is given by the
difference between the requirement of q triggered in jby the downstream demand of period t ðDt
j for j 2 D
and Uqpxtpj for j 2 M) and the available stock of q in
The variable ytqj is defined only for manufacturing
facilities. Unlike ztqj, the variable ytqj takes into account
the requirement of q triggered in j not only by
the downstream demand ðUqpxtpjÞ but also by the
stock replenishment ðUqpstpjÞ. Constraint (8) is used to
define ytqj.
ytqj ¼ maxðUqpðxtpj þ stpjÞ � htqj; 0Þj 2 M; q 2 InðjÞ; p 2 OutðjÞ; 1� t�T:
ð8Þ
4.3. Modeling the Lead Time ConstraintsWe first define the lead time variables and thenformulate the lead time constraints.
4.3.1. Lead Time Variables. We consider thefollowing new variables:
• Douttpj: elapsed time between the beginning ofperiod t and the time when we make availablein facility j (j 2 M), the quantity of product prequired to fill the downstream demand ofperiod t. The downstream demand of period tin facility j regarding product p is given byP
i=ðj;iÞ 2A
Ph;h� t w
thp;j!i:
• Xintqj: elapsed time between the beginning ofperiod t and the time when we receive in facil-ity j (j 2 F) the order of input product q associ-ated with period t. As a facility cannot receivemore than one order of each of its input itemsin a given period, the quantity of q received infacility j in period t is
Ph;h� t w
htq;i!j (where i is
the upstream facility of j regarding product q).
• Dintqj: elapsed time between the beginning of per-iod t and the time when the quantity ztqj of inputproduct q becomes available in facility j (j 2 F).
• Cintqj: elapsed time between the beginning ofperiod t and the time when the quantity ytqj ofinput product q becomes available in facility j(j 2 M).
We also consider the following binary variable:
• nhtpij: equals 1 if facility i places an order ofproduct p to facility j in period h and receivesthe order in period t (i.e., wht
p;i!j [ 0Þ; 0 other-wise.
Obviously, if ztqj (respectively, ytqj and
Ph;h� t w
htq;i!j)
is equal to zero, then Dintqj (respectively, Cintqj and
Xintqj) must take the null value. It is also important to
note that if ztqj [ 0 (respectively, ytqj [ 0), then the
lead time Dintqj associated with ztqj (respectively, Cintqj
associated with ytqj) is equal to Xintqj. Indeed, it was
assumed that all requirements of a given input item
in a given period are grouped together in the sameorder and received at the same time. We explain in
Table 2 the relationships between Dintqj;Cintqj, and
Xintqj for all possible situations.
4.3.2. Lead Time Constraints. We can now turnto modeling the lead time constraints. The main con-straint imposes that the delivery lead time must notexceed the CLT. The delivery lead time of the order ofperiod t to customer k is the sum of (i) the lead timerequired to make the final product p available in theupstream node j of customer k, which is given byDintpj, and (ii) the transportation lead time from j to kdenoted by �j!k. Thus, the satisfaction of CLTs isguaranteed by constraint (9), where K(t) representsthe set of customers who place orders in periodt (K(t) ⊂ K).
Dintpj þ �j!k �Ctk
1� t�T; k 2 KðtÞ; j 2 UpðkÞ; p 2 OutðjÞ: ð9Þ
The variable Dintpj will be defined through the vari-
able Xintpj. Therefore, we first focus on modeling the
constraints related to Xintpj: For given period t, prod-
uct q, and facility j, if there exists a period h (h � t)
such as whtq;i!j [ 0 (i.e., nhtqij ¼ 1), then Xintqj, which is
the elapsed time between the beginning of t and the
receipt of order whtq;i!j, is given by Douthqi þ �q;i!j �P
h� t0\t Lt0 . Indeed, we first need to make product q
available in upstream facility i, which requires the
lead time Douthqi, and then transport products from i to
j, which requires the lead time �q;i!j. Otherwise (i.e.,
for all h � t;whtq;i!j ¼ 0Þ, Xintqj ¼ 0. Hence, Xintqj is
Recall that facilities cannot receive more than oneorder of each product in a given period. Therefore, fora given period t, there exists at most only one period h(h � t) such as nhtqij ¼ 1. This is guaranteed by con-straint (11). Constraint (11) also means that therequirements of a given period must be grouped
together in a unique order.Xh;h� t
nhtqij � 1
j 2 F; i 2 UpðjÞ; q 2 InðjÞ \OutðiÞ; 1� t�T:ð11Þ
By definition, the order whtq;i!j must be received in
period t. This is imposed by constraint (12). Accordingto constraint (12), an order that is received just at thebeginning of t (end of t�1) must be considered in thematerial flows of period t�1 and not those of period t.
nhtqij � 1\ Xintqj �Lt
j 2 F; i 2 UpðjÞ; q 2 InðjÞ \OutðiÞ; 1� t�T:ð12Þ
The variables whtq;i!j and nhtqij are related by constraint
(13), where Ψ is a sufficiently big number.
1
Wwht
q;i!j � nhtqij �Wwhtq;i!j
j 2 F; i 2 UpðjÞ; q 2 InðjÞ \OutðiÞ; 1� t�T:ð13Þ
We define Xintqj in constraint (10) through Douttpj.We focus now on the variable Douttpj. In order to fillthe downstream demand of period t, facility j mustmanufacture the net required quantity xtpj, whichrequires the lead time dðxtpjÞ. Nevertheless, beforestarting the manufacturing process, two conditionsmust be satisfied:
1. The previous manufacturing order must becompleted. If Doutt�1
pj � Lt�1 � 0, then the pre-vious order is completed before the beginningof period t. Therefore, we can start manufac-turing xtpj at the beginning of t if all the otherconditions are satisfied. Otherwise, processingthe order xtpj cannot start before the lead time
Doutt�1pj � Lt�1.
2. All input products must be available, whichrequires the lead time max
q2InðjÞDintqj.
As a result, the lead time Douttpj is determinedaccording to constraint (14).
Douttpj � dðxtpjÞ þmax Doutt�1pj � Lt�1; max
q2InðjÞDintqj
� �j 2 M; p 2 OutðjÞ; 1� t�T:
ð14ÞThen, facility j can start manufacturing the quantity
stpj of output product p to replenish the stock for thenext period if the following two conditions are satis-fied:
1. The processing of the order xtpj is completed,which requires the lead time Douttpj.
2. All input products are available, whichrequires the lead time max
In addition, processing the order stpj must becompleted before the end of period t. Hence, the leadtime dðstpjÞ is subject to constraint (15).
dðstpjÞ� minðLt � maxq2InðjÞ
Cintqj;maxðLt �Douttpj;0ÞÞj 2M; p 2OutðjÞ; 1� t�T:
ð15Þ
Finally, we define the lead times Dintqj and Cintqjby adding constraints (16) and (17), respectively.These constraints represent the relationships between
Dintqj, Cintqj, and Xintqj that have been explained in
4.3.3. Illustration of the Calculation of LeadTimes. To help the reader understanding the calcula-tion of the different lead times, we consider the exam-ple of Figure 2. We assume that the flow rate of themanufacturing process in facility m is 100 units/dayand that the time required to obtain the first unit is0.1 day. The procurement lead time (from s to m) isequal to 3 days. The transportation lead times areequal to 2 days and 1 day from m to d and d to k,respectively. In Table 3, we calculate (in days) the dif-ferent lead times for the scenarios of Table 1. Weexplain between brackets how we obtained some ofthe lead time values.
5. Model Extensions and ParticularCases
In our model, we impose that all demand must be sat-isfied and that CLT must be respected. In some situa-tions, companies might prefer to backorder or/and tolose some sales if this provides cost benefits. In thissection, we show how to extend the model to considerbackorders and lost sales. Furthermore, we do notimpose particular manufacturing and inventory poli-cies in our model. We then explain how the modelcan consider some classical manufacturing and inven-tory policies under specific settings.
5.1. Backorders and Lost SalesTo consider lost sales, we add the new decision vari-
able atjk to denote the quantity of final product deliv-
ered by facility j to customer k regarding the order of
period t. We also denote by Dtjk the demand of cus-
tomer k in period t allocated to distribution facility
j ðPk2K Dtjk ¼ Dt
jÞ. Clearly, if lost sales are allowed,
thenP
k2K atjk may be smaller than Dt
j : Hence, we
replace Dtj by
Pk2K a
tjk in constraints (2) and (6). We
also add constraint (18) to the model. In addition, wehave to include the cost (penalty) of lost sales in theobjective function. Hence, we add
P1� t�T
Pj2DP
k2K BCtkðDt
jk � atjkÞ to the objective function to be
minimized where BCtk denotes the unit cost associated
with lost sales for the order of customer k in period t.Xk2K
atjk �Dtj j 2 D; 1� t�T: ð18Þ
Now, we turn to the case of backorders. We nowallow late delivery, but still assume that orders cannotbe split. Hence, if the delivery of an order is delayed,then this concerns the total quantity of the order. We
consider the new non-negative decision variable utk to
denote the delay in the shipment of the order of cus-
tomer k of period t. The variable utk is the elapsed time
between the due date and the delivery date. If there isno late shipment, then it must be set to the null value.Thus, we replace constraint (9) by constraint (19). Asin the case of lost sales, we need to include the cost ofbackorders in the objective function. Hence, we addP
1� t�T
Pk2K DCt
kutk to the objective function, where
DCtk denotes the cost of late shipment per time unit
for the order of customer k in period t.
Dintpj þ �j!k �Ctk þ ut
k
1� t�T; k 2 KðtÞ; j 2 UpðkÞ; p 2 OutðjÞ: ð19Þ
We finally note that lost sales and backorders maybe considered simultaneously or separately in themodel.
5.2. Particular Manufacturing and InventoryPoliciesUnder particular settings of the model, we can con-sider different classical manufacturing and inventorypolicies. For instance, the model can consider a make
Table 3. Example of Calculation of Lead Times
Din1qm Cin1qm Xin1qm Dout1pm Din1pd Xin1pd Delivery lead time
to stock policy in manufacturing facilities, whichmeans that, in every period t, the downstreamdemand is satisfied by the stock that is held at thebeginning of t. In this case, the variables xtpj and Douttpjare set to the null value for every period t. The modelcan also consider a make to order policy, whichmeans that the manufacturing facility does not holdstocks but fills the downstream demand of every per-iod t from what is manufactured after the beginningof t. In this case, the variables stpj and htpj are set to thenull value for every period t.Furthermore, if the same stock level of product p
must be kept in facility j at the beginning of everyperiod t ðhtpj ¼ hpj ∀t), then the inventory policy infacility j regarding product p can be considered asbase-stock policy. In that case, hpj is the base stock. If,in addition, all periods have the same length (i.e.,Lt ¼ L 8t), then we are in the context of periodicreview policy. Finally, if facility j adopts the economi-cal order quantity model, then we can impose that allorders placed by j ðwht
q;i!jÞ have the same size, whichis the economical quantity.
6. Computational Experiments
The computational experiments are guided by threemain goals: (i) study the impact of finite manufactur-ing capacity, (ii) show how one can use the model tomanage the relationship with customers, and (iii)evaluate the performance of the inventory manage-ment policy adopted in our model compared withother classical policies. We consider an example of amulti-echelon supply chain with one supplier, fourmanufacturing facilities, one distribution facility, andone customer as given in Figure 3. There are fiveproducts: one purchased product, three intermediateproducts, and one final product. We assume thatUqp ¼ 1 8 p; q 2 P. On each arc, we indicate theinvolved product, the unit inventory holding cost ofthis product in both upstream and downstream facili-ties (whenever necessary), and the correspondingpurchasing/transportation lead time. Note that theunit inventory holding cost is here the cost of holdingin stock one unit of product over one unit of time. Oneach node representing a manufacturing facility, we
give between brackets the values of TFUpj and FRpj. Inall experiments we impose that the stock levels at thebeginning of the first period must be equal to thestock levels at the end of the planning horizon.We used the commercial optimization software
Cplex to solve the model and perform the computa-tional experiments. Thus, we first linearized themodel. The linearization approach is straightforwardso it is not addressed in the study. Our algorithm wasimplemented in C++ by using ILOG Concert 2.5 andCplex 11.0. It was run on a 2.1 GHz Intel Core 2 DuoT8100 computer with 2 GB of memory. As Cplexparameters heavily affect computational perfor-mances, we tried to find the best parameters for ouralgorithm. We varied two Cplex parameters: MIPEmphasis and Probe. The best combination we havefound is to set MIP Emphasis to 4 (emphasize findinghidden feasible solutions) and Probe to 3 (veryaggressive probing level). Indeed, the average compu-tational time (calculated on a sample of 40 instances)was decreased from 233 seconds with the default set-ting to 97 seconds with the best setting for small–medium size instances (instances with less than 10time periods). Nevertheless, for (very) large sizeinstances that could not be solved with Cplex in anacceptable time, it would be necessary to develop anefficient solving approach. However, this is beyondthe scope of the present study.
6.1. Management of Manufacturing CapacityThe proposed inventory placement model considers afinite manufacturing capacity in a general multi-eche-lon supply chain context. Therefore, we allow manag-ers to study the impact of varying the capacity in agiven facility on the total cost. Such an analysis canhelp decision makers to determine the amount ofcapacity that should be installed in a given facility.We address this issue with both numerical and ana-lytical approaches.In our model, we consider the interactions of
manufacturing orders between periods as we allow amanufacturing order that starts being processed inperiod t to be completed in a subsequent period t0
ðt0 � tÞ and, therefore, to delay the subsequent orders.This complicates both of the formulation and the
Figure 3 Multi-echelon Supply Chain for Computational Experiments
solving of the model. A simpler approach would be toimpose that manufacturing orders of period t must becompleted in period t. We perform numerical experi-ments to evaluate the gain achieved by using ourmodel instead of this simpler approach. The resultsare given in the last part of this section.
6.1.1. Impacts of Finite Manufacturing Capa-city. We let the flow rate vary from 40 to a very largeamount (INF) in each of the manufacturing facilitieswhile keeping the capacity unchanged in the otherfacilities. We then solve the model and obtain the sys-tem cost in each situation. We conduct these experi-ments on two instances that have the followingcharacteristics: four time periods with a demand of100 units in each period for both instances,Lt ¼ Ct
k ¼ 7 8 t for the first instance (Figure 4), andLt ¼ Ct
k ¼ 3 8t for the second instance (Figure 5).Observing Figures 4 and 5, one can note the follow-
ing:
• In general, the inventory cost decreases whenthe capacity goes up. However, there are situa-tions where increasing the capacity does not
lead to a reduction in system cost. For instance,in Figure 4, the inventory cost is not sensitive tothe variation of capacity in m2. Increasing thecapacity of m2 has no value. We can evenreduce the capacity in m2 from its initial level(120) to 40 without any impact on the inventorycost. The same observation can be made for m1when the capacity is between 80 and 140.
• The sensitivity of inventory cost to the varia-tion in capacity does not depend only on thesupply chain structure but also on the charac-teristics of demand (in particular, Lt and Ct
kÞ.Indeed, unlike the case of Figure 4 (whereLt ¼ Ct
k ¼ 7 8 t), increasing the capacity in m2leads to a reduction in the inventory cost inthe case of Figure 5 (where Lt ¼ Ct
k ¼ 3 8 t).• One can also determine the maximum gain that
results from increasing the capacity in a givenfacility. This gain is the difference between thecost with the actual capacity and the cost withinfinite capacity. For instance, in case of Figure5, if we take as a reference the flow rates thatwere given in Figure 3 (i.e., 150, 120, 120, and100 in m1, m2, m3, and m4, respectively), thenthe maximum gain that can be achieved if weincrease the capacities in m1, m2, m3, and m4 is1, 147, 0, and 289, respectively. Clearly, thishelps to target facilities where an effort on thecapacity should be made.
6.1.2. Analytical Insights. The above experimentscan help decision makers to determine how muchthey should be willing to pay for an additional unit ofcapacity in a given facility and to identify facilitieswhere capacity can be reduced without any impact onthe total inventory cost. It is very difficult to deter-mine with an analytical approach the amount ofcapacity that should be implemented in a given facil-ity j in the general case. To derive analytical insightswe have to consider some specific settings regardingthe production policy, the inventory policy, and theposition of facility j in the supply chain. The followingpropositions can help managers to determine thecapacity to install in a manufacturing facility.
PROPOSITION 1. Consider a manufacturing facility j thatsupplies a distribution facility d in a given supply chain.We assume that facility j adopts a make to stock manu-facturing strategy. Moreover, both of facilities j and dadopt a base-stock inventory policy (i.e., stocks must bereplenished to the same levels in all periods). Therefore,
(i) the flow rate that must be installed in facility j,denoted by FRMTS; must not be smaller thanmin
1� t�T
Dtd� 1
Lt �TFUpj.
900100011001200130014001500160017001800
40 60 80 120 140 180 220 260 300 INF
m1m2m3m4
Flow rate
Inve
ntor
y C
ost
Figure 4 Inventory Cost as a Function of Flow Rate (1)
14001600180020002200240026002800300032003400
40 60 80 120 140 180 220 260 300 INF
m1m2m3m4
Flow rate
Inve
ntor
y C
ost
Figure 5 Inventory Cost as a Function of Flow Rate (2)
(ii) if, in addition, the upstream facilities of j are exter-nal Suppliers, then there is no benefit of having
FRMTS larger than max1� t�T
Dtd� 1
Lt � �max �TFUpj, where
�max ¼ maxð�q;i!j, i 2 Up(j),q 2 In(j)).
PROPOSITION 2. Consider a manufacturing facility j thatsupplies a distribution facility d in a given supply chain.We assume that facility j adopts a make to order manu-facturing strategy and facility d adopts a base-stockinventory policy (i.e., stocks must be replenished to thesame levels in all periods). Therefore,
(i) the flow rate that must be installed in facility j,
denoted by FRMTO; must not be smaller than
min1� t�T
Dtd� 1
gtmax �TFUpj, where
gtmax ¼ max Ctk � �p;j!d � �d!k;Lt � �p;j!d
� �.
(ii) if, in addition, the upstream facilities of j are exter-nal suppliers and the manufacturing order of everyperiod t in facility j must be finished in t, then thereis no benefit of having FRMTO larger than
max1� t�T
Dtd� 1
Ctk ��p;j!d ��d!k ��max �TFUpj
.
The above propositions provide lower and upper boundson the manufacturing capacity. If the values of these boundsare close, then we can get a good approximation of the capac-ity that must be installed in the manufacturing facility.
PROOF. See Appendices S1, S2, and S3 in the onlineversion of this article.
6.1.3 Impacts of the Interactions of Manufac-turing Orders between Periods. The model accountsfor the possibility that a manufacturing order thatstarts being processed in period t is finished inanother period h, such as h > t. This gives more flexi-bility to the model to optimize the stock levels com-pared with the case where we impose that the orderof a given period must be completed in this same per-iod. In inventory placement models with infinitecapacity, this interaction of manufacturing ordersbetween periods is implicitly neglected as there isenough capacity to finish a manufacturing order of agiven period in this same period. Thus, it is interest-ing to use our model to evaluate the gain achieved byconsidering the interactions of the manufacturingorders between periods.With this scope in mind, we evaluate the percent-
age cost increase when the manufacturing orders in a
given period t are constrained to be finished beforethe end of t. We define the percentage cost increase asfollows:
We conduct the experiments on 100 randomly gener-ated instances where we consider four periods,demand in each period is randomly generatedbetween 50 and 250, and Ct
k and Lt are randomly gen-erated between 1 and 15.As shown in Table 4, the percentage cost increase
has an average value of 8.35% and reaches a maxi-mum value of 34.09%. Thus, the consideration of theinteractions of the manufacturing orders betweenperiods can lead to a significant gain in inventorycost. However, when the available capacity is largeenough, the gain is very small or even null. In suchsituations, one can use a simpler version of our modelwhere the manufacturing orders of every period t areconstrained to be finished before the end of t.
6.2. Management of the Relationship withCustomersThe proposed model can be used to manage the rela-tionship with customers in many ways. Basically, itcan help to negotiate the frequency of orders and theCLT. Here, we first study the impact of the frequencyof customer orders on the system cost and deriveexperimental insights. To explain the experimentalresults, we derive analytical insights under particularsettings of the model. Then, we focus on the impact ofthe CLT on the system cost.
6.2.1. Impacts of the Customer Order Frequency.Many companies are nowadays required by their cus-tomers (especially those who adopt the just-in-timeapproach) to ensure a large number of deliveries withsmall quantities. This can help the customer to reduceits inventory cost of purchased products and toimprove its logistics performance. However, what isthe impact of this decision on the supplier’s perfor-mance in terms of inventory cost? To answer thisquestion, we evaluate the system cost where we varythe number of periods (i.e., frequency of customer
Table 4 Percentage Cost Increase When Orders of Period t MustFinish in t
MeanStandarddeviation
Minimumvalue
Maximumvalue
95% Confidenceinterval
8.35% 9.67% 0% 34.09% [6.46% , 10.25%]
100 � ðoptimal cost when orders of period t must finish in t� optimal cost of our modelÞOptimal cost of our model:
orders) while keeping constant the length of the plan-ning horizon and the total demand (TD). This meansthat we vary Lt and the amount of demand in everyperiod t. For instance, for a TD of 60 units over a plan-ning horizon of 30 days, Lt may be fixed to 15 days(which corresponds to 2 orders of 30 units), 10 days(which corresponds to 3 orders of 20 units), and soforth. In Figures 6 and 7, we consider a planning hori-zon of 30 days and we show how the inventory costvaries as a function of the number of orders (i.e., LtÞfor different scenarios of TD and CLT.It is interesting to observe that increasing the fre-
quency of orders, and consequently the number ofshipments, is generally profitable to the supplier froman inventory cost perspective. In most cases, theinventory cost decreases when we increase the num-ber of orders. The experiments show that the larger isthe initial number of orders the smaller the gainachieved by increasing the frequency becomes. Forinstance, if we increase the number of orders from 7to 8 or 9, then the cost decrease is very small. In thiscase, the supplier would not accept increasing thenumber of shipments if one considers the other addi-
tional costs (such as transportation and productioncosts). There are even situations when increasing thefrequency of orders leads to an increase in inventorycost. In this case, the model may be used to calculatethe incurred additional cost to better negotiate withthe customer.When we increase the frequency of orders, both Lt
and demand per period decrease. The decrease in thedemand per period pushes toward the decrease ininventory cost, whereas the decrease in Lt may leadto increasing the cost. Indeed, if we decrease Lt, thenwe have less time to replenish stocks between succes-sive periods even when lower stocks are required, asthe replenishment lead time includes the transporta-tion lead time, which does not depend on theinvolved quantity. To understand how the inventorycost varies as a function of the frequency of orders,we provide analytical insights under particular set-tings of the model.
PROPOSITION 3. Consider a supply chain with a manu-facturing facility j that has a set of external suppliers andone downstream facility d (d is a distribution facility).We assume that facility j adopts a make to stockmanufacturing strategy. Facility j follows a base-stockinventory policy regarding the final product p. Facility ddoes not hold stocks. The TD is split into several ordersover the planning horizon. These orders have the samesize and are placed at regular time intervals.We first consider T orders (i.e., T time periods). In this
case, the length of periods is denoted by L and the demandin every period is denoted by Dd.
(i) If the production strategy is feasible and, at theoptimal solution, facility j does not hold stocks ofinput products, then the optimal inventory cost isf0 ¼ TLICpjDd. The parameter ICpj denotes theinventory holding cost of one unit of product p infacility j over one time unit. We now assume thatthe conditions of (i) are satisfied and increase thenumber of orders from T to T + 1 while keeping thesame planning horizon and TD.
(ii) If Tþ 1T �p;j!d �L� Dd
FRpj, then the inventory cost
decreases (from f0 to f1 ¼ TTþ 1 f0).
(iii) If there exists an input item q such asL\ Tþ 1
T �q;i!j (i 2 Up(j)) andICqj
ICpj[ 2
Uqp
Tþ 1T then
the inventory cost increases.
PROOF. See Appendix S4 in the online version ofthis article.
6.2.2. Impacts of the Customer Lead Time. Toquote the best delivery lead time to their customers,companies would have to analyze the variation in the
inventory cost as a function of the CLT. One canexpect that the inventory cost decreases with theincreasing values of the CLT as lower inventorieswould be required to satisfy lead time constraints.Nevertheless, what is the amplitude of cost decreasewhen the CLT becomes larger? In addition, how doesthe frequency of orders impact on the cost variation inthis case?To address these issues, we consider the case study
of Figure 3 with four time periods, and we analyzethe variation in the inventory cost as a function ofCLT under three scenarios of frequency of orders: Lt
= 5, 10, and 15 ∀t. For each scenario, we randomlygenerate 40 instances of the model by randomlygenerating the demand in each period between 50and 250. We then calculate the percentage costdecrease when we increase Ct
k from 1 to 15 for eachinstance of each scenario. The percentage decrease isobtained by
We found that the average percentage decrease ininventory cost is 60.37%, 84.2 % and 87.47% for Lt =5, 10, and 15, respectively. From these experiments,we can conclude that the larger is the time intervalbetween orders the more beneficial the quotation oflarge CLTs becomes. However, this result cannot begeneralized, as it depends on different factors such asthe initial and final values of CLT as we can see inFigures 8 and 9.Figures 8 and 9 show how inventory cost varies as a
function of CLT for demand = (99,69,64,74) and(153,165,199,151), respectively. We observe that theinventory cost may either decrease or stay constantwith the increasing values of Ct
k. When the inventorycost is not sensitive to the variation in Ct
k, the
proposed model can be used to determine the shortestdelivery lead time that can be promised to the cus-tomer without additional inventory cost. Otherwise,managers can use the model to quantify the costincrease and negotiate lead times with customers.
6.3. Performance of the Adopted InventoryManagement PolicyThe proposed inventory model can consider differentmanufacturing and inventory policies through spe-cific settings of model variables and parameters. Thisallows managers to evaluate and compare differentstrategies. In this section, we first compare the perfor-mance of our model to a situation where we imposethe same stock levels in all periods. Another noveltyin our work as against many published inventoryplacement models is that we simultaneously considerthe inventories of input and output products in agiven facility. In the second part of this section, we
evaluate the gain a company may achieve in using theproposed model instead of a model that does not holdin stock input items.
6.3.1. Time-varying Stock Levels vs. ConstantStock Levels. The base-stock policy is widely used inpractice and adopted by many works in the literature(see, e.g., Simchi-Levi and Zhao 2005). According tothis policy, companies generally consider a constanttarget stock level that does not vary over time. In ourmodel, the stock of every product in every site mayvary from one period to another. To evaluate the gainachieved by considering the solution of the proposedmodel instead of the solution with constant stock lev-els, we assess the percentage cost increase when con-
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1220172022202720322037204220
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Length of periods=5Length of periods=10Length of periods=15
stant stock levels are imposed in our model. Wedefine the percentage cost increase as follows:
We evaluate the percentage cost increase for two dif-ferent scenarios: (i) constant demand where the sizeof customer orders, Lt, and Ct
k are constant in all peri-ods, and (ii) fluctuating demand where the size ofcustomer orders, Lt, and Ct
k may vary from one per-iod to another. To provide an estimate of the percent-age cost increase, we consider a sample of 100randomly generated instances for each scenario underthe following conditions: four time periods, demandrandomly generated between 50 and 250 in each per-iod, and Lt and Ct
k randomly generated between 1and 15 in each period. Recall that, in the computa-tional experiments, we imposed that the stocks at thebeginning of the first period must be equal to thestocks at the end of the planning horizon. For eachscenario, we calculate the mean of the percentage costincrease for the sample of 100 instances and give the95% confidence interval. The results are shown inTable 5.Although the gain under the scenario of constant
demand is smaller than the gain under fluctuatingdemand, Table 5 shows that the gain achieved by con-sidering time-varying stock levels instead of constantstock levels is very large for both scenarios. Even for
the scenario of constant demand, stock levels at theoptimal solution generally fluctuate over time. Thismay have different causes. For instance, if the pro-curement lead time from a supplier is larger than thelength of the period, then the downstream manufac-turing facility may have interest in placing one orderevery two periods. In this case, the order quantity willserve to cover the requirements of two periods, which
leads to fluctuating stock levels of this input productin the manufacturing facility.
6.3.2. Impacts of the Integration of InputProducts. In the existing literature, most inventoryplacement models consider only one product in eachstage or facility (e.g., Graves and Willems 2008,Kaminsky and Kaya 2008, Simchi-Levi and Zhao2005). These models manage only the stock of theprocessed output product in each stage and do notexplicitly consider the stocks of input products. Inpractice, many companies focus only on the stocks offinal or/and purchased products and ignore thestocks of intermediate products. The proposed modelcan help managers improve the performance of theirinventory management systems by coordinating thestocks of the different products at the different eche-lons of the supply chain in the different time peri-ods.In this section, we evaluate the gain achieved by
considering the solution of our model instead of thesolution obtained when the stocks of input productsin manufacturing facilities are ignored. Hence, weassess the percentage cost increase when the stocks ofinput items in manufacturing facilities are forced totake the null value. The percentage cost increase iscalculated as follows:
We conduct the experiments on a sample of 100 ran-domly generated instances under the following condi-tions: four time periods, demand randomly generatedbetween 50 and 250 in each period, and Lt and Ct
k ran-domly generated between 1 and 15 in each period. InTable 6, we calculate the mean of the percentage costincrease for the sample of 100 instances and give the95% confidence interval.The results of Table 6 prove that considering the
interactions between input and output items in manu-facturing facilities and simultaneously optimizing the
Table 5 Percentage Cost Increase with Constant Stock Levels
MeanStandarddeviation
MinImumvalue
Maximumvalue
95% Confidenceinterval
Scenario 1:Constantdemand
178% 82% 33% 300% [162%, 194%]
Scenario 2:Fluctuatingdemand
405% 254% 156% 955% [355%, 455%]
Table 6 Percentage Cost Increase with Stocks of Input Products Fixedto the Null Value
MeanStandarddeviation
Minimumvalue
Maximumvalue
95% Confidenceinterval
31.46% 19.82% 7.49% 72.43% [27.58%, 35.35%]
100 � ðoptimal cost with constant stock levels � optimal cost of our modelÞoptimal cost of our model:
100 � ðoptimal cost when the stocks of input items are null � optimal cost of our modelÞoptimal cost of our model:
stocks of all products in the different echelons of thesupply chain, as our model does, leads to a significantcost saving.
7. Conclusion
We developed an inventory placement model wherethe objective is to minimize the total inventory costwhile satisfying the CLT. The model deals with acomplex context that is close to many real-world situ-ations. Indeed, we consider multi-echelon supplychains of general structures with finite manufactur-ing capacities. We take into account the interactionsof manufacturing orders between time periods. Wemanage the inventories of both output and inputproducts in each facility. The amount of customerorders, their arrival dates, and due dates areassumed to be known as common in the B2B context.The delivery lead time is calculated based on theavailable stocks in the different facilities and takesinto account the purchasing lead times, the manufac-turing lead times, and the transportation lead times.The proposed model can consider different classicalmanufacturing and inventory policies throughspecific settings of model variables and parameters.This is, to the best of our knowledge, the first studythat explores the inventory placement problem fromthis perspective.We performed extensive computational experi-
ments to show the theoretical and practical relevanceof our model to the inventory management science.We studied the impact of considering finite manufac-turing capacity and provided analytical insights onthe amount of capacity that should be installed in agiven facility. We showed how the model can be usedto negotiate the frequency of orders and the quotationof delivery lead time with customers. Under someconditions, we analytically showed how the systemcost varies as a function of the frequency of orders.We compared the performance of our model in whichwe consider time-varying stock levels with the case offixed stock levels. We also evaluated the gain that canbe achieved by considering the stocks of input prod-ucts in the different facilities and not only the stocksof output products as done by the majority of pub-lished studies.One of the main extensions of this work is to con-
sider the case of stochastic demand. We can use a sce-nario-based approach to model stochastic demand.However, this will make the model very difficult tounderstand and to interpret, which would require thesimplification of some aspects of the model. If there issmall fluctuation of demand, then we can ratherdevelop rolling horizon heuristics. For instance,assuming that demand is stochastic and, therefore,the size of the customer order cannot be known
before the order placement date. If the size of ordersis upper bounded, then a heuristic approach wouldbe to run the model at the beginning of every period twhile using the real demand for period t and themaximum demand for the subsequent periods. Thedecisions undertaken before period t (in particular,production and procurement decisions) cannot berevised, but are considered as input parameters. Wethen obtain the inventory levels that must be held atthe beginning of period t + 1 when solving the modelof period t. In the future, we will work on developingefficient methods to consider the case of stochasticdemand.
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