Operations Research for Health Care 2 (2013) 52–64

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Operations Research for Health Care

journal homepage: www.elsevier.com/locate/orhc

Pharmaceutical supply chain and inventory management strategies:Optimization for a pharmaceutical company and a hospitalR. Uthayakumar, S. Priyan ∗

Department of Mathematics, The Gandhigram Rural Institute, Deemed University, Gandhigram 624 302, Dindigul, Tamil Nadu, India

a r t i c l e i n f o

Article history:Received 28 January 2013Accepted 14 August 2013Available online 28 August 2013

Keywords:Pharmaceutical supply chainInventory managementService level constraint

a b s t r a c t

A high level of service for medical supplies and effective inventory policies are essential objectives forall health care industries. Medicine shortages and improper use of pharmaceuticals can not only leadto financial losses but also have a significant impact on patients. Many health systems and hospitalsexperience difficulties in achieving these goals as they have not addressed how medicines are managed,supplied, and used to save lives and improve health. Studies are essential to understand operationsin health care industries and to offer decision support tools that improve health policy, public health,patient safety, and strategic decision-making in thepharmaceutical supply chain.Wepresent an inventorymodel that integrates continuous review with production and distribution for a supply chain involvinga pharmaceutical company and a hospital supply chain. The model considers multiple pharmaceuticalproducts, variable lead time, permissible payment delays, constraints on space availability, and thecustomer service level (CSL). We develop a procedure for determining optimal solutions for inventorylot size, lead time, and the number of deliveries to achieve hospital CSL targets with a minimum total costfor the supply chain. A numerical example illustrates the model application and behavior.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Research into supply chain and inventory management hasbeen extensive in the field of health care. The main goal of this re-search is to reduce health care costs without sacrificing customerservice. Pharmaceuticals represent a significant part of health carecosts, account for approximately 10% of annual health care expen-diture in the USA and about $600 billion globally in 2009 [1]. Phar-maceutical products can be expensive to purchase and distribute,but shortages of essential medicines, improper use of medicines,and spending on unnecessary or low-quality medicines also have ahigh costs in terms ofwasted resources and preventable illness anddeath. Almarsdóttir and Traulsen identified a number of reasonswhy pharmaceutical deserve special consideration in the control ofinventory [2]. In the current economic crisis, increasing attentionis being focused on the rising costs of health care and specificallypharmaceuticals.

Careful management of pharmaceutical is directly related to acountry’s ability to address public health concerns. Aptel and Pour-jalali stated that management of pharmaceutical supplies is one ofthe most important managerial issues in health care industries [3].However, many health care industries experience difficulty inmanaging their pharmaceutical products. A pharmaceutical supply

∗ Corresponding author. Tel.: +91 451 2452371; fax: +91 451 2453071.E-mail addresses: uthayagri@gmail.com (R. Uthayakumar),

jaisilpriyan@gmail.com (S. Priyan).

2211-6923/$ – see front matter© 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.orhc.2013.08.001

chain (PSC) can be defined as ‘‘the integration of all activities asso-ciated with the flow and transformation of drugs from rawmateri-als through to the end user, aswell as associated information flows,through improved supply chain relationships to achieve a sus-tainable competitive advantage’’ [4]. The PSC comprises three ma-jor players: producers, purchasers, and pharmaceutical providers.Producers consist of pharmaceutical companies, medical surgicalproduct companies, device manufacturers, and manufacturers ofcapital equipment and information systems. Purchasers includegrouped purchasing organizations (GPOs), pharmaceutical whole-salers, medical surgical distributors, independent contracted dis-tributors, and product representatives. Providers include hospitals,hospital systems, integrated delivery networks (IDNs), and alterna-tive site facilities [5].

The PSC is very complex and carries high responsibility in en-suring that the right drug reaches the right people at the right timeand in the right condition to fight against disease and suffering. It isa highly sensitive supply chain in which anything less than a cus-tomer service level (CSL) of 100% is unacceptable because of thedirect impact on health and safety. The solution adopted by manypharmaceutical industries is to carry a huge inventory in the sup-ply chain to ensure a fill rate close to 100%.However, ensuring 100%product availability at an optimal cost represents a huge challengeunless the supply chain processes are streamlined towards cus-tomer needs and demands. Product perishability is another criticalPSC issue. Outdated or expired items may be overlooked and dis-pensed to patients, which could have potentially disastrous effects

R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64 53

on both patient care and public relations. In a 2003 survey, the es-timated cost for expiration of branded products in supermarketsand drug stores was over 500 million dollars [6]. Apart from thisperishability issue, health care managers are challenged with de-veloping inventory policies given changing demands, limited spacecapacity, CSL, patient safety, and regulations affecting supply.

In considering a PSC, health care managers have to decide or-der quantities and purchasing dates and the inventory level theycarry to effectively serve their customers. They also have tomanagetheir interactions with pharmaceutical companies tominimize theintegrated total cost for PSC and inventory management withoutsacrificing CSL. Therefore, PSCs require effective inventory man-agement policies and coordination among producers, purchasers,and providers. PSC decisions are significant because a shortage ofmedicines and improper use of pharmaceutical products lead to fi-nancial losses and have a significant impact on patients. Therefore,PSC decision-makers require expert knowledge to make the bestuse of their organizational resources and improve customer satis-faction without negatively affecting public health, patient safety,or relations with PSC members.

Unlike many industries, hospital administrators and pharmacymanagers have to manage very complicated distribution networksand inventory management problems without proper guidance onefficient practices. This is because most hospital administratorsand pharmacy managers are doctors with expert knowledge inmedicine, and are not supply chain professionals [7]. Hence, giventhe high costs, coordination, constraints, and perishability of phar-maceuticals, more study is necessary to help health care managersin setting optimal PSC and inventory management policies. Oper-ations research (OR) provides a wide range of methodologies thatcan help hospitals and other health care systems to significantlyimprove their operations. A number of studies have consideredproblems related to health care using OR techniques [8–11]. Herewe develop an OR model for PSC and inventory management for apharmaceutical company and a hospital. The next section reviewsthe literature on supply chain management issues from the per-spective of health care industries.

2. Literature review

Management of the procurement, storage, and distribution ofpharmaceutical supplies is crucial for hospitals and pharmaceuti-cal companies from economic and organizational points of view.PSC issues have been addressed by several authors from differ-ent point of views. Norris investigated cost reductions for hospi-tals by considering the total delivered cost of a product rather thanjust the unit cost [12]. This involves quantifying every cost asso-ciated with a product, including the unit cost and costs related toordering, inventory, distribution, preparation and use, and paper-work. Lapierre and Ruiz presented a strategy for improving hospi-tal logistics by focusing on scheduling decisions and a supply chainapproach rather than the more common multi-echelon inventorymanagement [13]. They placed an emphasis on scheduling deci-sions, such as when to buy a product, when to deliver to each careunit, when each employee should work, and what task should bedone. Scott and Graham proposed that implementation of an off-sitewarehouse to pool resourceswould result in a great savings forhospital supply chains [14]. The savings would be achieved as a re-sult of dramatic reductions in inventory holding costs and on-handinventory.

Bevilacqua et al. compared two different management ap-proaches for procurement of medical items by hospitals in a regionof central Italy [15]. They focused on a selected range of medicalsupplies according to their prevalence in three separate hospitalbudgets. Comparisons were made for parameters such as bufferstock, the reorder point, and relative costs. Dellaert andVan de Poel

developed a simple inventory rule for joint ordering in a univer-sity hospital, but they ignored capacity constraints [16]. Kelle et al.provided decision support tools that improve operational, tactical,and strategic decision-making in PSC and inventory managementunder an inventory policy that involves periodic review [17]. Theinventorymodelsmentioned above are single-echelonmodels thatconsider a constant lead time, which is not a controllable factor un-der a periodic review environment.

Outsourcing is allocation of specific business processes to anexternal specialist service provider and can yield organizationalflexibility. Although outsourcing has a variety of organizationalbenefits, it can also pose difficulties if a suitable service provideris not identified. As in other industries, outsourcing has becomean important strategy for pharmaceutical companies owing to in-creasing competitive pressures to reduce costs and the time tomarket. Regardless of the specific product, partnerships amongsuppliers and distributors to combine services have generated ben-efits for health care providers [18]–[20]. However, outsourcingleads to longer and more complex PSCs and reduces their trans-parency. Nicholson et al. developed analytical models to study andanalyze the impact of outsourcing of inventory management de-cisions in health care to a third-party provider [19]. They com-pared inventory costs and service levels for non-critical inventoryitems of an in-house three-echelon distribution network to an out-sourced two-echelon distribution network. According to Veral andRosen, a long-term benefit of outsourcing is the ability to reducethe number of suppliers in the system, whichwill eventually lowerprocurement costs for downstream members of the supply chain[20]. Chasin et al. investigated medical errors arising from out-sourcing of laboratory and radiology services [21]. As for any com-plicated area of study, the limitations of an investigation must beconsidered. The lack of outsourcing is a limitation of this study.

All steps from the supply of raw materials to the finished prod-ucts can be included in a supply chain connecting raw mate-rial suppliers, manufacturers, retailers, and the customer/hospital.Multi-echelon inventory management is the management of in-ventory and coordination of the distribution process at more thanone level of a supply chain network.Multi-echelon inventorymod-els have attractedmuch attention and the integrated approach hasbeen extensively studied. This approach was first implemented inretail and manufacturing industries, but has since spread to healthcare industries. Kim presented an explanation of an integratedsupply chain management system developed to specifically ad-dress issues related to pharmaceuticals in the health care sector[22]. Meijboom and Obel addressed supply chain coordination fora pharmaceutical company with a multi-location and multi-stageoperations structure [23]. Amir et al. developed a generalized net-work oligopoly model for PSC competition that takes into accountproduct perishability, brand differentiation, and discard costs [24].Their generalized network-based framework captures competitionamong firms in the various supply chain activities of manufactur-ing, storage, and distribution.

Lead time is an essential factor in any supply chain and inven-torymanagement system. Pharmaceutical manufacturers typicallyset a specific time for delivery products to customers that consid-ers factors such as labor strikes and natural disasters. This time iscalled the inventory lead time and comprises components such asorder preparation, order transit, supplier lead time, delivery time,and set-up time [25]. Inmanypractical situations, lead times can bereduced by paying an additional crash cost. According to Hsu andLee, crash costs could be expenditure on equipment improvement,information technology, order expedition, or special shipping andhandling [26]. Using this viewpoint, Liao and Shyu devised a prob-abilistic inventory model in which lead time was considered as adecision variable [27]. Several researchers extended this approachin integrated production–inventorymodels for lead time reductionin a single-vendor, single-buyer supply chain [28–30].

54 R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64

In practice, a periodic-review inventory policy is not applicablefor health care inventorymanagement because customer demandsand patient arrivals are uncertain. Thus, efficient management ofhealth care inventory systems requires a different approach than aperiodic-review reorder point model. According to Woosley, thereare at least three limitations for use of the continuous replenish-ment model in the context of health care supply systems [7]. Themodel does not account for limited human resources or physicalstorage capacity, CSL which is critical in most hospitals, and deci-sions are based only on costs and do not consider inventory controlactivities and restricted capacity. Because of the above-mentionedlimitations in health care industries, a continuous-review inven-tory policy is more suitable than a periodic review approach inhealth care inventory management.

Multi-echelon inventory problems for perishable products havebeen widely studied for general applications. However, existinginventory models are not applicable to pharmaceutical productsfor several reasons. Pharmaceutical products can be more expen-sive than other products to purchase and distribute, and short-ages and improper use of essential medicines can have a high costin terms of wasted resources and preventable illness and death.Therefore, special care should be taken in pharmaceutical inven-tory decisions to ensure 100% product availability at the right time,at the right cost, and in good condition to the right customers. Thequality of health care industries strongly depends on the availabil-ity of pharmaceuticals on time. If a shortage occurs at a hospital,an emergency delivery is necessary, which is very costly and canbe implications for patient health. Inventory management strate-gies that are unsuitable for health care industries may lead to largefinancial losses and a significant impact on patients. Hence, inven-tory strategies for pharmaceutical products are more critical thanthose for other products. Thus, a specific inventory model is nec-essary for control of pharmaceutical products to save patient livesand reduce unnecessary inventory costs.

Here we investigate a two-echelon supply chain inventorymodel for multiple pharmaceutical products under realistic prob-lems in health care industries. These include effective inventorypolicies and decisions, constraints and limitations, customer sat-isfaction, permissible payment delays, inventory lead time, and aminimum expected total inventory cost. We present a continuous-review integrated production–distribution inventory model forsupply chain involving a pharmaceutical company and a hospi-tal. We consider multiple pharmaceutical products, a variable leadtime that can be controlled via crash cost, permissible paymentsdelays, and realistic constraints such as space availability and CSL.We develop a procedure for determining the optimal solutions forinventory lot size, lead time and the total number of deliveries fromthe pharmaceutical company to the hospital to achieve the targethospital CSL at a minimum supply chain cost.

The remainder of the paper is organized as follows. Section 3provides the notations and assumptions used. Derivation of the ORmodel is described in Section 4 and the solution procedure is pre-sented in Section 5. A numerical example is provided in Section 6.Section 7 concludes.

3. Notations and assumptions

We develop an OR model using the notations and assumptionslisted below. Additional notations and assumptions are providedwhen required.

3.1. Notations

M: Number of products controlled in the supply chainDecision variablesQi: Order quantity for the ith product (i = 1, 2, 3, . . . ,M)L: Lead time (days) for all products

n: Total number of lots of M products delivered by the phar-maceutical company to the hospital in one production cycle, apositive integerOther parameters for the hospitalDi: Average demand for the ith product per yeardi: Expiry rate for the ith producthbi: Holding cost per unit per year excluding interest charges forthe ith productAi: Ordering cost per order for the ith producttc : Common trade credit period for all products offered by thepharmaceutical company in yearsId: Common deposit interest rate for all products per yearIc : Interest charge paid per $ in stock to the bank for all productsper yearpi: Purchase price per unit for the ith productsi: Selling price per unit for the ith productri: Reorder point for the ith productXi: Lead time demand for the ith product, which follows a nor-mal distribution with finite mean DiL and standard deviationσi

√L, where σi is the standard deviation for the demand per

unit time for the ith productE(Xi − ri)+: Expected demand shortage for the ith product atthe end of a cycleθi: Fraction of demand for the ith product that is not met fromstock, so 1 − θi is the service levelF : Fixed transportation cost for all products per deliveryfi: Storage space for the ith productW : Total space available for theM productsOther parameters for the pharmaceutical companyhvi: Holding cost per unit time for the ith finished productSi: Set-up cost for the ith finished productPi: Production rate for the ith finished productIv: Interest rate for calculating the opportunity interest loss forthe pharmaceutical company due to delayed payment per yeardci: Expiration rate for the ith finished productCdci: Cost of expiry for the ith finished productQwi: Replenishment quantity for the ith raw material for pro-duction in each production cycleAwi: Ordering cost for the ith raw materialhwi: Holding cost per unit time for the ith raw materialFw: Fixed transportation cost for all raw materialsvwi: Labor cost for order handling and receipt for the ith rawmaterialαi: Defect rate for the ith raw material in an order lot, αi ∈

[0, 1), a random variablesci: Screening cost per unit time for the ith raw materialrsi: Screening rate per unit time for the ith raw materialIETC(.): PSC integrated expected total cost for all products

3.2. Assumptions

1. The supply chain consist of a single pharmaceutical companyand a single hospital and they deal with multiple (M) pharma-ceutical products. For the ith product, the hospital orders a lotof size Qi (i = 1, 2, 3, . . . ,M) and the pharmaceutical companyproduces nQi units at a finite production rate of Pi (Pi > Di) perunit time in one production cycle, but ships in quantityQi to thehospitaln times. All products for thehospital are in shortage andcompletely backordered.

2. For the ith rawmaterial, all orders are delivered to the pharma-ceutical company in one shipment by an external supplier. Inother words, the quantity of the ith raw material required forproduction in each production cycle is instantaneous.

3. All expired pharmaceutical products held in inventory bythe pharmaceutical company are a constant fraction of theaccumulated inventory.

R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64 55

4. The hospital uses a continuous-review inventory policy for allproducts and the order quantity Qi for the ith product is placedwhen its inventory level falls to the reorder point ri.

5. The reorder point for the ith product is ri = expected demandfor product i during the lead time (DiL)+ safety stock of i (ssi),where ssi = ki x (standard deviation for the lead time demandfor i), that is,

ri = DiL + kiσi√L, (1)

where ki is the safety factor for product i and satisfies Pr(Xi >ri) = qi, and qi is the allowable stock-out probability for i duringthe lead time.

6. The pharmaceutical company offers a certain trade credit pe-riod (permissible payment delay) for all products to cooperatewith the hospital in an integrated strategy. Thus, the hospitaldoes not have to pay immediately on receipt of products.

7. The credit period tc is less than the reorder interval for eachproduct, which means that the credit period cannot be longerthan the time at which another order is placed. This is inagreement with usual practice in health care industries.

8. The hospital deposits sales income in a bank with an annualinterest rate of Id before payment is due. At the time ofpayment, the hospital pays the pharmaceutical company forproducts purchased. The hospital has a loan from a bank forunpaid purchase costs for unsold units. During the paymentdelay period, the pharmaceutical company has an opportunityinterest loss at an annual rate of Iv , where Iv = Id.

9. Crash costs increase as an approximate exponential function ofthe lead time. Therefore, the lead-time crash cost R(L) per orderfor all products is assumed to be an exponential function of Land is defined as

R(L) =

0 L = L0,eC/L Lb ≤ L < Lb−1,

where C is a positive constant and L0 and Lb represent theexisting and shortest lead times, respectively.

4. OR model development

OR provides methodologies that can support logistical opera-tions in health care industries and help in process optimization.We examine an ORmodel for PSC and inventorymanagement for asingle pharmaceutical company and a single hospital. We considera two-echelon supply chain inventory model for multiple pharma-ceutical products under realistic conditions. In this section, we for-mulate an ORmodel to identify the optimal inventory lot size, leadtime, and total number of deliveries in a production cycle by mini-mizing the integrated expected total cost while satisfying hospitalspace availability and CSL constraints.

4.1. Inventory model for the hospital

For product i, the hospital administrator places an order of Qiunits. Therefore, for an average cycle time of Qi/Di, the expectedorder cost is AiDi

Qiand the lead time crash cost per unit time is DiR(L)

Qi.

The order cost involves the cost of preparing the order or invoice,stationery and postage, wages, telephone charges, and travel ex-penses.

The expected net inventory level just before arrival of a procure-ment quantity Qi is only the safety stock ssi = ri − DiL = kiσi

√L

according to Eq. (1). The expected net inventory level immediatelyafter arrival of procurement quantity Qi is Qi + kiσi

√L. Hence, the

expected on-hand inventory over the cycle is Max. on hand + Min. on hand2

=(Qi+kiσi

√L)+kiσi

√L

2 =Qi2 + kiσi

√L.

To accommodate a more realistic inventory situation, we addthe effects of trade credit finance. In real life, business via sharemarketing, trade credit finance, or permissible payment delayscan improve sales in health care industries. Many pharmaceuticalcompanies offer hospitals an interest-free credit period to promotemarket competition and improve health policy, patient safety,and public health. Before the end of the trade credit period, thehospital can sell the goods and accumulate revenue and earninterest. However, higher interest is charged if the payment is notsettled by the end of the trade credit period. Therefore, it makeseconomic sense for the hospital to delay payment to the end of thepermissible delay period allowed by the pharmaceutical company.We assume that the pharmaceutical company offers a commontrade credit period for all products to attract hospital cooperationin our integrated strategy.

Let tc be the common credit period for all products and let hbi bethe unit stock-holding cost per unit time excluding interest chargesfor stock financing. The expected inventory for product i over thecycle is Qi

2 +kiσi√L, where Qi

2 is the expected cycle stock. Therefore,the holding cost per unit time for the cycle stock of i is hbiQi

2 . A safetystock of kiσi

√L is held throughout the cycle. Therefore, according

to the trade credit policy, the pharmaceutical company chargesinterest at rate Ic for this portion of the stock and the hospital mustpay the corresponding holding cost. Hence, the total cost of thesafety stock per unit time is the sumof the holding cost and interestcharged: (hbi + piIc)kiσi

√L.

The credit period tc is less than the reorder interval for eachproduct. This is a reasonable assumption because previous ordersshould be paid for before another order is placed. Therefore, thehospital can sell products and earn interest at the common rate ofId up to the end of the credit period. Hence, the interest earned by

the hospital per unit time for product i is siIdDiQi

tc0 Ditdt =

D2i t

2c siId2Qi

.The expected shortage for i, E(Xi − ri)+, is completely backloggedin the previous cycle and is cleared at the beginning of the currentcycle. Therefore, the hospital earns interest of sitc IdDi

QiE(Xi − ri)+ per

unit time during the credit period.Conversely, the hospital still has (Qi − Ditc) unsold units at the

end of the permissible delay period. The pharmaceutical companycharges interest for this portion of the stock. However, the hospitalhas a loan from a bank for unpaid purchase costs for unsold units,at the common interest rate of Ic .

Therefore, the opportunity interest cost per unit time for unsold

units of product i is piIcDiQi

QiDi

tc (Qi − Dit)dt =(Qi−Ditc )2piIc

2Qi.

In general, pharmaceutical products have a deterministic shelflife: if a product is not usedwithin its lifetime, it is considered to beexpired and must be destroyed. Accordingly, we assume that di isthe proportional probability for expiry of product i at the hospital.Consequently, the total amount of expired product for i is diQi. Weassume that the expiry cost applies immediatelywhen the hospitalorders the product from the pharmaceutical company. Hence, theexpiry cost Cdi(L) is a function of the lead time. Therefore, theexpected expiry cost is diDiCdi(L), where Cdi(L) is a linear functionof lead time L, that is, Cdi(L) = Γi + Γ0iL, where 0 ≤ Γ0i ≤ 1 andΓi is a positive constant.

In practice, the hospital pays a fixed transportation cost andlabor costs for order handling and receipt from the pharmaceuticalcompany. We assume that the hospital pays a fixed transportationcost of F for all products and a labor cost of vi for product i. Hence,the expected transportation and labor costs per unit time are FDi

Qiand viDi, respectively.

Accordingly, the expected total cost per unit time for product iis

ETChi(Qi, L) =Di

Qi(Ai + F + R(L)) +

(Qi − Ditc)2piIc2Qi

56 R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64

+hbiQi

2+ (hbi + piIc)kiσi

√L + Di(diCdi(L) + vi)

−D2i t

2c siId

2Qi−

sitc IdDi

QiE(Xi − ri)+.

The expected total cost per unit time for all (M) products is then

ETCh(Q , L) =

Mi=1

[ETChi(Qi, L)] =

Mi=1

Di

Qi(Ai + F + R(L))

+(Qi − Ditc)2piIc

2Qi+

hbiQi

2+ (hbi + piIc)kiσi

√L

+ Di(diCdi(L) + vi) −D2i t

2c siId

2Qi−

sitc IdDi

QiE(Xi − ri)+

(2)

where Q = (Q1,Q2,Q3, . . . ,QM).

4.2. Inventory model for the pharmaceutical company

Most production–distribution inventory models assume thatthe total cost for the manufacturer comprises holding, set-up, andproduction costs for finished goods. However, this total cost isnot suitable for real-life production situations because a manufac-turing company generally procures raw materials from externalsuppliers and holds them in inventory before starting production.Therefore, the inventory for a manufacturing company should in-clude raw materials and associated costs such ordering, purchas-ing, and holding costs. Here we assume that the pharmaceuticalcompany procures multiple raw materials from supplier and pro-duces multiple finished products. Therefore, we derive expectedtotal costs for both finished products and raw materials.

4.2.1. Finished productsIn our integrated inventory system, the pharmaceutical com-

pany begins production once the hospital orders l products of lotsize Ql, where the production rate is Pi (i = 1, 2, 3, . . . ,M) anda constant number of units are added to the inventory until theproduction run has been completed. The pharmaceutical companyproduces product i in lot size nQi in each production cycle of lengthnQiDi

, and the hospital receives the product in n lots each of sizeQi (i = 1, 2, 3, . . . ,M). The first lot of size Qi is ready for ship-ment after time Qi/Pi and then the pharmaceutical company con-tinues to the deliver the product on average every Qi/Di units oftime until the inventory level falls to zero. Hence, the expectedon-hand inventory of product i for the pharmaceutical companyis evaluated as the difference in accumulated inventory betweenthe pharmaceutical company and the hospital. According to Fig. 1,the inventory of product i for the pharmaceutical company is

nQi

QiPi

+ (n − 1) QiDi

−

n2Q 2i

2Piunits and the accumulated inventory

for the hospital is Q 2iDi

(1+2+3+· · ·+(n−1)) units. Hence, the ex-pected inventory per unit time for the pharmaceutical company is

nQi

Qi

Pi+ (n − 1)

Qi

Di

−

n2Q 2i

2Pi

−

Q 2i

Di(1 + 2 + 3 + · · · + (n − 1))

Di

nQi

=

nQ 2

i

Di

Di

Pi+ (n − 1)

−

nDi

2Pi

−

Q 2i n(n − 1)

2Di

Di

nQi

=Qi

2

n

1 −

Di

Pi

− 1 +

2Di

Pi

.

Therefore, the holding cost per unit time for the pharmaceuticalcompany is hviQi

2

n

1 −

DiPi

− 1 +

2DiPi

.

According to Assumption 3, the total number of expired prod-ucts held by the pharmaceutical company is

= dci

nQi

Qi

Pi+ (n − 1)

Qi

Di

−

n2Q 2i

2Pi

.

Therefore, the expiration cost per unit time

= dciQiCdci

Di

Pi+ (n − 1)

−

nDi

2Pi

.

Consequently, the total production cost is PiT1iPci(Qi), whereT1i =

nQiPi

is the production cycle for product i. The unit productioncost Pci(Qi) is a linear function of the order quantity Qi, that is,Pci(Qi) = δi + δoiQi, where δi is the fixed cost per unit finishedproduct and δoiQi is the tool/die cost per unit finished product,which is proportional to the lot size Qi. Hence, the total productioncost per unit time for product i is DiPci(Qi). Since Si is the set-upcost and the production quantity for a lot is nQi units, the expectedset-up cost per unit time is given by SiDi

nQi.

The hospital has a loan from a bank for unsold products. There-fore, during the trade credit period, the pharmaceutical companyhas an opportunity interest loss at an annual rate of Iv , whereIv = Id. Hence, the expected opportunity interest loss per unit timefor product i is IvpitcDi.

Let ETCfi(n,Qi) denote the expected total cost per unit time forfinished product i for the pharmaceutical company, which equalsthe sum of set-up, holding, production, and product expiry costsand opportunity interest losses for product i. That is,

ETCfi(Qi, n) =SiDi

nQi+

hviQi

2

n

1 −

Di

Pi

− 1 +

2Di

Pi

+DiPci(Qi) + IvpitcDi

+QidciCdci

Di

Pi+ (n − 1)

−

nDi

2Pi

.

The expected total cost per unit time for all finished products isthen

ETCf (Q , n) =

Mi=1

ETCfi(n,Qi)

=

Mi=1

SiDi

nQi+

hviQi

2

n

1 −

Di

Pi

− 1 +

2Di

Pi

+DiPci(Qi) + IvpitcDi

+ QidciCdci

Di

Pi+ (n − 1)

−

nDi

2Pi

. (3)

4.2.2. Raw materialsUnder Assumption 2, replenishment quantity Qwi for raw

material i is received instantaneously at the beginning of each cycletime nQi

Di. Hence, the expected order cost per unit time is given by

AwiDinQi

. For a practical perspective, we assume that each quantityQwi

contains defective raw materials at a rate of αi, which is a randomvariable. Therefore, raw materials lots are screened at a rate of rsito separate perfect and imperfect raw material. At the end of thescreening process, imperfect rawmaterials are sold as a single lot atthe lowest sales price per unit item to the external supplier. Thus,the screening cost is sciQwiDi

nQiand the revenue from imperfect raw

materials per unit time is sdiαiQwiDinQi

.Without loss of generality, we assume that (1 − αi)Qwi = PiT1i,

where T1i =nQiPi

. In other words, the replenishment quantity is

Qwi =nQi

(1 − αi)∀i = 1, 2, 3, . . . ,M. (4)

R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64 57

Fig. 1. Inventory pattern for a supply chain between a pharmaceutical company and a hospital.

Hence, the holding cost per unit time is given by hwi(1−αi)QwiDinQi

for perfect raw materials and hwiαiQwiDinQi

Qwirsi

=

hwiαiQ 2wiDi

rsinQifor

imperfect raw materials.The pharmaceutical company pays fixed a transportation cost

and labor costs for order handling and receipt, given by FwDinQi

andDiQwivwi

nQi, respectively.

Let ETCwi(Qi,Qwi, n) denote the expected total cost for rawmaterial i per unit time for the pharmaceutical company, whichequals the sum of expected ordering, holding, defect, screening,transportation, and labor costs minus revenue. That is,

ETCwi(Qi,Qwi, n) =AwiDi

nQi+

Di(1 − E(αi))Qwihwi

nQi

+sciQwiDi

nQi+

hwiE(αi)Q 2wiDi

rsinQi+

FwDi

nQi

+DiQwivwi

nQi−

sdiE(αi)QwiDi

nQi. (5)

Using Eq. (4), we can transform Eq. (5) to

ETCwi(Qi, n) =Di(Awi + Fw)

nQi+ hwiDi +

hwiE(αi)nQiDi

rsi(1 − E(αi))2

+sciDi

(1 − E(αi))+

vwiDi

(1 − E(αi))−

sdiDiE(αi)

(1 − E(αi)). (6)

The expected total cost for all raw materials per unit time is

ETCw(Q , n) =

Mi=1

[ETCwi(Qi, n)] =

Mi=1

Di(Awi + Fw)

nQi

+ hwiDi +hwiE(αi)nQiDi

rsi(1 − E(αi))2+

sciDi

(1 − E(αi))

+vwiDi

(1 − E(αi))−

sdiDiE(αi)

(1 − E(αi))

. (7)

Let ETCp(Q , n) denote the expected total cost per unit timefor all inventory products held by the pharmaceutical company,which equals the sum of expected total costs for all finishedproducts, given by Eq. (3), and for all rawmaterials, given by Eq. (7).That is,

ETCp(Q , n) = ETCf (Q , n) + ETCw(Q , n)

ETCp(Q , n) =

Mi=1

Di(Awi + Fw + Si)

nQi

+hviQi

2

n

1 −

Di

Pi

− 1 +

2Di

Pi

+Di(Pci(Qi) + hwi) + IvpitcDi

+QidciCdci

Di

Pi+ (n − 1)

−

nDi

2Pi

+

hwiE(αi)nQiDi

rsi(1 − E(αi))2+

Di(sci + vwi − sdiE(αi))

(1 − E(αi))

. (8)

Therefore, the integrated expected total cost for the supplychain, per unit time for all products, IETC(Q , L, n), can be expressedas the sum of expected total costs for the hospital, given byEq. (2), and for the pharmaceutical company, given by Eq. (8).

58 R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64

That is,

IETC(Q , L, n) = ETCh(Q , L) + ETCp(Q , n)

IETC(Q , L, n) =

Mi=1

Di(Ai + F + R(L))

Qi+

Di(Awi + Fw + Si)nQi

+(Qi − Ditc)2piIc

2Qi+ (hbi + piIc)kiσi

√L +

hbiQi

2+Di [diCdi(L) + vi + Pci(Qi) + hwi]

+hviQi

2

n

1 −

Di

Pi

− 1 +

2Di

Pi

−

D2i tc

2siId2Qi

+ IvpitcDi

−sitc IdDi

QiE(Xi − ri)+ + QidciCdci

Di

Pi+ (n − 1)

−

nDi

2Pi

+

hwiE(αi)nQiDi

rsi(1 − E(αi))2

+Di(sci + vwi − sdiE(αi))

(1 − E(αi))

, L ∈ [Lj, Lj−1]. (9)

4.3. Constraints

In practice, when a hospital carries inventory it may facelimitations such as the availability of floor space, total investmentin inventory, the total number of orders to be placed per year, thenumber of deliveries that can be accepted, the delivery size thatcan be handled, and CSL maintenance. Specifically, most hospitalshave an undersized central area for pharmacy storage that canhold very limited quantities of the numerous products required. Inaddition, the health care industry is service-oriented, and customerservice and satisfaction are paramount. Therefore, segmentationby market or customer is a very important step in developingan appropriate supply chain strategy for the target customer.Therefore, we consider hospital issues in relation to floor space andCSL constraints.

Consider first a case in which the hospital floor space is limitedto W square feet and one unit of inventory product i occupies fisquare feet. If Qi is the lot size, then the floor space constraintmeans thatMi=1

fiQi ≤ W . (10)

For the CSL constraint, certain goals are defined and CSL is thepercentage that should be achieved for each goal. This is used insupply chain and inventory management to measure the perfor-mance of inventory replenishment policies. Service levels θi areassociated with products according to the number of occasions ofshortage that managers are willing to accept during a period oftime. If a hospital pharmacy is out of a product, it can place anemergency order with the pharmaceutical company to replenishthis product. However, hospitals would like to avoid these emer-gency orders if possible because they are very costly [7]. We con-sider the service level θi for each item separately. We assume thatthe hospital has set a target service level in terms of the fill rate cor-responding to the proportion of all product demands to be satisfieddirectly fromavailable stock. Therefore, the CSL constraint imposesa limit on the proportion of demands that are not met from stock.For product i this can be expressed asExpected demand shortages of product i at the end of the cycle for a given safety factor

Quantity of product i available to satisfy the demand per cycle≤ θi.

That is,

E(Xi − ri)+

Qi≤ θi. (11)

Weassume that the lead time demandXi follows a normal prob-ability distribution function (p.d.f.) f (xi) with mean DiL, standarddeviation σi

√L, and reorder point ri = DiL + kiσi

√L, where ki is

the safety factor for product i.For product i, shortages occur when Xi > ri. Therefore, the

expected shortage at the end of the cycle time for the hospital isgiven by

E(Xi − ri)+ =

∝

ri(xi − ri)f (xi)dxi = σi

√LΨ (ki), (12)

where Ψ (ki) = ϕ(ki) − ki[1 − Φ(ki)] > 0 and ϕ, Φ are thestandard normal p.d.f. and cumulative distribution function (c.d.f),respectively, for product i.

Hence, using Eqs. (11) and (12), the CSL constraint for product ican be written as

σi√LΨ (ki)Qi

≤ θi, L ∈ [Lj, Lj−1].

The CSL constraint for all products is then given by

Mi=1

θiQi ≥

Mi=1

σi√LΨ (ki), L ∈ [Lj, Lj−1]. (13)

Our problem involves finding the optimal Qi, lead time L, andtotal number of deliveries n for all products in a production cyclethat minimize the integrated expected total cost expressed byEq. (9) and satisfy both the floor space constraint expressed byEq. (10) and the CSL constraint expressed by Eq. (13):

Min IETC(Q , L, n) =

Mi=1

Di(Ai + F + R(L))

Qi

+Di(Awi + Fw + Si)

nQi+

(Qi − Ditc)2piIc2Qi

+ (hbi + piIc)kiσi√L

+hbiQi

2+ Di [diCdi(L) + vi + Pci(Qi) + hwi]

+hviQi

2

n

1 −

Di

Pi

− 1 +

2Di

Pi

−

D2i tc

2siId2Qi

+ IvpitcDi −sitc IdDi

Qiσi

√LΨ (ki) + QidciCdci

Di

Pi+ (n − 1)

−

nDi

2Pi

+

hwiE(αi)nQiDi

rsi(1 − E(αi))2+

Di(sci + vwi − sdiE(αi))

(1 − E(αi))

subject toMi=1

fiQi ≤ W ,

Mi=1

θiQi ≥

Mi=1

σi√LΨ (ki),

(14)

where Q = (Q1,Q2,Q3, . . . ,QM), L ∈ [Lj, Lj−1] and Ψ (ki) =

ϕ(ki) − ki[1 − Φ(ki)] > 0.

5. Solution

The solution to Eq. (14) is complex because of the number ofvariables and constraints in this nonlinear problem. Therefore, weapply a Lagrangian multiplier algorithmic approach. The solutionis discussed in detail for the following cases.

Case 1.In this case, we temporarily ignore the constraints

Mi=1 fiQi

≤ W andM

i=1 θiQi ≥M

i=1 σi√LΨ (ki) and then determine the

Qi, L, and n that minimize the integrated expected total costIETC(Q , L, n).

R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64 59

Initially, for fixed Qi and L ∈ [Lj, Lj−1], we can prove thatIETC(Q , L, n) is a convex function of n. Hence, the search for anoptimal number of lots, n∗, is reduced to finding a local minimum.

Property 1. For fixed Qi and L, IETC(Q , L, n) is convex in n

Proof. Taking the first and second partial derivatives of IETC(Q , L,n) with respect to n, we have

∂ IETC(Q , L, n)∂n

=

Mi=1

−

Di(Awi + Fw + Si)n2Qi

+hviQi

2

1 −

Di

Pi

+ dciQiCcdi

1 −

Di

2Pi

+

hwiE(αi)QiDi

rsi(1 − E(αi))2

(15)

and

∂2IETC(Q , L, n)∂n2

=

Mi=1

2Di(Awi + Fw + Si)

n3Qi

> 0.

Therefore, for fixed Qi and L, IETC(Q , L, n) is convex in n.This completes the proof of Property 1. �

For fixed n, we take the first partial derivative of IETC(Q , L, n)with respect to Qi and L ∈ [Lj, Lj−1], respectively, to obtain

∂ IETC(Q , L, n)∂Qi

= −Di(Ai + F + R(L))

Q 2i

−Di(Awi + Fw + Si)

nQ 2i

+piIc2

−D2i t

2c (piIc − siId)

2Q 2i

+hbi

2+ Diδ0i

+hvi

2

n

1 −

Di

Pi

− 1 +

2Di

Pi

+

sitc IdDiσi√LΨ (ki)

Q 2i

+ dciCdci

Di

Pi+ (n − 1)

−

nDi

2Pi

+

hwiE(αi)nDi

rsi(1 − E(αi))2(16)

and

∂ IETC(Q , L, n)∂L

=

Mi=1

−

DiCeC/L

QiL2+

kiσi(hbi + piIc)

2√L

−sitc IdDiσiΨ (ki)

2Qi√L

+ DidiΓ0i

.

By examining the second-order sufficient conditions (SOSC) forfixed n, it can be verified that IETC(Q , L, n) is a convex functionof Qi and L ∈ [Lj, Lj−1] because the second partial derivatives ofIETC(Q , L, n) with respect to Qi and L ∈ [Lj, Lj−1] are

∂2IETC(Q , L, n)∂Q 2

i=

2Di(Ai + F + R(L))Q 3i

+2Di(Awi + Fw + Si)

nQ 3i

+

DitcDitcpiIc − siId

Ditc + 2σ

√LΨ (ki)

Q 3i

> 0

and

∂2IETC(Q , L, n)∂L2

=

Mi=1

2DiCeC/L

QiL3+

DiC2eC/L

QiL4

+σi(sitc IdDiΨ (ki) − ki(hbi + piIc))

4L3/2

> 0,

respectively, when sitc IdDiΨ (ki) > ki(hbi + piIc).For fixed n and L ∈ [Lj, Lj−1], we obtain optimal Qi (see Eq. (17)

given in Box I) by setting Eq. (16) to zero.Thus, for fixed n and L ∈ [Lj, Lj−1], when all the constraints

are ignored, Eq. (17) gives an optimal value of Qi such that theintegrated expected total cost is minimum. Using the convexitybehavior of the objective function with respect to the decisionvariables, we propose the following algorithm to find optimalvalues of Qi, L, and n.

Algorithm 1. Step 1. Set n = 1.Step 2. For each Lj, j = 0, 1, . . . , b, perform (2.1) and (2.2).

Step 2.1. Compute Qi (i = 1, 2, . . . ,M) from Eq. (17).Step 2.2. Compute the corresponding IETC(Q , Lj, n), where

Q = Q1,Q2, . . . ,QM , by putting Qi in Eq. (9).Step 3. FindMinj=0,1,...,b IETC(Q , Lj, n).Step 4. Set IETC(Q •, L•, n) = Minj=0,1,...,bIETC(Q , Lj, n). Then the

set (Q •, L•) is the optimal solution for fixed n.Step 5. Setn = n+1 and repeat steps 2–4 to obtain IETC(Q •, L•, n).Step 6. If IETC(Q •, L•, n) ≤ IETC(Q •, L•, n − 1), then go to step 5;

otherwise go to step 7.Step 7. Set (Q ∗, L∗, n∗) = (Q •, L•, n − 1). Then set (Q ∗, L∗, n∗) is

the set of optimal solutions.

Case 2. In this case, we consider the floor space constraint andignore CSL constraint. To solve this problem, we optimize thefollowing function by adding a Lagrange multiplier β:

Min IETC(Q , L, n, β) =

Mi=1

Di(Ai + F + R(L))

Qi

+Di(Awi + Fw + Si)

nQi+

(Qi − Ditc)2piIc2Qi

+ (hbi + piIc)kiσi√L

+hbiQi

2+ Di [diCdi(L) + vi + Pci(Qi) + hwi]

+hviQi

2

n

1 −

Di

Pi

− 1 +

2Di

Pi

−

D2i tc

2siId2Qi

+ IvpitcDi −sitc IdDi

Qiσi

√LΨ (ki)

+QidciCdci

Di

Pi+ (n − 1)

−

nDi

2Pi

+

hwiE(αi)nQiDi

rsi(1 − E(αi))2

+Di(sci + vwi − sdiE(αi))

(1 − E(αi))+ β [fiQi − W ]

. (18)

In this case, for fixed n and L ∈ [Lj, Lj−1], the optimal Qi can bedetermined by solvingm+1 equations inm+1 unknown variablesgiven by ∂ IETC(Q ,L,n,β)

∂Qi= 0 and ∂ IETC(Q ,L,n,β)

∂β= 0.

7)

Qi =

2Di

Ai + F + R(L) + (Awi + Fw + Si)/n + Dit2c (piIc − siId)/2 − sitc Idσi

√LΨ (ki)

piIc + hviG0(n) + hbi + 2Di

δ0i +

hwiE(αi)nrsi(1−E(αi))2

+ 2G1(n)

, L ∈ [Lj, Lj−1], (1

where G0(n) =

n

1 −

DiPi

− 1 +

2DiPi

and G1(n) = dciCdci

DiPi

+ (n − 1)

−nDi2Pi

Box I.

60 R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64

9)

0)

Qi =

2Di

Ai + F + R(L) + (Awi + Fw + Si)/n + Dit2c (piIc − siId)/2 − sitc Idσi

√LΨ (ki)

piIc + hviG0(n) + hbi + 2Di

δ0i +

hwiE(αi)nrsi(1−E(αi))2

+ 2(G1(n) + βfi)

, L ∈ [Lj, Lj−1], (1

where the β value can be determined by solving the following equation:

Mi=1

fi

2Di

Ai + F + R(L) + (Awi + Fw + Si)/n + Dit2c (piIc − siId)/2 − sitc Idσi

√LΨ (ki)

piIc + hviG0(n) + hbi + 2Di

δ0i +

hwiE(αi)nrsi(1−E(αi))2

+ 2(G1(n) + βfi)

− W = 0 (2

Box II.

Without loss of generality, we only present the final results asEqs. (19) and (20) given in Box II.

Furthermore, as in the first case, we can prove that IETC(Q , L, n,β) is a convex function of n and L ∈ [Lj, Lj−1]. Using the convexitybehavior of the objective function with respect to the decisionvariables, we propose the following algorithm to find optimalvalues of Qi, L, n, and β .

Algorithm 2. Step 1. Set n = 1.Step 2. For each Lj, j = 0, 1, . . . , b, perform (2.1) to (2.3).

Step 2.1. Calculate the β value solving Eq. (20).Step 2.2. ComputeQi (i = 1, 2, . . . ,M) fromEq. (19) using

the value of β .Step 2.3. Compute the corresponding IETC(Q , Lj, n, β) (Q

= Q1,Q2, . . . ,QM ) by puttingQi andβ in Eq. (18).Step 3. FindMinj=0,1,...,b IETC(Q , Lj, n, β).Step 4. Set IETC(Q •, L•, n, β•) = Minj=0,1,...,bIETC(Q , L, n, β).

Then the set (Q •, L•, β•) is the optimal solution for fixed n.Step 5. Set n = n + 1 and repeat steps 2–4 to obtain IETC(Q •, L•,

n, β•).Step 6. If IETC(Q •, L•, n, β•) ≤ IETC(Q •, L•, n − 1, β•), then go to

step 5; otherwise go to step 7.Step 7. Set (Q ∗, L∗, n∗, β∗) = (Q •, L•, n − 1, β•). Then the set

(Q ∗, L∗, n∗, β∗) is the set of optimal solutions.

Case 3.In this case, we consider the CSL constraint and ignore the floor

space constraint. To solve this problem, we optimize the followingfunction by adding a Lagrange multiplier γ :

Min IETC(Q , L, n, γ ) =

Mi=1

Di(Ai + F + R(L))

Qi

+Di(Awi + Fw + Si)

nQi+

(Qi − Ditc)2piIc2Qi

+ (hbi + piIc)kiσi√L

+hbiQi

2+ Di [diCdi(L) + vi + Pci(Qi) + hwi]

+hviQi

2

n

1 −

Di

Pi

− 1 +

2Di

Pi

−

D2i tc

2siId2Qi

+ IvpitcDi −sitc IdDi

Qiσi

√LΨ (ki)

+QidciCdci

Di

Pi+ (n − 1)

−

nDi

2Pi

+

hwiE(αi)nQiDi

rsi(1 − E(αi))2

+Di(sci + vwi − sdiE(αi))

(1 − E(αi))+ γ

Qiθi + σi

√LΨ (ki)

. (21)

In this case, for fixed n and L ∈ [Lj, Lj−1], the optimal Qi can bedetermined by solvingm+1 equations inm+1 unknown variablesgiven by ∂ IETC(Q ,L,n,γ )

∂Qi= 0 and ∂ IETC(Q ,L,n,γ )

∂γ= 0.

Without loss of generality, we only present the final results asEqs. (22) and (23) given in Box III.

Furthermore, as in the first case, we can prove that IETC(Q , L, n,γ ) is a convex function of n and L ∈ [Lj, Lj−1]. Using the convexitybehavior of the objective function with respect to the decisionvariables, we propose the following algorithm to find optimalvalues for Qi, L, n, and γ .

Algorithm 3. Step 1. Set n = 1.Step 2. For each Lj, j = 0, 1, . . . , b, perform (2.1) to (2.3).

Step 2.1. Calculate γ value by solving Eq. (23).Step 2.2. ComputeQi (i = 1, 2, . . . ,M) fromEq. (22) using

the value of γ .Step 2.3. Compute the corresponding IETC(Q , Lj, n, γ ) (Q

= Q1,Q2, . . . ,QM)byputtingQi andγ in Eq. (21).Step 3. FindMinj=0,1,...,b IETC(Q , Lj, n, γ ).

Step 4. Set IETC(Q •, L•, n, γ •) = Minj=0,1,...,bIETC(Q , Lj, n, γ ).Then the set (Q •, L•, γ •) is the optimal solution for fixedn.

Step 5. Set n = n + 1 and repeat steps 2–4 to obtainIETC(Q •, L•, n, γ •).

Step 6. If IETC(Q •, L•, n, γ •) ≤ IETC(Q •, L•, n− 1, γ •), then go tostep 5; otherwise go to step 7.

Step 7. Set (Q ∗, L∗, n∗, γ ∗) = (Q •, L•, n − 1, γ •). Then the set(Q ∗, L∗, n∗, γ •) is the set of optimal solutions.

2)

3)

Qi =

2Di

Ai + F + R(L) + (Awi + Fw + Si)/n + Dit2c (piIc − siId)/2 − sitc Idσi

√LΨ (ki)

piIc + hviG0(n) + hbi + 2Di

δ0i +

hwiE(αi)nrsi(1−E(αi))2

+ 2(G1(n) + θiγ )

, L ∈ [Lj, Lj−1], (2

where the γ value can be determined by solving the following equation:

Mi=1

2Di

Ai + F + R(L) + (Awi + Fw + Si)/n + Dit2c (piIc − siId)/2 − sitc Idσi

√LΨ (ki)

piIc + hviG0(n) + hbi + 2Di

δ0i +

hwiE(αi)nrsi(1−E(αi))2

+ 2(G1(n) + θiγ )

θi + σi√LΨ (ki)

= 0 (2

Box III.

R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64 61

Case 4.In the final case, we consider both the floor space and CSL

constraints. To solve this problem, we optimize the followingfunction by adding Lagrange multipliers β and γ :

Min IETC(Q , L, n, β, γ ) =

Mi=1

Di(Ai + F + R(L))

Qi

+Di(Awi + Fw + Si)

nQi+

(Qi − Ditc)2piIc2Qi

+ (hbi + piIc)kiσi√L

+hbiQi

2+ Di [diCdi(L) + vi + Pci(Qi) + hwi]

+hviQi

2

n

1 −

Di

Pi

− 1 +

2Di

Pi

−

D2i tc

2siId2Qi

+ IvpitcDi −sitc IdDi

Qiσi

√LΨ (ki)

+QidciCdci

Di

Pi+ (n − 1)

−

nDi

2Pi

+

hwiE(αi)nQiDi

rsi(1 − E(αi))2

+Di(sci + vwi − sdiE(αi))

(1 − E(αi))+ β[fiQi − W ]

+ γQiθi + σi

√LΨ (ki)

. (24)

In this case, for fixed n and L ∈ [Lj, Lj−1], the optimal Qican be determined by solving m + 2 equations in m + 2 un-known variables given by ∂ IETC(Q ,L,n,β,γ )

∂Qi= 0, ∂ IETC(Q ,L,n,β,γ )

∂β=

0 and ∂ IETC(Q ,L,n,β,γ )

∂γ= 0.

Without loss of generality, we only present the final results asEqs. (25)–(27) given in Box IV.

Furthermore, as in the first case, we can prove that IETC(Q , L, n,β, γ ) is a convex function of n and L ∈ [Lj, Lj−1]. Using the convex-ity behavior of the objective function with respect to the decisionvariables, we propose the following algorithm to find optimal val-ues for Qi, L, n, β , and γ .

Algorithm 4. Step 1. Set n = 1.Step 2. For each Lj, j = 0, 1, . . . , b, perform (2.1) to (2.3).

Step 2.1. Calculate β and γ values by solving Eqs. (26) and(27), respectively.

Step 2.2. ComputeQi (i = 1, 2, . . . ,M) fromEq. (25) usingthe values of β and γ .

Step 2.3. Compute the corresponding IETC(Q , Lj, n, β, γ )by putting Qi, β , and γ in Eq. (24).

Step 3. FindMinj=0,1,...,b IETC(Q , Lj, n, β, γ ).

Step 4. Set IETC(Q •, L•, n, β•, γ •) = Minj=0,1,...,bIETC(Q , Lj, n, β,γ ). Then the set (Q •, L•, n, β•, γ •) is the optimal solutionfor fixed n.

Step 5. Set n = n + 1 and repeat steps 2–4 to obtain IETC(Q •, L•,n, β•γ •).

Step 6. If IETC(Q •, L•, n, β•, γ •) ≤ IETC(Q •, L•, n − 1, β•, γ •),then go to step 5; otherwise go to step 7.

Step 7. Set (Q ∗, L∗, n∗, β∗, γ ∗) = (Q •, L•, n−1, β•, γ •). Then theset (Q ∗, L∗, n∗, β∗, γ ∗) is the set of optimal solutions.

5.1. Main computational procedure

When the two constraints are imposed simultaneously, themain computational procedure to solve the problem is as follows.Step 1. Determine the optimal values ignoring both constraintsusing Algorithm1. IfQi satisfies both constraints, then the values ofQi, L, and n obtained are optimal solutions such that the integratedexpected total cost is minimum; go to step 5.Step 2. Else solve the optimization problem subject to the floorspace constraint and ignore the CSL constraint. That is, determinethe optimal values using Algorithm 2. If Qi satisfies the CSL con-straint, then the values of Qi, L, n, and β obtained are optimal solu-tions such that the integrated expected total cost is minimum; goto step 5.Step 3. Else solve the optimization problem subject to the CSLconstraint and ignore the floor space constraint. That is, determinethe optimal values using Algorithm 3. If Qi satisfies the floor spaceconstraint, then the values of Qi, L, n, and γ obtained are optimalsolutions such that the integrated expected total cost is minimum;go to step 5.Step 4. If none of the first three steps is applicable, both con-straints are active. Then solve the optimization problem subject toboth constraints. That is, determine the optimal values using Algo-rithm 4 and obtain the optimal Qi, L, n, β , and γ values such thatthe integrated expected total cost is minimum; go to step 5.Step 5. Stop.

6. Numerical analysis

In this section we conduct a numerical analysis to illustrate theprocedure. Consider a pharmaceutical company–hospital supplychain for three products (isM = 3) that have identical parametersof F = $100, Fw = $400, tc = 0.1 year, Id = 0.02, Iv = 0.02,Ic = 0.06, C = 10, W = 500 square feet, L0 = 5, and Lb = 2. Theother parameters for the hospital and for finished goods and rawmaterials for the pharmaceutical company are listed in Tables 1–3,respectively. In addition, the defect rates α1 and α2 for the first two

5)

6)

7)

Qi =

2Di

Ai + F + R(L) + (Awi + Fw + Si)/n + Dit2c (piIc − siId)/2 − sitc Idσi

√LΨ (ki)

piIc + hviG0(n) + hbi + 2Di

δ0i +

hwiE(αi)nrsi(1−E(αi))2

+ 2(G1(n) + βfi + θiγ )

, L ∈ [Lj, Lj−1], (2

where the β and γ values can be determined by solving the following equations simultaneously:

Mi=1

fi

2Di

Ai + F + R(L) + (Awi + Fw + Si)/n + Dit2c (piIc − siId)/2 − sitc Idσi

√LΨ (ki)

piIc + hviG0(n) + hbi + 2Di

δ0i +

hwiE(αi)nrsi(1−E(αi))2

+ 2(G1(n) + βfi + θiγ )

− W = 0, (2

Mi=1

2Di

Ai + F + R(L) + (Awi + Fw + Si)/n + Dit2c (piIc − siId)/2 − sitc Idσi

√LΨ (ki)

piIc + hviG0(n) + hbi + 2Di

δ0i +

hwiE(αi)nrsi(1−E(αi))2

+ 2(G1(n) + βfi + θiγ )

θi + σi√LΨ (ki)

= 0 (2

Box IV.

62 R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64

Table 1Numerical parameters for the hospital.

Product (i) Di Ai hbi pi si di vi ki σi Γi Γ0i fi θi (1 − θi)

1 600 20 4 5 10 0.3 1 0.5 2 18 0.3 0.3 0.015 98.5%2 800 25 6 15 25 0.15 0.5 0.75 5 20 0.1 1 0.01 99%3 1100 30 3 10 20 0.4 0.8 0.25 4 15 0.2 0.5 0.05 95%

Table 2Numerical parameters for finished products for the pharmaceutical company.

Product(i) Pi Si hvi δi δ0i Cdci dci

1 1000 400 2 3 0.0001 12 0.152 1200 300 1.5 3 0.0005 28 0.13 1800 500 3 2 0.0002 22 0.2

Table 3Numerical parameters for raw materials for the pharmaceutical company.

Raw material (i) Awi sdi sci hwi vwi rsi

1 150 5 0.4 7 0.5 180002 200 6 0.6 9 1.5 200003 100 3 0.2 10 1 10000

products of the pharmaceutical company have a beta distributionwith parameters x0 = 1 and y0 = 4, and x1 = 1 and y1 = 3,respectively.

The p.d.f. for α1 is g(α1) =

4(1 − α1)

3, 0 < α1 < 10 otherwise.

Hence, the mean of α1 is E(α1) =x0

x0+y0= 0.2.

Then the p.d.f. of α2 is g̃(α2) =

3(1 − α2)

2, 0 < α2 < 10 otherwise.

Hence, the mean of α2 is E(α2) =x1

x1+y1= 0.25. The defect rate for

the third product is a constant: α3 = 0.12.If the hospital runs out of a product, it can place an emergency

order with the pharmaceutical company to replenish the product.However, the hospital would like to avoid emergency ordersbecause they are very costly for the organization. For this objectivewe consider the service level θi for each product separately. Weconsider that the service levels required by the hospital are 98.5%,99%, and 95% for the three different products.

6.1. Computation for the numerical example

Calculation of optimal inventory policies for multiple pharma-ceutical products with multiple constraints in a multi-echelon in-ventory system requires efficient solution procedures suitable forlarge-scale inventory systems with short computation time andlow model complexity. We proposed computational algorithmsbased on a Lagrangian multiplier approach. The computational ef-fort and time are small for the proposed algorithm and it is simpleto implement. The algorithms were coded in MATLAB.

We performed computational testing to evaluate the algorithmperformance for our numerical example. Twelve different inte-grated expected total costs were computed using the algorithmsto obtain optimal values. The procedure for the numerical exam-ple as described in Section 5.1 is as follows.First, ignore both constraints and do the following:1. Set n = 1.

Run step 2 of Algorithm 1 to determine the IETC(.) values.(i) For L = 5, we obtain Q1 = 433,Q2 = 383, and Q3 = 531

using Eq. (17).(ii) Substituting the values of Q1, Q2, and Q3 in Eq. (9), we

obtain IETC(.) = $57157.(iii) For L = 4, we obtain Q1 = 436,Q2 = 391, and Q3 = 534

using Eq. (17).

(iv) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $57062.

(v) For L = 3, we obtain Q1 = 439,Q2 = 394, and Q3 = 537using Eq. (17).

(vi) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $56985.

(vii) For L = 2, we obtain Q1 = 462,Q2 = 416, and Q3 = 564using Eq. (17).

(viii) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $57459. Inspection of the solutionsobtained for all the L values reveals that the optimalsolutions for n = 1 are L = 3, Q1 = 439, Q2 = 394,Q3 = 537, and IETC(.) = $56 985

2. Set n = 2.Run step 2 of Algorithm 1 to determine the IETC(.) values.

(i) For L = 5, we obtain Q1 = 263, Q2 = 244, and Q3 = 291using Eq. (17).

(ii) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $56531.

(iii) For L = 4, we obtain Q1 = 266,Q2 = 247, and Q3 = 294using Eq. (17).

(iv) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $56483.

(v) For L = 3, we obtain Q1 = 269,Q2 = 250, and Q3 = 297using Eq. (17).

(vi) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $56465.

(vii) For L = 2, we obtain Q1 = 294,Q2 = 274, and Q3 = 324using Eq. (17).

(viii) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $57351. Inspection of the solutionsobtained for all the L values reveal that the optimalsolutions for n = 2 are L = 3, Q1 = 269, Q2 = 250,Q3 = 297, and IETC(.) = $56 465.

3. Set n = 3.Run step 2 of Algorithm 1 to determine the IETC(.) values.

(i) For L = 5, we obtain Q1 = 195, Q2 = 185, and Q3 = 207using Eq. (17).

(ii) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $56601.

(iii) For L = 4, we obtain Q1 = 198,Q2 = 187, and Q3 = 209using Eq. (17).

(iv) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $56593.

(v) For L = 3, we obtain Q1 = 201,Q2 = 191, and Q3 = 213using Eq. (17).

(vi) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $56627.

(vii) For L = 2, we obtain Q1 = 226,Q2 = 215 and Q3 = 237using Eq. (17).

(viii) Substituting the values of Q1, Q2, and Q3 in Eq. (9), weobtain IETC(.) = $57865. Inspection of the solutionsobtained for all the L values reveals that the optimalsolutions for n = 3 are L = 4, Q1 = 198, Q2 = 187,Q3 = 209, and IETC(.) = $56 593.

R. Uthayakumar, S. Priyan / Operations Research for Health Care 2 (2013) 52–64 63

Table 4Illustration of the solution for given parameters (lead time in days).

n L ∈ [Lj, Lj−1] j = 0, 1, 2, 3j = 0, Lj = 5 j = 1, Lj = 4 j = 2, Lj = 3 j = 3, Lj = 2Q1 Q2 Q3 IETC Q1 Q2 Q3 IETC Q1 Q2 Q3 IETC Q1 Q2 Q3 IETC

1 433 383 531 57157 436 391 534 57062 439 394 537 56985 462 416 564 574592 263 244 291 56531 266 247 294 56483 269 250 297 56465a 294 274 324 573513 195 185 207 56601 198 187 209 56593 201 191 213 56627 226 215 237 57865a Minimum integrated expected total cost.

The integrated expected total cost IETC(Q , L, n) is clearly lowerfor n = 2 than for n = 1 and n = 3, and the algorithm reportsthis as an approximate minimum. In other words, from the abovesolution procedure we can verify that

IETC(Q1,Q2,Q3, L, n = 1) > IETC(Q1,Q2,Q3, L, n = 2)< IETC(Q1,Q2,Q3, L, n = 3).

Hence, we can choose the optimal lot sizes of Q1 = 269, Q2 =

250, and Q3 = 297, a lead time L = 3 and a delivery number ofn = 2 when both constraints are ignored.

Now we consider both floor space and CSL constraints. Then

f1Q1 + f2Q2 + f3Q3 < 500

θ1Q1 + θ2Q2 + θ3Q3 >√L (σ1Ψ (k1) + σ2Ψ (k2) + σ3Ψ (k3)) .

Here, the optimal solutions are not affected by the constraintsas the lot sizes of Q1 = 269, Q2 = 250, and Q3 = 297 satisfy bothconstraints. Suppose that the optimal lot sizes do not satisfy thefloor space constraint and satisfy the CSL constraint and we applythe same procedure to find the optimal solution using Algorithm 2.Similarly, if the optimal lot sizes do not satisfy the CSL constraintand satisfy floor space constraint, then we apply the same proce-dure to find the optimal solution using Algorithm 3. If the optimallot sizes do not satisfy either of the constraints, we apply the sameprocedure to find the optimal solution using Algorithm 4. In thisexample, the optimal lot sizes satisfy both constraints, so the con-straints can be ignored.

Hence, for CSL of 98.5%, 99%, and 95% for three different prod-ucts, the optimal solutions for an integrated supply chain are lotsizes of Q ∗

1 = 269, Q ∗

2 = 250, and Q ∗

3 = 297, respectively, a leadtime of L∗

= 3, and a delivery number of n∗= 2. The minimum

integrated expected total cost is then IETC(.) = $56 465. The solu-tion procedure is illustrated in Table 4.

We can conclude from the numerical results that the hospitaladministrator can achieve CSL targets of 98.5%, 99% and 95% forthree products at $56465 under the conditions considered. Inaddition, computational experiments showed that the algorithmis accurate and efficient.

7. Conclusion

Management of pharmaceutical inventory has become a majorchallenge for health care industries as they simultaneously tryto reduce costs and improve CSL in an increasingly competitivebusiness environment. OR has a long history of successfulapplication for better decision-making in many industrial sectors(e.g. airline, telecommunication, and manufacturing industries).Although health care OR is not a new field, the number of ORapplications and their impact lag behind other service industries.We outlined and discussed an OR model for PSC and inventorymanagement for health care facilities such as pharmaceuticalcompanies and hospitals. We developed a two-echelon PSCinventory model under realistic conditions. We used a Lagrangianmultiplier algorithmic approach to determine the optimal lot size,lead time, and total number of deliveries from a pharmaceuticalcompany to a hospital in a production cycle. This yields a PSC that

achieves a target hospital CSL for a minimum integrated expectedtotal cost.

In practice, pharmaceutical companies carry a huge inventoryto ensure close to 100% CSL. However, ensuring 100% productavailability at an optimal cost is a huge challenge unless supplychain processes are streamlined towards customer needs anddemands. This paper offers decision support tools that improveoperational, health policy, and strategic decision-making for PSCand inventory management. The proposed model can achievethe target CSL at minimum total PSC inventory cost. Our studyimproves current inventory management policy in health care andoffers managerial support via the decision support tool developed.It can be used (i) to maintain medical/pharmaceutical inventorywithout overstocking or expiration and (ii) to achieve a targetCSL at a minimum PSC inventory cost. This approach can be usedin health care industries, especially hospitals, to provide goodmedical services to customers/patients at a minimum inventorycost.

Acknowledgments

This research was supported by the University Grants Commis-sion (UGC-BSR Fellowship and UGC-SAP-DRSII), Government of In-dia.

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