joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with...

11
Production, Manufacturing and Logistics Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints W.J. Guerrero a , T.G. Yeung b,, C. Guéret c a PyLO, Depto. Ingeniería Industrial, School of Engineering, Universidad de Los Andes, Bogotá, Colombia b Ecole des Mines de Nantes, IRCCyN, France c Université d’Angers, LISA (EA 4094 - CNRS), 4 bd Lavoisier 49016 Angers CEDEX, France article info Article history: Received 28 November 2011 Accepted 18 May 2013 Available online 1 June 2013 Keywords: Multi-echelon Joint optimization Health care logistics Markov processes Inventory/distribution problem OR in health services abstract This paper presents a methodology to find near-optimal joint inventory control policies for the real case of a one-warehouse, n-retailer distribution system of infusion solutions at a University Medical Center in France. We consider stochastic demand, batching and order-up-to level policies as well as aspects partic- ular to the healthcare setting such as emergency deliveries, required service level rates and a new con- straint on the ordering policy that fits best the hospital’s interests instead of abstract ordering costs. The system is modeled as a Markov chain with an objective to minimize the stock-on-hand value for the overall system. We provide the analytical structure of the model to show that the optimal reorder point of the policy at both echelons is easily derived from a simple probability calculation. We also show that the optimal policy at the care units is to set the order-up-to level one unit higher than the reorder point. We further demonstrate that optimizing the care units in isolation is optimal for the joint multi- echelon, n-retailer problem. A heuristic algorithm is presented to find the near-optimal order-up-to level of the policy of each product at the central pharmacy; all other policy parameters are guaranteed optimal via the structure provided by the model. Comparison of our methodology versus that currently in place at the hospital showed a reduction of approximately 45% in the stock-on-hand value while still respecting the service level requirements. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction Healthcare spending has been continuously growing to more than 2.5 trillion USD in the United States in 2009 (CMS, 2011) and up to 234.1 billion euros in France in 2010 (DREES, 2011). Facing the in- crease of these expenditures, and under pressure from governments, insurance companies and communities, hospital administrators have begun to utilize operations research methodologies to improve service quality and reduce costs (Li and Benton, 1996). Aptel and Pourjalali (2001) provide insight into how hospitals in the US and France have improved their logistic activities and decreased costs. This article addresses the pharmacy inventory control problem for the supply chain network of hospitals of the University Medical Center in Nantes, France. We consider a one-warehouse, n-retailer inventory system with emergency replenishments and target ser- vice level constraints under stochastic demand. The objective is to optimize the inventory policy of the supply chain of infusion solutions including the special constraints and operational differ- ences that the healthcare industry demands. To be precise, healthcare supply chain models are often service- oriented (Samuel et al., 2010) in contrast to traditional approaches which include a penalty or monetary cost in the objective function for the chosen ordering policy. In our problem, imposing a subjec- tive (and potentially biased) cost for ordering would result in sub- optimal solutions. Instead, the limited and constant workforce restrains the ordering policy which is a distinguishing feature of this problem. The remainder of this paper is organized as follows. In Section 2, the current situation of the hospital is described, and in Section 3 a literature review on the topic is made. Section 4 presents the mod- el. Section 5 presents special properties of our model and our near- optimal solution methodology. Results are presented in Section 6 followed by conclusions in Section 7. 2. Current situation For the hospital pharmacy problem, we focus on infusion solu- tions as they are very common medicine for various treatments. 0377-2217/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2013.05.030 Corresponding author. Tel.: +33 2 51 85 86 45. E-mail addresses: [email protected] (W.J. Guerrero), Thomas.Yeung@ mines-nantes.fr (T.G. Yeung), [email protected] (C. Guéret). European Journal of Operational Research 231 (2013) 98–108 Contents lists available at SciVerse ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Upload: c

Post on 18-Dec-2016

216 views

Category:

Documents


2 download

TRANSCRIPT

Page 1: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

European Journal of Operational Research 231 (2013) 98–108

Contents lists available at SciVerse ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Production, Manufacturing and Logistics

Joint-optimization of inventory policies on a multi-productmulti-echelon pharmaceutical system with batching and orderingconstraints

0377-2217/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2013.05.030

⇑ Corresponding author. Tel.: +33 2 51 85 86 45.E-mail addresses: [email protected] (W.J. Guerrero), Thomas.Yeung@

mines-nantes.fr (T.G. Yeung), [email protected] (C. Guéret).

W.J. Guerrero a, T.G. Yeung b,⇑, C. Guéret c

a PyLO, Depto. Ingeniería Industrial, School of Engineering, Universidad de Los Andes, Bogotá, Colombiab Ecole des Mines de Nantes, IRCCyN, Francec Université d’Angers, LISA (EA 4094 - CNRS), 4 bd Lavoisier 49016 Angers CEDEX, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 November 2011Accepted 18 May 2013Available online 1 June 2013

Keywords:Multi-echelonJoint optimizationHealth care logisticsMarkov processesInventory/distribution problemOR in health services

This paper presents a methodology to find near-optimal joint inventory control policies for the real caseof a one-warehouse, n-retailer distribution system of infusion solutions at a University Medical Center inFrance. We consider stochastic demand, batching and order-up-to level policies as well as aspects partic-ular to the healthcare setting such as emergency deliveries, required service level rates and a new con-straint on the ordering policy that fits best the hospital’s interests instead of abstract ordering costs.The system is modeled as a Markov chain with an objective to minimize the stock-on-hand value forthe overall system. We provide the analytical structure of the model to show that the optimal reorderpoint of the policy at both echelons is easily derived from a simple probability calculation. We also showthat the optimal policy at the care units is to set the order-up-to level one unit higher than the reorderpoint. We further demonstrate that optimizing the care units in isolation is optimal for the joint multi-echelon, n-retailer problem. A heuristic algorithm is presented to find the near-optimal order-up-to levelof the policy of each product at the central pharmacy; all other policy parameters are guaranteed optimalvia the structure provided by the model. Comparison of our methodology versus that currently in place atthe hospital showed a reduction of approximately 45% in the stock-on-hand value while still respectingthe service level requirements.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Healthcarespendinghas been continuouslygrowingto morethan2.5 trillion USD in the United States in 2009 (CMS, 2011) and up to234.1 billion euros in France in 2010 (DREES, 2011). Facing the in-crease of these expenditures, and under pressure from governments,insurance companies and communities, hospital administratorshave begun to utilize operations research methodologies to improveservice quality and reduce costs (Li and Benton, 1996). Aptel andPourjalali (2001) provide insight into how hospitals in the US andFrance have improved their logistic activities and decreased costs.

This article addresses the pharmacy inventory control problemfor the supply chain network of hospitals of the University MedicalCenter in Nantes, France. We consider a one-warehouse, n-retailerinventory system with emergency replenishments and target ser-vice level constraints under stochastic demand. The objective isto optimize the inventory policy of the supply chain of infusion

solutions including the special constraints and operational differ-ences that the healthcare industry demands.

To be precise, healthcare supply chain models are often service-oriented (Samuel et al., 2010) in contrast to traditional approacheswhich include a penalty or monetary cost in the objective functionfor the chosen ordering policy. In our problem, imposing a subjec-tive (and potentially biased) cost for ordering would result in sub-optimal solutions. Instead, the limited and constant workforcerestrains the ordering policy which is a distinguishing feature ofthis problem.

The remainder of this paper is organized as follows. In Section 2,the current situation of the hospital is described, and in Section 3 aliterature review on the topic is made. Section 4 presents the mod-el. Section 5 presents special properties of our model and our near-optimal solution methodology. Results are presented in Section 6followed by conclusions in Section 7.

2. Current situation

For the hospital pharmacy problem, we focus on infusion solu-tions as they are very common medicine for various treatments.

Page 2: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

Fig. 2. New Two-echelon distribution system of infusion solutions.

W.J. Guerrero et al. / European Journal of Operational Research 231 (2013) 98–108 99

Infusion solutions are also generally non-critical and have longexpiration dates. Nevertheless, stockout events imply highly vari-able costs (some patients might be treated if similar products areavailable, other patients require specific products and replacementis forbidden. For such cases, stockout might lead to a patient’sdeath or other serious health consequences). As an alternative ofmonetizing the stockout costs of medicines, the hospital agreesthat imposing a minimum service level constraint is more realisticand emergency deliveries are always available to fulfill the demandas a last resource on stockout events.

In the current situation, the central warehouse (referred to asthe CP for Central Pharmacy) of the hospital delivers infusion solu-tions along with a myriad of other pharmaceutical and medicalsupplies to the three hospital pharmacy depots (HP) that belongto the hospital’s network. They are then delivered to the corre-sponding Care Unit (CU) and to other periphery hospitals integrat-ing a three-echelon system. Fig. 1 shows the current supply chainin the hospital scheme. The hospital currently handles 74 infusionsolution products that are delivered to 250 CUs.

The CP has a periodic review, order-up-to level inventory con-trol policy with batching that is known in the literature as the(nQ, r, T) policy, formally described first by Hadley and Whitin(1963). The review period happens once every T periods (once aweek), where pharmacists place an order in multiples of Q(batches), if and only if the inventory position is less than or equalto the re-order point r (denoted as s from now on), to raise inven-tory to some value S. These values are currently set by ‘‘expertknowledge’’ according to the price and the average weekly demandof the product. There is also a limit on the number of product ref-erences that the CP is allowed to order each week due to staff lim-itations to process all of the orders and the associated paperwork.This constraint does not justify an eventual stockout situation, butif the limit on the number of product references is violated system-atically, the payment to suppliers can be delayed. Therefore, whatbest fits the hospital’s interest is to limit the expected number ofproduct references that the CP orders weekly rather than the ‘‘ac-tual’’ number. Further, suppliers request a fixed-size batch-order-ing policy and orders arrive one week after they are placed.

Daily trucks deliver to the HPs from the CP according to fixedschedules to supply the hospitals with clothes, food and medicine.These transportation activities limit the use of vehicles for medi-cine transportation to once a week, so replenishment has to becoordinated to this schedule. The HPs follow a weekly (R, s, S) pol-icy as defined by Scarf (1959). Every R time units (the review pointis performed once a week), pharmacists check the stock-on-handquantities using automated software and trigger orders to suppli-ers to take the quantity in stock up to a pre-established value S ifthere is less than a re-order point s. As with the CP, policy param-

Fig. 1. Current three-echelon distribution system of infusion solutions.

eters are set by empirical methods with storage capacity and targetservice level constraints. Once the order is delivered to the HP, it isdistributed to the CUs. Every CU also follows an (R, s, S) policy andis replenished weekly. CUs have very limited storage space andcannot generally store more than one week of inventory.

If a stock out happens at any CU, emergency replenishments aremade from the HP within a negligible lead time. If the product isnot available at the HP, the common strategy is transshipment be-tween CUs in the same facility, but this disturbs the supply chainnetwork due to unregistered movements of products. The hospitalwould like to forbid transshipment actions between CUs except incase of emergency. In some cases, the CP might have to send anemergency replenishment directly to the CU within few hours onsmall compartments that vehicles have available for such cases.This is done while vehicles perform other transportation activities.Therefore, there is not an actual extra cost. In the rare case that theCP lacks the product, suppliers must make the emergency deliver-ies. Stock out at the suppliers is extremely unlikely. Over thecourse of 9 months there have only been 2 shortage cases, neitherof which were infusion solutions. It is important to note that in thehealth care setting there is no concept of ‘‘lost sales’’; the demandmust be filled in one way or another. However, in our model, if thedemand is fulfilled by an outside supplier it is considered as lostwithin our model.

Moving forward, the hospital wishes to redesign its supplychain network to distribute infusion solutions directly to each ofthe CUs from the CP, eliminating the individual hospital pharma-cies. The goal for the medical center is to optimize the supply chainin this new distribution system without reducing the current ser-vice level and eliminating transshipments. We characterize thisnew problem as a two-echelon inventory distribution system withemergency replenishments, stochastic demand, target service le-vel, limited storage capacity and restricted ordering policy on thefirst level of the supply chain by limiting the expected number ofproduct references ordered to suppliers per week by the CP.Fig. 2 shows the proposed configuration of the supply chain. In fact,Nicholson et al. (2004) address the issue of determining which dis-tribution network is better. A two-echelon system and a three-ech-elon system were compared to evaluate the possibility ofoutsourcing the distribution of non-critical medical supply inven-tory items. The paper concluded that the two-echelon system ismore cost efficient, even for an approximate solution. We presenta different approach, different cost structure and constraint setsto test the hypothesis on a real case.

3. Literature review

Different approaches have been developed to solve similarproblems in the health care industry. Details on the special charac-

Page 3: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

Fig. 3. Timeline of events.

100 W.J. Guerrero et al. / European Journal of Operational Research 231 (2013) 98–108

teristics of inventory systems in a healthcare setting are discussedby de Vries (2011), de Vries and Huijsman (2011).

One typical approach to solve similar problems is guided by thesupply chain optimization approach. In this way, the benefits ofcoordinating the activities of every echelon in the system and theoptimization of the policies at several points of the supply chainare simultaneously considered. Axsäter and Marklund (2008),Cachon (2001), Geng, Qiu, and Zhao (2010), Haji, Neghab, andBaboli (2009), Hernandez, Velasco, and Amaya (2008), van derHeijden (1997), Muckstadt and Roundy (1987) and Tagaras andVlachos (2001) are examples of papers that consider this approachto deal with single-item, 1-warehouse and n-retailer distributionsystems with stochastic or deterministic demand.

In addition, the joint-replenishment perspective is another rel-evant approach as it evaluates the benefits of coordinating the pol-icies of different products across the supply chain. Recently, Zhouet al. (2013) present a non-linear model for the multi-product,multi-echelon inventory control problem using a (T, S) inventorycontrol strategy solved by a genetic algorithm. According to theirexperiments, the joint replenishment strategy reduces the costby up to 21% compared to the unified replenishment strategy.For further information on joint-replenishment approaches seeAharon et al. (2009), Axsäter and Zhang (1999), Kang and Kim(2010), Khouja and Goyal (2008) and Narayanan and Robinson(2010). However, we have not found articles that optimize themulti-product, multi-echelon system with a framework that fitsthe concerns of the hospital, constraints on the service level andon the ordering policy like limiting the percentage of product ref-erences that are expected to be ordered every week rather thanimposing stockout and ordering costs.

Regarding single-facility models, Kapalka et al. (1999) describea single-warehouse, multiple-retailers system with multiple prod-ucts that sequentially optimizes one product and one facility at atime. Their objective was to determine the optimal (s, S) valuesfor each location (retailer) and each product with a service levelrequirement constraint. They modeled the system with a dis-crete-time Markov chain while considering a fixed ordering costand daily holding cost rate. They optimize the long run averagecost of inventory by a monotone search algorithm that purportssignificant time savings. However, there is no guarantee that theirapproach of optimizing the retailers first and then the warehouseseparately is equivalent to the joint optimization solution. In fact,such a decomposition approach will almost always be suboptimal.

Nicholson et al. (2004) deal with a similar problem to the oneaddressed in this paper, but less complex as they do not considerstorage capacity constraints and demands are deterministic. Theiroptimization model considers zero lead time and backorders satis-fied by emergency deliveries. The decision variables are the levels sfor the central warehouse, the hospital pharmacy and the CU. Theobjective is to minimize the total expected cost of one period sub-ject to a target service level. The proposed solutions to these mod-els are approximations using mixed-integer programming andgreedy heuristics. They do not consider the joint-optimization ofthe whole system.

To our knowledge, there are no papers in the open literaturethat fulfill all the special features of the multi-echelon, multi-inventory healthcare (R, s, S) inventory supply chain such as emer-gency deliveries, target service level, limited storage capacity,batch ordering and a constraint on the ordering policy as a limiton the expected number of product references that are orderedto suppliers weekly with a joint-optimization approach. In addi-tion, solving similar problems by dynamic or stochastic program-ming is not realistic as the dimensions of the problem growexponentially with the number of decisions, states and periods(Aharon et al., 2009).

4. Model and assumptions

4.1. Assumptions

We assume that demand faced by each CU is stochastic, Poissondistributed and independent between products and CUs. The inde-pendence assumption is justified by analyzing historical demanddata and finding no relevant correlation among CUs or amongproducts (correlation coefficients are smaller than 0.1). A mini-mum inventory position to satisfy a service level a of 92% or higheris required for every product in the CU. In addition, every product iat the CU j employs a periodic review (R, sij, Sij) policy with reviewevery R periods and replenishments within a fixed lead time L.Transshipments between CUs are not allowed.

Let j = 0 represent the index for the CP, and j > 0 the index forthe corresponding CU j. The following assumptions were madeabout the CP: (1) there is no economic benefit from suppliers tocoordinate the ordering policy among different products; (2) thereis a review period every R periods for all products at the same time;(3) the CP is replenished from a trustable source in a fixed leadtime which equals one period length (i.e., L = one week); and (4)employs a periodic review (nQ, r = si0, R) order-up-to Si0 policy. Fur-ther, the CP is required to follow a batch-ordering policy withbatch sizes bi fixed by the suppliers and there is a restriction onthe maximum expected percentage of product references c thatcan be ordered by the CP every week.

We also assume that products do not have an expiration date,emergency deliveries can be made from the CP to the CU immedi-ately and instantaneously any time (see Section 2 on trade-offanalysis between time until next regular order arrives and emer-gency delivery). When demand is not supplied by the CU or emer-gency deliveries, it is supplied by outside sources, therefore it islost in the supply chain model. All demands must be supplied ifthere is stock. Further, there is a maximum storage capacity atthe CP (mi0) and at the CUs (mij) for each product reference. Fixedordering costs are not considered but the ordering policy must re-spect the limitation on the expected number of product referencesper order.

Each period, these events take place as shown in Fig. 3:

(1) CP makes emergency deliveries to CUs as necessary.(2) CP receives the orders from suppliers from last period.(3) CUs review inventories and place orders to the CP (assumed

instantaneous).(4) CP dispatches orders to the CUs and updates inventory

(assumed instantaneous).(5) CP places orders to suppliers for next period (assumed

instantaneous).(6) CUs receive orders from the CP.

The following notation is utilized:

Parameters

p Number of products (infusion solutions) n Number of CUs supplied by the CP ci Cost of product i (Euros) kij Average demand of product i in CU j (units/week) L Lead time for the CUs R Period length (weeks)
Page 4: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

W.J. Guerrero et al. / European Journal of Operational Research 231 (2013) 98–108 101

Dij

Cumulative demand of product i in CU j in a givenperiod (units)

Daij

Cumulative demand of product i in CU j in a given

period up to time a (units)

DR�L

ij

Demand of product i in CU j in the non-lead time part ofthe period (units)

DEi0

Demand of product i in the CP supplied by emergencydelivery (units)

DWi0

Weekly orders of product i from CUs to CP at the reviewperiod (units)

mij

Maximum storage capacity of product i at CU j (units) mi0 Maximum storage capacity of product i at the CP (units) bi Number of units of product i in a batch (units) a Minimum service level required by the hospital (%) c Maximum expected percentage of products ordered per

week (%)

qi Probability of triggering an order to suppliers of

product i

pkij Probability of having k units of product i in CU j at the

review period

gkij Probability of having k units of product i in CU j just

after receiving orders

Decision variables

Sij Order-up-to level for CU j and product i, j > 0 (units) sij Re-order level for CU j and product i, j > 0 (units) Si0 Order-up-to level for the CP and product i (units) si0 Re-order level for the CP and product i (units)

4.2. Model

The goal of the model is to minimize the stock-on-hand value Zof the system equal to: Z = Z0 + Z1 where, Z1 represents the stock-on-hand value for the CUs and Z0 represents the stock-on-hand va-lue for the CP. The minimization is subject to a service level con-straint and a constraint on the expected number of productreferences ordered by the CP. Let Xi0 be a random variable repre-senting the stock-on-hand of product i in the CP at the review per-iod, let Xij be a random variable for the stock-on-hand of product iin CU j, and let Xi0 and Xij be their corresponding expected values.Let ci be the holding cost of product i. The structure of the model isdemonstrated in the following stochastic program:

min Z ¼Xp

i¼1

ci � Xi0 þXp

i¼1

Xn

j¼1

ci � Xij ð1Þ

subject to Si0 þ bi � 1 6 mi0; 8i ¼ f1; . . . ; pg ð2ÞSij 6 mij; 8i ¼ f1; . . . ; pg; 8j ¼ f1; . . . ;ng ð3ÞPrfDij 6 sij þ 1gP a; 8i ¼ f1; . . . ; pg;8j ¼ f1; . . . ;ng ð4Þ

Pr DWi0 þ DE

i0 6 si0 þ 1n o

P a; 8i ¼ f1; . . . ; pg ð5ÞPpi¼1qi

p6 c ð6Þ

sij; Sij P 0; 8i ¼ f1; . . . ;pg; 8j ¼ f0; . . . ;ng ð7Þ

where

qi ¼Xsi0

k¼0

pki0; 8i ¼ f1; . . . ; pg ð8Þ

With this model structure, we propose an approach to minimize theexpected stock-on-hand value for the overall distribution systemwith Eq. (1). Constraints (2) and (3) represent the limited capacityof the system to store products. Since the CP is subject to a batch-

ordering policy, Eq. (2) implies that if the CP is required to raiseits stock up to Si0 with batches of size bi, in the worst case, the CPwill surpass Si0 by an amount equal to bi � 1 but in any case it mustnot surpass mi0. Constraints (4) and (5) represent the need of theCUs and the CP to assure that the minimum inventory position ful-fills some target rate a. Finally, constraint (6) limits the percentageof products that are expected to be ordered every week by the CP.The probability of ordering any product is defined by Eq. (8).

4.3. Transition probabilities

To calculate the values of expected stock-on-hand in any CU jand the probability of triggering an order of some product i, letXij(t) be the stock-on-hand for the CU j at the review period t. Then,Xij(t) in the process: {Xij(t); t = 0, 1, 2, . . .}, describes a Markov chainwith Space State: Xi0 = [0, Si0 + bi � 1] for the CP, and Xij = [0, Sij] forany CU j. Under the assumption that the CUs are replenished with-in a lead time L and the CP is replenished after the lead time R, thebalance equations that describe the Markov chain are as follows:

For the CUs, j = {1, . . ., n}:

Xijðt þ 1Þ ¼XijðtÞ � DL

ij

� �þþ ðSij � XijðtÞÞ � DR�L

ij

� �þif Xij 6 sij

XijðtÞ � DRij

� �þif Xij > sij

8>><>>:

ð9Þ

For the CP, j = 0:

Xi0ðt þ 1Þ ¼Xi0ðtÞ � DE

i0

� �þþ Sij�Xi0ðtÞ

bi

l m� bi � DW

i0

� �þif Xi0 6 si0

Xi0ðtÞ � DWi0 � DE

i0

� �þif Xi0 > si0

8>><>>:

ð10Þ

Eq. (9) shows the typical description of the (s, S) policy with lostsales for the care units, while Eq. (10) includes the concept of emer-gency replenishments delivered by the CP and the batch orderingpolicy for the CP. Whenever a CU is not available to meet a demand,the CP must supply it. As a consequence, we focus first on the CUs asalso proposed by Kapalka et al. (1999). However, we will ensurethat our decomposition is optimal for the joint problem. With thisapproach we derive the one-period transition probabilities fromstate u to state v. These denote the probability of having v unitsof product i at CU j on hand given that there were u units one periodbefore, and are given as follows:

For 0 6 u 6 sij

pijuv ¼

Xu�1

k¼0

Pr DLij ¼ k

n oPr DR�L

ij P Sij�kn oh i

þPr DLij P u

n oPr DR�L

ij P Sij�un o

if v ¼0

Xu�1

k¼0

Pr DLij ¼ k

n oPr DR�L

ij ¼ Sij�v�kn oh i

þPr DLij P u

n oPr DR�L

ij ¼ Sij�u�vn o

if 16v 6 Sij�u

XSij�v

k¼0

Pr DLij ¼ k

n oPr DR�L

ij ¼ Sij�v�kn oh i

if Sij�u<v 6 Sij

8>>>>>>>>>><>>>>>>>>>>:

ð11Þ

and for sij < u 6 Sij

pijuv ¼

PrfDij P ug if v ¼ 0PrfDij ¼ u� vg if 0 < v 6 u

0 if u < v 6 Sij

8><>: ð12Þ

Given that the CP is constrained to order by batches, it is possible tohave in stock up to Si0 + bi � 1 of product i. In fact, considering thatin the review period, for a particular product i, the stock level at theCP is u, the minimum number of batches required to raise the stocklevel over Si0 is Si0�u

bi

l m. To simplify the following equations, we will

define the lot size ordered to suppliers for product i given a stocklevel u in the CP as:

Page 5: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

102 W.J. Guerrero et al. / European Journal of Operational Research 231 (2013) 98–108

Q iu ¼Si0 � u

bi

� �� bi ð13Þ

Therefore, the transition probabilities for the Markov chain are:For 0 6 u 6 si0

pi0uv ¼

Xu�1

k¼0

Pr DEi0¼k

n oPr DW

i0 PQiuþu�kn oh i

þPr DEi0 Pu

n oPr DW

i0 PQ iu

n oif v¼0

Xu�1

k¼0

Pr DEi0¼k

n oPr DW

i0 ¼Qiuþu�v�kn oh i

þPr DEi0 Pu

n oPr DW

i0 ¼Q iu�vn o

if 16v6Q iu

XQ iuþu�v

k¼0

Pr DEi0 ¼k

n oPr DW

i0 ¼Q iuþu�v�kn oh i

if Q iu <v6Si0þbi�1

8>>>>>>>>>><>>>>>>>>>>:

ð14Þ

And for si0 < u 6 Si0 + bi � 1

pi0uv ¼

Pr DEi0 þ DW

i0 P un o

if v ¼ 0

Pr DEi0 þ DW

i0 ¼ u� vn o

if 0 < v 6 u

0 if u < v 6 Si0 þ bi � 1

8>>><>>>: ð15Þ

Now, to evaluate Xi0;Xij and constraint (6), the limiting probabilitiespkij of having k units of the product i in CU j at the review period arerequired. Then, pkij are obtained after solving the following systemof equations:

pkij ¼XSij

l¼0

plijpijlk ð16Þ

XSij

k¼0

pkij ¼ 1 ð17Þ

Similiar to Kapalka et al. (1999), we require to calculate the shiftedlimiting probabilities gkij to compute accurately the expected stock-on-hand for every product (see Eq. (18)). These represent the limit-ing probabilities of having an inventory level of k units of product iin CU j when the order has just arrived. They are calculated with thelimiting probabilities of having a certain stock level at the beginningof the period and the probability density function of the demand. Toexplain further, the stock-on-hand after the orders have just arrivedequals the stock-on-hand at the review period minus the demandduring lead-time plus the stock just delivered. Bear in mind theassumption of lost sales implies that the stock is never below zero.If the stock is zero at the review period (p0ij), no matter the value ofthe demand, the stock just after the order has arrived is gkij = Sij.

gkij ¼

XSij

l¼0

plijPr DLij P l

n oif k ¼ 0

XSij

l¼maxðk;sijþ1ÞplijPr DL

ij ¼ l� kn o

if 1 6 k < Sij � sij

XSij

l¼maxðk;sijþ1ÞplijPr DL

ij ¼ l� kn o

þ psij ijPr DLij P sij

n oif k ¼ Sij � sij

XSij

l¼maxðk;sijþ1ÞplijPr DL

ij ¼ l� kn o

þXSij

l¼Sij�kþ1

plijPr DLij ¼ Sij � k

n oþ pSij�k;ijPr DL

ij P Sij � kn o

if Sij � sij < k < Sij

Xsij

l¼1

plijPr DLij ¼ 0

n oþ p0ij þ pSij ijPr DL

ij ¼ 0n o

if k ¼ Sij

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

ð18Þ

With these equations and representing the expected stock of prod-uct i in the CU j at the moment after the orders have just arrived(shifted review period) as lgij, we conclude that the expectedstock-on-hand for any CU will be equal to the expected value ofproducts at the end of the rth day, for r = {1, 2, . . ., 6} and lgij forday 7, i.e.,

Xij ¼1R

XR�1

r¼1

XSij

k¼0

gkij

Xk

l¼1

l � Pr Drij ¼ k� l

n oþ lgij

" #ð19Þ

Further, let Xi0 be the expected stock of product i in the CP at thereview period and kE

i0 and kWi0 be defined as the rates in which emer-

gency and weekly deliveries are respectively made. Define mij as thelimiting probability of being in stock-out of product i at CU j whenthe day ends; the values sij and Sij are the values for the policies ofthe CUs. Mathematically:

kEi0 ¼

Xn

j¼1

kij � mij8i ¼ f1; . . . ;ng ð20Þ

mij ¼1R

g0ij þXR�1

r¼1

XSij

k¼0

gkij � Pr Drij P k

n o" #ð21Þ

kWi0 ¼

Xn

j¼1

Xsij

k¼0

ðSij � kÞ � pkij ð22Þ

Xi0 ¼XSi0þbi�1

k¼0

pki0 � k ð23Þ

Eq. (20) shows the rate of emergency deliveries. This is given by theportion of the Poisson process for the demand faced by the CU dur-ing stock-out. The probability of stock-out is defined by (21) repre-senting the average probability of being in stockout at any dayr 2 {0, 1, . . ., R � 1}. The rate in which regular orders are triggeredfrom the CUs to the CP is defined by Eq. (22). Finally, the expectedinventory in the CP is as defined by (23).

5. Solution methodology

We now demonstrate that optimizing independently the CUswill provide the same solution as the joint-optimization case. Wefirst formally recall two well-known mathematical properties.

Definition 1. If Xj ¼ f gjk

� �8k ¼ f0;1; . . . ; Sjg;gj

k ¼ g pjk

� �8k ¼

f0;1; . . . ; Sjg and pjk ¼ hðsj; Sj; kjÞ there exists a function such that

Xj ¼ f 0ðsj; Sj; kjÞ.

Definition 2. In the optimization problem min z = f(x), with @z@x > 0,

the optimal value of z, z⁄, will be z⁄ = f(x⁄) when x⁄ is minimized.

Next, we show that we can optimize the values of sij for all CUsand the CP and all products by finding smallest values that do notviolate constraint (4) (Theorem 5.1). Theorem 5.2 then demon-strates that these values are optimal for the overall jointly opti-mized system. Finally, we show that the optimal policy for a CUis always (s, s + 1) (Theorem 5.3).

Page 6: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

W.J. Guerrero et al. / European Journal of Operational Research 231 (2013) 98–108 103

Theorem 5.1. The minimum quantity required to satisfy an apercentage of the demand for at least one period length will providethe optimal value of sij for a given product i in a given CU j when thelead time is less than or equal to the period length. Hence, the optimalsij is the minimum value that satisfies the equation:

PrfDij 6 sij þ 1gP a ð24Þ

Proof. By definition of the (s, S) inventory control policy, the min-imum inventory position that can remain in CU j after replenish-ment will be s + 1. Therefore, the solution of (24) provides theminimum point where the decision of triggering an order mustbe taken. h

Theorem 5.2. Let eij be a matrix of zeros and at least one value 1 atsome particular position and~a1;~a2;~a3;~a4 be column vectors of properdimension with constant values. If:

1. Z = Z0 + Z1, where Z0 ¼ f ð si0�!

; Si0�!

; sij!; Sij!Þ and Z1 ¼ gðsij

!; Sij!Þ

2. The optimization problem has the following structure:

min Z

subject to the set of constraints U:

si0�!P~a18i ¼ f1; . . . ;pg

Si0�!

P~a28i ¼ f1; . . . ;pgsij!P~a38i ¼ f1; . . . ; pg; 8j ¼ f1; . . . ;ng

Sij!

P~a48i ¼ f1; . . . ; pg; 8j ¼ f1; . . . ;ng

3.

f ð si0�!

; Si0�!

; sij!þ eij; Sij

!Þ � f ð si0

�!; Si0�!

; sij!; Sij!Þ > 0

8i ¼ f1; . . . ;pg; 8j ¼ f1; . . . ; ng

4.

gðsij!þ ei;j; Sij

!Þ � gðsij!; Sij!Þ > 0; 8i ¼ f1; . . . ;pg;

8j ¼ f1; . . . ;ng

Then:(A) Optimizing the problem: min Z1 subject to U, will provide

the optimal values sij!� for the problem min Z0 subject to U.

(B) The same optimal value sij!� will be the optimal solution to

the problem min Z subject to U.

Fig. 4. Simplified structure of the supply chain.

Proof. Using definition (2) we can conclude that, given the fourthclause in Theorem 5.2, the optimal value of sij

!� for the problem minZ1 subject to U, will be the smallest possible value within the rangeof sij! such that sij

!� ¼ a3�!. Therefore, if the same condition is utilized

for the problem: min Z0 subject to U, as provided by condition (3),the same optimal value of sij

!� is found. We have optimized thedecomposed problem with the value sij

!�. We now show that giventhese conditions, the joint problem is optimized with the samevalue sij

!�. Observe:

Z ¼ hð si0�!

; Si0�!

; sij!; Sij!Þ ¼ f ð si0

�!; Si0�!

; sij!; Sij!Þ þ gðsij

!; Sij!Þ

Followed by the substraction:

hð si0�!

; Si0�!

; sij!þ eij; Sij

!Þ � hð si0

�!; Si0�!

; sij!; Sij!Þ

¼ f ð si0�!

; Si0�!

; sij!þ eij; Sij

!Þ � f ð si0

�!; Si0�!

; sij!; Sij!Þ þ gðsij

!þ eij; Sij!Þ

� gðsij!; Sij!Þ

> 0

The inequality is strictly greater than zero under conditions (3) and(4). Thus, using definition (2), we demonstrate that optimizing theproblem min Z subject to U with the conditionhð si0�!þ eij; Si0

�!; sij!; Sij!Þ � hð si0

�!; Si0�!

; sij!; Sij!Þ > 0 will find its optimal

solution when sij!� takes the smallest possible values such that

sij!� ¼ a3

�!h

Theorem 5.3. The optimal policy for a CU is an (s, s + 1) policy.

Proof. It suffices to show that hð si0�!

; Si0�!

; sij!; Sij!þ ei�j�Þ�

hð si0�!

; Si0�!

; sij!; Sij!Þ > 0 is always true and repeat the same procedure

developed previously in Theorem 5.2. Moreover, it is obvious thatgðsij!; Sij!þ eijÞ � gðsij

!; Sij!Þ > 0 because the rationality of the system

implies that if the order size triggered by a CU increases, theexpected stock-on-hand in that CU will increase as well. h

However, to further illustrate Theorem 5.3, we show that thelast statement is true by emulating the performance of the system.We aim to show that the rationality of the system implies that thecost function increases as Sij increases. We begin by, without loss ofgenerality, simplifying our system to one CP, two identical CUs andonly one product. The simplified system is shown in Fig. 4. We maygeneralize the simplified system to the larger one due to the factthat we analyze the effect of the variation of one specific parameterSij on the overall system, regardless of the other CUs. Given that theCUs are independent, their costs will not be affected.

We compare the behavior of the system for a given CU policy,the sij

!0; S0ij!� �

against some other policy sij!0; S0ij!þ eij

� �. This analysis

is performed by the following procedure for every CU j:

Step 1: Calculate s0ij with Eq. (24).Step 2: Set S0ij equal to s0ij þ 1.Step 3: Calculate transition probability matrix P for the system

s0j; S0j

� �Step 4: Find limiting probabilities pkij (Eqs. (15) and (16)).Step 5: Calculate gkij (Eq. (17)).Step 6: Find the probability of having emergency deliveriesusing gkij and the ratio of weekly orders triggered to the CPusing pkij (Eq. (19)).Step 7: Estimate the cost for the s0ij; S

0ij

� �policy as the average

expected stock-on-hand value for CUs, denoted Z1.

After repeating this procedure for all CUs, we analyze thebehavior of the CP as follows:

Step 1: Find the ratio of emergency and regular deliveries perproduct as the sum of the emergency delivery ratios and theregular weekly order ratios.Step 2: Estimate parameters s0i0; S

0i0

that will describe the

inventory control policy for the CP with Eq. (24) and by settingS0i0 ¼ s0i0 þ 1.Step 3: Calculate the transition probability matrix P for the sys-tem s0i0; S

0i0

(Eqs. (13) and (14)).

Step 4: Find the limiting probabilities p0 (Eqs. (15) and (16)).

Page 7: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

104 W.J. Guerrero et al. / European Journal of Operational Research 231 (2013) 98–108

Step 5: Estimate the cost of the s0i0; S0i0

policy as the expected

stock-on-hand value at the CP in the review period, denoted Z0.

Knowing the values of Z1 and Z0, we modify the policy adopted byone of the retailers and observe the result. That is, we now repeat the

procedure, but one of the retailers will have a s0ij; S0ij þ 1

� �inventory

policy. There are two main effects resulting from the increment ofparameter Sij by one in CU j: (1) increasing the size of the weekly or-ders and (2) reduction of emergency deliveries. It is obvious thatincreasing the order-up-to level in the CU will increase the expectedstock-on-hand value because the sizes of the orders will be largerand less frequent. Also, the emergency delivery ratio will decreasebecause the probability of stock-out is smaller.

Consequently, these effects will impact the CP performance.Increasing the size of the orders and decreasing the emergencydeliveries modify the demand behavior faced by the CP. Specialfeatures of the Poisson distribution and the definition of the systemallow us to conclude the following:

Theorem 5.4. Increasing parameter Sij for some CU within a small �(which is discrete and strictly positive) decreases the expected averagestock-on-hand for the CP if and only if the CP does not have to modifyits inventory policy to maintain the target service level.

Proof. The proof also follows from the procedure above in theproof in Theorem 5.3. h

Theorem 5.5. The cost advantage obtained from increasing parame-ter Sij on the CP’s expected stock-on-hand is always less than the incre-ment on the average stock-on-hand on the corresponding CU j.

Proof. The proof follows from the procedure above in Theorem 5.3.Increasing the expected stock-on-hand at any CU one unitdecreases the average stock on the CP by less than one unit consid-ering the target fill rate constraint and constant demand. h

Fig. 5 shows the results obtained by the analysis for one productand 2 CUs. We can see that objective function Z (which can bedecomposed into Z0, the average stock-on-hand at the CP, plusZ1, the average stock-on-hand at the CUs) is strictly increasingand convex as result of the effects previously discussed. Moreover,we can see the behavior of the decomposed cost function whenincreasing the order-up-to level for any CU.

As a consequence of Theorem 5.2, the optimization of the CU sub-problem is also optimal for the joint system. Therefore, we proceedwith the optimization of the inventory control policy for the CP. Con-straint (6) forces us to consider simultaneously all the products inthe CP. Absent this constraint, we would simply conclude that the(s, s + 1) policy would be optimal for every product in the CU andthe CP. There is no issue with the CU ordering every product every

Fig. 5. Variation of the stock-on-hand value of the overall system when increasingthe order-up-to level for one CU.

week, however, the number of orders per week is restricted at theCP due to human resource constraints. Nevertheless, we may stilloptimize parameter si0 easily by finding the smallest value that sat-isfies constraint (5) as dictated by Theorem 5.1. The procedure todemonstrate this is analog to the one developed for the CUs.

It remains only to optimize parameter Si0 for the CP. Recall thatour objective is to minimize the stock-on-hand value while stillrespecting a constraint on the expected number of orders perweek. Theorem 5.1 showed us that the service level constraint isrespected through si0 and the constraint on the expected numberof orders per week will be respected through Si0. It follows thatone should choose to minimize the stock-on-hand of the mostexpensive products (ordering them more frequently), while keep-ing larger inventories of cheaper products to reduce the overallnumber of products ordered per week. Therefore, we propose thefollowing heuristic to minimize the cost. Better parameters ofSi0"i = {1, . . ., p} are found when larger expected order sizes are as-signed to those products with the largest b ratio until feasible solu-tions are found. We define the b ratio as:

bi ¼qi � q0i

ci � X0i0 � Xi0 ð25Þ

qi represents the ordering probability of any (si0, Si0) policy that hasan expected stock-on-hand of Xi0, while q0i represents the orderingprobability of the (si0, Si0 + bi) policy that is expected to have X0i0units of inventory.

The first step of the heuristic is to relax constraint (6). By doingso, a lower bound on Si0 is obtained by finding the minimum valuethat satisfies the following constraints for each product i, where dis some positive integer:

Pr DEi0 þ DW

i0 6 si0

n oP a ð26Þ

Si0 P si0 þ 1 ð27ÞSi0 ¼ d � bi; d ¼ f1;2;3; . . .g ð28Þ

Recall that the objective function of the model is convex with re-spect to the order-up-to level of the inventory control policy. There-fore, we should iteratively increase the parameter Si0 for thoseproducts i that give the greatest dual benefit in reducing the cost in-curred through holding more product and decreasing the expectednumber of orders per week. This is accomplished through the bi ra-tio. The bigger the cost ci and the difference X0i0 � Xi0, the less effi-cient is the decision of increasing the Si0 parameter for thatproduct i. Moreover, as the demand of the products are indepen-dent, the bi ratio is invariable with respect to changes in the policiesof other products.

The proposed heuristic algorithm is as follows:

Algorithm 1. Main Algorithm (Overview)

1: Initialize Sij, sij"i = {1, . . ., p}, "j = {1, . . ., n} with Eq. (24) andresults of Theorem 5.3

2: Initialize Si0, si0"i = {1, . . ., p} with Eqs. (26)–(28)3: for i = 1 to p do4: Calculate the probability of making an order for every

product qi

5: Calculate the benefit factor bi resulting from incrementingthe parameter Si0

6: end for

7: whilePp

i¼1qi

p > c do

8: i = arg maxj{bj}9: Si0 Si0 + bi

10: Update bi, qi

11: end while

Page 8: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

Table 1Parameters for numerical example.

Weekly demand rate (CU Storage capacity)

CU Prod. 1 Prod. 2 Prod. 3 Prod. 4

CU 1 1 (5) 40 (80) 200 (400) 12 (50)CU 2 10 (50) 50 (200) 150 (300) 24 (60)CU 3 20 (60) 60 (200) 150 (350) 35 (100)CU 4 30 (100) 70 (200) 401 (1000) 62 (300)

Cost and batch sizeci 0.5 0.63 0.68 1bi 2 10 12 10mi0 1000 1000 1000 1000

Operational requirementsService level s 92%Max. exp. % of products c 50%

Table 2Results from numerical example.

Results

Mean orders frequency (weeks) 5.41Stock value (UBCP) 676.39 EurosLBCP 197.09 Euros

Table 3Inventory control Policy for the CP.

Product s S Frequency

Prod. 1 70 999 16.22Prod. 2 229 500 1.99Prod. 3 859 989 1.06

W.J. Guerrero et al. / European Journal of Operational Research 231 (2013) 98–108 105

Prod. 4 143 390 2.41

6. Numerical results

For a first numerical example, the model was implemented inJAVA and was tested using a 1.50 GHz Intel Celeron processor with224 MB of RAM. The instance was composed of 4 CUs and 4 prod-ucts. The storage capacity at the CUs was assumed to be enough tostore the historical average of product requested. The running timewas 1642 s. The parameters and results are displayed in Tables 1–3. It becomes clear that more expensive products are not supposedto be stored in large quantities and therefore, they will be the ones

Fig. 6. Reduction of the expected fraction of prod

ordered in small quantities but more frequently. Also, batching hasan important effect on the decision variables as it affects the tran-sition probabilities and if the product has high demand, optimalinventory policy parameters are elevated too. Further, the con-straint (6) plays a critical role in the computation of inventory pol-icies at the CP. Table 2 presents the stock value when Eq. (6) isconsidered (UBCP), and when is not (LBCP). Note that by includingthis ordering constraint, the inventory holding cost at the CP is in-creased by 243%.

In the real case instance, the central pharmacy (CP) at the Uni-versity Hospital of Nantes has 74 products of infusion solutionsthat require 360 m2 of the warehouse. This equates to 41.86% ofthe available stocking space (CETI, 2007). About 1.5 million unitsof infusion solutions are delivered by the CP annually (CETI, 2007).

Precise historical demand data faced by the CP is not available.However, data of demand in the last 20 months made by each ofthe 250 CUs was provided in order to estimate the monthly de-mand of each product in each CU (as each CU does not order everyproduct, we account for 3184 orders from CUs for different prod-ucts). Following the literature Vernimmen et al. (2008), we assumea Poisson distributed demand as statistical analysis did not yieldany other feasible distribution. The mean demand per CU per prod-uct is 9.88 units with a standard deviation of 25.64. The 90th per-centile of the weekly demand per CU per product is 27 units andthe 95th percentile is 53 units per week. The maximum value ofthe mean demand per product per CU is 402.2 units. The aggre-gated demand of the CUs is used as the demand on the CP as de-mands are assumed independent.

The mean ratio of storage capacity over the average weekly de-mand per CU per product is 25.92 with a standard deviation of52.77. The minimum ratio is 0.86 (requiring more than 1 replenish-ment per week) and the maximum ratio is 500. On average, prod-uct unitary cost ci is 1.02 euros and the standard deviation is 1.24euros. In all cases, ci 2 [0.04, 8.57] euros. Most of the products aresold in batches of 6, 10, 12, 20, 30 or 48 units. Only 2 products arenot sold in batches (bi = 1). A particular product is sold in batches of250 units. The CP has an average storage capacity of 111.5 batchesper product ranging from 4 to 1000 batches. As for dispersion, thestandard deviation is 179.68 batches, the first quartile is 23batches, the median is 50 batches and the third quartile is 120batches. Only two products have storage capacity at the CP of1000 batches, that is when bi = 1.

For the real situation of the hospital, p = 74, n = 250, s = 92%,and c = 20%. The computational time of the algorithm is 5 h. Thesolution of our model (UB) yielded an average stock-on-hand valueof 33,439 euros. The current stock-on-hand value of the hospital

uct references ordered weekly per iteration.

Page 9: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

Fig. 7. Expected stock-on-hand value per iteration.

106 W.J. Guerrero et al. / European Journal of Operational Research 231 (2013) 98–108

using their current methodology is approximately 60,740 euros.Our methodology provides a reduction of 44.9% while still respect-ing the service level requirements. As expected, more expensiveproducts will be ordered more frequently in lower quantities whilethe cheaper ones will be kept on-hand in larger quantities to avoidordering more products per week than dictated by the constraints.

The expected percentage of products to be ordered weekly ateach iteration of the while loop in the heuristic algorithm is shownin Fig. 6. At first, the model under the relaxed ordering constraintproposes to order every product every week. This policy providesa lower bound (LB) that in the real case equals to 22,042 euros.The gap between LB and UB is 34.08% computed as (UB-LB)/LB.

Table 4Results for random instances.

Instance LB [euros] UB [euros] GAP (%) CPU (s)

10-10-a 2429.12 8869.66 72.61 882010-10-b 1799.82 6007.80 70.04 118710-10-c 2162.05 7949.82 72.80 2,02010-50-a 12,063.40 32,619.26 63.02 10,24510-50-b 10,770.00 32,040.33 66.39 12,79510-50-c 12,003.22 31,323.82 61.68 8,27950-10-a 4831.31 10,271.46 52.96 693650-10-b 3402.08 7026.86 51.58 996250-10-c 4407.91 10,322.86 57.30 12,93250-50-a 20,109.93 37,573.57 46.48 62,96250-50-b 20,090.75 47,386.87 57.60 73,43350-50-c 23,862.27 61,179.96 61.00 20,38850-100-a 36,503.81 81,630.50 55.28 12,48250-100-b 37,922.34 80,358.03 52.81 10,33650-100-c 38,205.00 76,128.36 49.82 90,005100-50-a 21,399.66 33,858.19 36.80 7854100-50-b 23,460.76 35,350.51 33.63 11,265100-50-c 23,749.43 34,746.46 31.65 8868100-100-a 44,150.33 66,698.83 33.81 9000100-100-b 45,448.81 68,510.02 33.66 7687100-100-c 51,239.44 81,752.48 37.32 7466100-200-a 96,097.17 143,834.09 33.19 15,679100-200-b 92,076.55 137,273.10 32.92 14,178100-200-c 93,090.77 141,053.88 34.00 16,945180-180-a 150,027.65 220,538.21 31.97 71,580180-180-b 158,725.64 223,442.78 28.96 71,859180-180-c 143,717.60 211,018.85 31.89 70,085200-100-a 91,512.84 130,158.55 29.69 75,706200-100-b 92,211.81 126,221.00 26.94 33,886200-100-c 86,470.33 114,313.25 24.36 36,949Average 48,131.39 76,648.65 45.74 26,726.27

The iterations of the algorithm progress to respect constraint (6).The larger improvements in the early iterations are due to the aug-mentation of the products with the largest beta values first. Thespeed of the improvement is significantly reduced in the later iter-ations. This is also due to the decreasing benefit of larger order-up-to levels as the constraint is designed only to reduce the number ofproducts ordered, not the quantity in the order. Naturally, theobjective function has an opposite behavior as shown in Fig. 7.The first iterations have little impact on the expected stock-on-hand value.

Finally, 30 random instances were generated with 10, 50, 100,180 or 200 CUs and products. The instances are labeled asn � p � x. Recall that n is the number of CUs and p the number ofproducts. x is used to itemize. The mean demand faced at the CUi for product j, kij, is randomly generated with Normal (1.88,5.64) probability density function. Let mij �max(5, kij � Normal (2,5)) to randomly generate the storage capacity at CU i for productj. For each product i, ci is generated with a truncated normal distri-bution with parameters 1.02 for the mean, and 1.24 for the stan-dard deviation. The size of each batch bi is chosen from the set{1, 6, 10, 20, 30, 48} while the depot storage capacity m0i is selectedfrom the set {200, 400, 600, 800, 100, 1500, 3000} with uniformprobabilities. Again, for the experiments we considered s = 92%and c = 50%. Table 4 presents the results on the generated in-stances. Column LB presents the cost of the policy when constraint(6) is relaxed. Column UB presents the solution cost for our policyand the percentage gap between both is computed as 100 � (UB-LB)/UB. Computation times in seconds are presented in the columnCPU. On one hand, optimizing each product independently withoutconsidering constraint (6) does not provide tight lower bounds. Onthe other hand, the computation time of our method remains rea-sonable considering the operational constraints on instances withup to 200 products and 100 CUs. Fig. 8 presents the mean compu-tation time (CPU) per instance size. It increases with the number ofCUs and products, but more importantly, with the dimension of thespace state Xi0 for every product i.

Note that a new policy is designed to minimize the expectedstock-on-hand value for the long term. Nonetheless, as new prod-ucts might be included by the hospital or others might be with-drawn, the decision is only updated at most on a quarterly basiswith the new set of products and updated estimations of the de-mand. There is little interest in updating the inventory policy morefrequently as demand forecast remains unchanged for severalmonths. In practice, the decision maker might take about one

Page 10: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

Fig. 8. Instance Size (n-p) versus Mean computation time.

W.J. Guerrero et al. / European Journal of Operational Research 231 (2013) 98–108 107

day to optimize the system. Thus, the methodology responds to theoperational requirements.

7. Conclusion

In this paper we study the problem of determining optimalparameters for the order-up-to level inventory control policies onthe one-warehouse, n-retailer, multi-product system with batch-ing and a new constraint on the ordering policies for the healthcareindustry and the benefits of having a two-echelon distribution sys-tem (versus three) on a hospital network of CUs. The objective is tominimize the stock-on-hand value at the supply chain subject tocapacity constraints, requirements on minimal inventory positions,batch-ordering policies, and a limit on the maximum expectednumber of products per order made to suppliers. The model is spe-cially made for non-critical products with long expiry dates such asinfusion solutions. Ordering costs and stock-out costs are notconsidered since ordering costs are not relevant to the hospital(but a constraint on the ordering policy was imposed instead)and prevention of stockout is handled via the service level con-straint rather than an associated cost that in the healthcare settingis difficult to calculate given its high variability. A Markov Chainmodel is used to find the weekly demand ratio for each productfaced by the central depot. Analysis is made to provide severalimportant properties of the model: (1) The s of the policy may bederived from a simple probability calculation. (2) The optimal solu-tion of the decomposed problem is equivalent to the joint optimi-zation solution, and (3) it is proven that the optimal solution forthe CUs is (s, s + 1) given the cost structure. We have also provideda heuristic algorithm to find the near optimal policy for the Sij valueat the CP. All other policy parameters are guaranteed optimalthrough the model structure.

For the real case of the university hospital in Nantes, the modelis expected to reduce the stock-on-hand value up to 45% and willbe implemented at the hospital within the next year. The servicelevel will remain the same, even when HP are eliminated. Futureresearch might involve expanding the model to include every typeof product with expiry dates and special storing conditions. Itwould also be interesting to consider the routing decisions of theinventory in the supply chain in order to further improve the over-all system performance.

Acknowledgement

We thank the anonymous referee for his valuable commentsand suggestions that helped improving the paper.

References

Aharon, B.-T., Boaz, G., Shimrit, S., 2009. Robust multi-echelon multi-periodinventory control. European Journal of Operational Research 199 (3), 922–935.

Aptel, O., Pourjalali, H., 2001. Improving activities and decreasing costs of logisticsin hospitals: A comparison of US and french hospitals. The International Journalof Accounting 36 (1), 65–90.

Axsäter, S., Marklund, J., 2008. Optimal position-based warehouse ordering indivergent two-echelon inventory systems. Operations Research 56 (4), 976–991.

Axsäter, S., Zhang, W.-F., 1999. A joint replenishment policy for multi-echeloninventory control. International Journal of Production Economics 59 (13), 243–250.

Cachon, G.P., 2001. Exact evaluation of batch-ordering inventory policies in two-echelon supply chains with periodic review. Operations Research 49 (1), 79–98.

CETI (2007). Rapport d’état des lieux, centralization et Distribution Globale desmédicaments – Pharmacies du CHU-Nantes. Centre d’Evaluation desTechnologies Médicales Innovantes-CETI.

CMS (2011). National Health Expenditure Data. Centers for Medicare & MedicaidServices. US Department of Health & Human Services. http://www.cms.gov.

de Vries, J., 2011. The shaping of inventory systems in health services: A stakeholderanalysis. International Journal of Production Economics 133 (1), 60–69.

de Vries, J., Huijsman, R., 2011. Supply chain management in health services: Anoverview. Supply Chain Management 16 (3), 159–165.

DREES (2011). Direction de la recherche, des études, de l’évaluation et desstatistiques. Ministère du travail, de l’emploi et de la santé. June 6th 2011.http://www.sante.gouv.fr.

Geng, W., Qiu, M., Zhao, X., 2010. An inventory system with single distributor andmultiple retailers: Operating scenarios and performance comparison.International Journal of Production Economics 128 (1), 434–444.

Hadley, G., Whitin, T., 1963. Analysis of inventory systems. Prentice-Hall,Englewood Cliffs, NJ.

Haji, R., Neghab, M.P., Baboli, A., 2009. Introducing a new ordering policy in a two-echelon inventory system with poisson demand. International Journal ofProduction Economics 117 (1), 212–218.

Hernandez, P., Velasco, N., & Amaya, C. (2008). Modelo de coordinación deinventarios en la cadena de abastecimiento de medicamentos de un hospitalpúblico. Universidad de los Andes. <http://hdl.handle.net/1992/1080>.

Kang, J.-H., Kim, Y.-D., 2010. Inventory replenishment and delivery planning in atwo-level supply chain with compound poisson demands. The InternationalJournal of Advanced Manufacturing Technology 49 (9–12), 1107–1118.

Kapalka, B.A., Katircioglu, K., Puterman, M.L., 1999. Retail inventory control withlost sales, service constraints, and fractional lead times. Production andOperations Management 8 (4), 393–408.

Khouja, M., Goyal, S., 2008. A review of the joint replenishment problem literature:1989–2005. European Journal of Operational Research 186 (1), 1–16.

Li, L., Benton, W., 1996. Performance measurement criteria in health careorganizations: Review and future research directions. European Journal ofOperational Research 93 (3), 449–468.

Muckstadt, J.A., Roundy, R.O., 1987. Multi-item, one-warehouse, multi-retailerdistribution systems. Management Science 33 (12), 1613–1621.

Narayanan, A., Robinson, P., 2010. Evaluation of joint replenishment lot-sizingprocedures in rolling horizon planning systems. International Journal ofProduction Economics 127 (1), 85–94.

Nicholson, L., Vakharia, A.J., Erengüç, S.S., 2004. Outsourcing inventory managementdecisions in healthcare: Models and application. European Journal ofOperational Research 154 (1), 271–290.

Samuel, C., Gonapa, K., Chaudhary, P., Mishra, A., 2010. Supply chain dynamics inhealthcare services. International Journal of Health Care Quality Assurance 23(7), 631–642.

Scarf, H., 1959. The optimality of (s,s) policies in the dynamic inventory problem.Mathematical Methods in the Social Sciences, 49.

Page 11: Joint-optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints

108 W.J. Guerrero et al. / European Journal of Operational Research 231 (2013) 98–108

Tagaras, G., Vlachos, D., 2001. A periodic review inventory system with emergencyreplenishments. Management Science 47 (3), 415–429.

van der Heijden, M., 1997. Supply rationing in multi-echelon divergent systems.European Journal of Operational Research 101 (3), 532–549.

Vernimmen, B., Dullaert, W., Willem, P., Witlox, F., 2008. Using the inventory-theoretic framework to determine cost-minimizing supply strategies in a

stochastic setting. International Journal of Production Economics 115 (1), 248–259.

Zhou, W.-Q., Chen, L., Ge, H.-M., 2013. A multi-product multi-echelon inventorycontrol model with joint replenishment strategy. Applied MathematicalModelling 37 (4), 2039–2050.