The effect of human factors on the performance of a two level supply chain

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The effect of human factors on the performance of atwo level supply chainM. Khan a , M.Y. Jaber a & A.L. Guiffrida ba Department of Mechanical & Industrial Engineering , Ryerson University , Toronto , ON,M5B 2K3 , Canadab Department of Management and Information Systems , Kent State University , Kent , OH44242 , USAPublished online: 27 Jun 2011.

To cite this article: M. Khan , M.Y. Jaber & A.L. Guiffrida (2012) The effect of human factors on the performance of a twolevel supply chain, International Journal of Production Research, 50:2, 517-533, DOI: 10.1080/00207543.2010.539282

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International Journal of Production ResearchVol. 50, No. 2, 15 January 2012, 517–533

The effect of human factors on the performance of a two level supply chain

M. Khana, M.Y. Jabera* and A.L. Guiffridab

aDepartment of Mechanical & Industrial Engineering, Ryerson University, Toronto,ON, M5B 2K3, Canada; bDepartment of Management and Information Systems,

Kent State University, Kent, OH 44242, USA

(Received 15 July 2010; final version received 24 October 2010)

Many studies have addressed the issue of coordination in a supply chain. Coordinating mechanisms such asjoint lot sizing models, quantity discounts and delay in payments have been used to achieve coordination ina supply chain. An important omission in this literature is the role of human factors, in particular inspectionerrors and learning, as a tool to improve coordination in a supply chain. In this paper, two coordinationmechanisms found in the literature are integrated into a model for a two-level supply chain in which theincoming quality level of raw materials provided to a vendor by a set of suppliers is not perfect. The modeladdresses supply chain coordination by specifically investigating the role of different human factors on thetotal cost of the supply chain. These factors are: (a) type I and type II inspection errors; (b) learning in theproduction process; and (c) learning in quality at the suppliers’ end. Numerical examples are used to comparethe costs of the three extensions with the base model (with no defectives).

Keywords: EOQ/EPQ; imperfect quality; supply chain coordination; logistic learning curve

1. Introduction

The economic order quantity/economic production quantity (EOQ/EPQ) model is perhaps the most popularinventory model among researchers and industries since its inception in 1913 (Simpson 2001). This model can besummarised as determining an order quantity that makes a balance or trade-off between the annual costs ofordering and holding inventory. Despite a wide acceptance, this basic model has several weaknesses due to itsunderlying assumptions, one of which is that all items received from a supplier are non-detective. The quality-inventory relationship initiated a huge arena of research for many in the industry and academics. The literaturecontain a plethora of studies that investigate the EOQ/EPQ model under different situations, e.g., Silver (1976),Porteus (1986), and Rosenblatt and Lee (1986).

In this paper, a two-level (supplier-vendor) supply chain is considered. The vendor (manufacturer) followsan EPQ policy to manufacture a single product. The coordination mechanism is such that: (i) the vendor receives themarket demand and orders for the different components that are needed to make the single product; (ii) everysupplier provides a single type of a part/item required for the product; and (iii) all the suppliers replenish the ordersat the same time, i.e., at the beginning of the vendor’s production cycle. The raw material from the suppliers isassumed to follow the assumptions of Salameh and Jaber (2000) where each shipment contains imperfect qualityitems. These defective items received from the suppliers may be a result of weak process control, deficient plannedmaintenance, inadequate work instructions and/or damage in transit (Ouyang et al. 2006). Two coordinationmechanisms are considered here, as in Khouja (2003). An optimal production quantity and the annual cost of thewhole supply chain are determined for each of the mechanisms. The model is then extended to introduce humanfactors such as inspection error, and learning in production and quality. In the first extension, the screening processis assumed to have type I and type II errors. In the second extension, the vendor’s production process is assumedto follow Wright’s (1936) learning curve, thus affecting the production time. In the third extension, the quality ofsuppliers’ items is assumed to follow learning. An optimal lot size and the annual cost of the supply chain aredetermined for each of these extensions. Thus, the major contributions of this paper can be summarised as:

(1) It brings in the concept of defective items in a supplier-vendor supply chain whereas Huang (2002) andGoyal et al. (2003) have done the same for a vendor-retailer supply chain.

*Corresponding author. Email: [email protected]

ISSN 0020–7543 print/ISSN 1366–588X online

� 2012 Taylor & Francis

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(2) It extends the coordination mechanisms in Khouja (2003) for defective items from suppliers (Salameh andJaber 2000), learning in vendor’s production process (Jaber and Bonney 1999, 2011), inspection error invendor’s screening process (Raouf et al. 1983) and learning in suppliers’ defective items (Jaber et al. 2008).

(3) Results of this research have endorsed Khouja’s (2003) finding for all the extensions, that the integer-multiplier mechanism remains better than the equal cycle time mechanism.

The rest of the paper is organised as follows. A review of the related literature is given in Section 2. In Section 3,a list of the notation and the description and formulation of the models are given. An approximate solutionprocedure to determine the multipliers of the cycle time is also presented in this section. Section 4 describes a numberof extensions pertaining to different human factors. Section 5 presents numerical examples and the sensitivityanalysis of the different parameters used in the models. Section 6 presents conclusions, limitations and somesuggestions for future research.

2. Literature review

In this section we review three areas in the literature that provide a foundation for the two-level supply chain modelpresented in this paper. These areas are: (i) supply chain coordination; (ii) supply chain quality; and (iii) humanfactors in lot sizing. Our intent is to provide the reader with background information that highlights the integrativenature of the supply chain model presented herein.

Various types of coordination mechanisms have been used in supply chain literature such as joint replenishment(Goyal et al. 2003), quantity discount (Munson and Rosenblatt 2001), buy-back and revenue sharing contracts(Cachon 2003), common replenishment epochs (Viswanathan and Piplani 2001), and permissible delay in payments(Jaber and Osman 2006). Readers may refer to the work of Sarmah et al. (2006) and (Jaber and Zolfaghari 2008) for areview. Many researchers have studied the joint replenishment problem as a tool for coordination in supply chainmanagement, e.g., Goyal and Gupta (1989), Sarmah et al. (2006), Ben-Daya et al. (2008), and Jaber et al. (2008).Khouja (2003) derived closed form solutions for the optimal cycle time of a three-level supply chain and illustratedthree mechanisms for coordinating the cycle time. In the first mechanism, the same cycle time is used throughoutthe chain; in the second mechanism, each stage has a cycle time which is an integer-multiplier of that of the adjacentdownstream stage; in the third mechanism, each stage has a cycle time which is an integer-power-of-two multiplier ofthat of the adjacent downstream stage. Khouja concluded that the third mechanism performs better in terms of cost.

Considering quality issues in supply chains became an interest to many researchers and practitioners in recentyears. Several authors have noted the need to link and integrate supply chain management and quality (see forexample, Robinson and Malhotra 2005, Kannan and Tan 2007). Quality as a coordinating mechanism for supplychains can be illustrated both from a modelling perspective and from empirical evidence.

Comeaux and Sarker (2005) presented a model-based coordination strategy for the two-level (manufacturer–retailer) supply chain. They relaxed the assumptions of 100% product quality inspections, zero scrap generation,and no inspection errors. Although the work in this paper touches upon similar aspects of a supply chain there aresome key differences in the two models. That is, Comeaux and Sarker: (i) considered a manufacturer-retailer supplychain while ours is a supplier-vendor one; (ii) accounted for return of the non-conforming products and scrappinga fraction of them while we do not; (iii) did not consider learning in the manufacturer’s production and thesuppliers’ quality while we did; and (iv) accounted for capital investment for quality investment while we do not.

Further model-based contributions on coordination have evolved from the work of Salameh and Jaber (2000)which used a joint replenishment coordination policy. For example, Huang (2002) is believed to be the first toinvestigate the model of Salameh and Jaber (2000) in a two-level (vendor–buyer) supply chain context. Goyal et al.(2003) extended their model for determining an integrated vendor–buyer inventory policy. Chung and Huang (2006)incorporated both Goyal (1985) and Salameh and Jaber (2000) to develop a production/inventory model for aretailer to allow items with imperfect quality under permissible delay in payments.

Sila et al. (2006) examined quality as a coordinating mechanism between manufacturers and suppliers throughthe empirical testing of a set of six hypotheses which addressed the impact of supply chain quality management onproduct quality. The results of the study suggest a need for better supply chain coordination among the firmssurveyed.

An important aspect of the EPQ model is ignoring the role of human factors like inspection errors, fatigue andlearning (both in production and quality). These factors have never been modelled in the context of supply chainmanagement though they play a vital role in measuring the performance of a supply chain.

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Inspection of incoming materials is still used in many industries. Deros et al. (2008) report on the improvementin delivery time and product quality for electrical and electronic products as a result of incoming sampling.Balachandran and Radhakrishnan (2005) utilise incoming inspection in their modelling of the quality implicationsof product warranties in a two-level supply chain. All inspection plans are subject to human error hence the financialconsequences of errors rates are an important facet of modelling quality in supply chains. Raouf et al. (1983)developed a model for determining the optimal number of repeat inspections for multi-characteristic componentswhile there are type I and type II errors in the inspection process. Duffuaa and Khan (2002) extended Raouf et al.’s(1983) model for the case where the inspector classifies an item as non-defective, reworkable and scrap rather thanclassifying them as defective and non-defective only.

Another interesting human aspect that is important in the area of inventory management is learning. This aspectis not new in determining the optimal lot sizes but it has not been given its due place while modelling a supply chainthough the impact of learning effects has been recognised in modelling inspection plans and inventory management.Learning is inherent when there are workers involved in a repetitive type of production process. A review ofinventory models with learning can be found in Jaber and Bonney (2011).

The temporal nature of learning based improvements in quality has been studied by Jaber and Bonney (2003).From the human factors perspective, as workers become more and more accustomed to a manufacturing process,the quality level of the product improves as a result of learning, over time. Using empirical data from the automotiveindustry, Jaber et al. (2008) extended the model of Salameh and Jaber (2000) for the case where the quality level ofsupplier’s raw material improves according to a learning curve.

Salameh and Jaber (2000) exposed a new course of research to the field of supply chain management that ensuresquality of the suppliers’ items. This paper has been receiving more and more attention recently. They assumed that:(i) an EOQ lot contains a certain percentage of defective items, which is a continuous random variable with knownprobability density function; (ii) a 100% inspection of the lot is conducted; (iii) demand occurs parallel to theinspection process and is fulfilled from goods found to be perfect by the inspection process; (iv) there are noshortages; and (v) items of poor quality are kept in stock and sold prior to receiving the next shipment as a singlebatch at a discounted price. Many researchers have extended this model for a number of production situations.A few of these are Goyal and Cardenas-Barron (2002), Goyal et al. (2003), Wang (2005), Papachristos andKonstantaras (2006), Eroglu and Ozdemir (2007), Konstantaras et al. (2007), Wee et al. (2007), Maddah and Jaber(2008), and Khan et al. (2010). Readers may refer to Khan et al. (2011) for a comprehensive review.

3. Model description and development

Consider a two-level supply chain scenario where a vendor (manufacturer) has to make Q (a deterministic quantity)assembled items of a product in each production segment of length Tp at a rate P, which is consumed at a rate D(P4D) over the cycle time T ¼ Q=D. Each of the finished items needs us parts from supplier s, where s¼ 1, 2, . . . ,m.A fixed percentage �s of these parts is believed to be defective. For this, the vendor institutes a 100% inspection andscreens out all the defective items from the lots provided by the suppliers, at a rate of x per unit time. Though wetake the same inspection rate x here, it is quite reasonable to assume different screening rates for a number of partsprovided by the suppliers. The rationale for this is that the parts may have a different level of complexity (Duffuaaand Khan 2005). Since every part is assumed to have a different rate of being defective, the vendor would end upwith some parts left in each cycle that would be utilised in the subsequent cycles.

An optimal production quantity for each mechanism, as described above, will be determined by minimising thetotal costs experienced by all the stakeholders of the supply chain. The costs considered in the model are ordering/setup cost, screening cost, and the inventory carrying cost. The nomenclature used throughout the model will beprovided in this section. The objective of the study is to minimise the total annual cost through: (i) an optimalproduction quantity; and (ii) an optimal multiplier for each supplier.

Figure 1 illustrates the behaviour of inventory for raw material (dotted lines) and finished goods (solid lines).The behaviour of raw material inventory is similar to that described in Salameh and Jaber (2000). Productionand screening start at time zero. The figure represents the inventory of the raw material from two suppliersbut the notation is given for the first supplier only. The inspection process for material received from supplier 1takes time t11, results in a defective sub-lot, B1. As each supplier may have a different percentage of defectives,the vendor may have a number of unused parts l1, left at the end of each cycle. These parts are used in thesubsequent cycles.

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Notation

s a notation representing suppliers, s¼ 1, . . . ,m;Q optimal lot size in a production cycle (a decision variable);Ks integer-multiplier for the second mechanism (a decision variable);T vendor’s cycle time;us number of parts from supplier s required for a product;�s percentage of defective parts supplied by supplier s (a known fraction);

�max the highest percentage of defectives in suppliers’ items;�s the fraction used to accommodate the leftovers before ordering parts of type s, in a cycle

(¼1� ð�max � �sÞ);D demand for the vendor;Qs demand for the supplier s (Qs¼Qus);Z1s inventory level of raw material from supplier s, after finishing its screening;t1s time to screen raw material from supplier s;Z2s inventory level of raw material from supplier s, after screening out the defectives;t2s remaining time in production after screening raw material from supplier s;Bs items screened out (¼Z1s�Z2s);Tp vendor’s total time for production;Td remaining time in the cycle, after production;P vendor’s production rate;c vendor’s production cost per unit time;

T1 vendor’s time to produce the first unit, in case of learning (¼1=P);ys number of non-defective items provided by suppliers s; ys ¼ Qusð1� �sÞ;z minimum number of items that can be produced in a production cycle; z ¼ minðintðQð1� �sÞÞÞ;ls number of unused non-defective parts of type s in a cycle; ls¼ ys� zus;ds unit screening cost for the parts provided by supplier s;x screening rate for the vendor;

Av vendor’s fixed ordering/setup cost;avs vendor’s variable cost of ordering an item from supplier s;c vendor’s production cost per unit time;cf vendor’s cost per unit of making a defective product;As suppliers’ setup cost;

B1

Qu1

Inventory level

Z11

Z21

l1

Timet21t11

Tp Td

T

Figure 1. Vendor’s inventory level for raw material and finished product (two suppliers).

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e1 probability of type I error in the vendor’s screening process (a known fraction);e2 probability of type II error in the vendor’s screening process (a known fraction);Ms percentage of defective parts supplied by supplier s, accommodating the inspection errors at the

vendor’s end (¼ ð1� �sÞ e1 þ �sð1� e2Þ);Mmax the highest percentage of defectives in suppliers’ items, accommodating the inspection errors at the

vendor’s end;�se the fraction used to accommodate the leftovers before ordering parts of type s, in a cycle,

accommodating inspection errors (¼ 1� ðMmax �MsÞ);hv2 vendor’s unit holding cost for finished products;hs suppliers’ unit holding cost.

3.1 Equal cycle time mechanism

A just-in-time (JIT) manufacturing environment advocates that the vendor/suppliers deliver in small batches tominimise the overall holding cost of a supply chain. Following an equal cycle time for all the stakeholders in sucha system, the total annual cost will be computed in this section. It should be noted that the vendor has toaccommodate the minimal number of parts left, ls in each cycle. For simplicity, we relax the integer numberrestriction and compute ls as:

ls ¼ Ds �max � �sð Þ ¼ Qus �max � �sð Þ:

Therefore, the order quantity of the raw material/parts of type s, in a cycle, would be:

Qs ¼ Qus � ls

or

Qs ¼ Qus 1� �max � �sð Þ� �

¼ Qus�s, where �max ¼ max �s, s ¼ 1, 2, . . . ,m� �

: ð1Þ

The fraction �s balances for the leftovers of type s in a cycle. Therefore, the raw material in each cycle wouldconsist of: (i) non-defective parts; (ii) defective parts; and (iii) the leftovers. It should be noted that production andscreening processes, both, start at time zero in each cycle. The defective raw material from supplier s is screened outat time t1s, whereas the inventory level for this raw material drops from Z1s to Z2s, as described in Figure 1. Thedifferent costs of the raw material/parts of type s, for the vendor, in a cycle, are given in the following:

ordering cost ¼ avsQus�s

holding cost ¼ hv1sQs þ Z1sð Þt1s

2þZ2sðTp � t1sÞ

2þ lsT

� �,

where avs and hv1s are respectively the vendor’s unit variable cost for ordering and storing one unit and fromsupplier s. It should be noticed that the leftovers are actually carried for the depletion time (time after production) ina cycle although they remain in the system for time after screening. As defined in the notation, the terms in the aboveexpression are:

t1s ¼Qs

x

Z1s ¼ Qs �Dt1s ¼ Qs 1�D

x

� �and

Z2s ¼ Z1s � �sQs ¼ Qs 1� �s �D

x

� �:

Using Equation (1), it can be simplified as:

holding cost ¼hv1sQ

2

2

u2s�2s 1þ �sð Þ

xþ

us�sP

� 1� �s �

D

x

� �� þ hv1sQus 1� �sð Þ T�

Q

P

� �screening cost ¼ dsQus�s,


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where ds is the cost to screen one item received from supplier s. Thus, the vendor’s total cost of the raw material,in a cycle, would be:

Costvr ¼Xms¼1

avs þ dsð ÞQus�s þhv1sQ

2

2


xþ

us�sP

� 1� �s �

D

x


Q

P

� � �ð2Þ

Similarly, the vendor’s cost of the finished products, in a cycle, is the sum of its setup cost, production cost andholding cost. That is:

Costvf ¼ Av þcQ

Pþhv2QT

21�

D

P

� �,

where Av, c and hv2 are, respectively, the vendor’s setup cost, unit production cost, and the unit holding cost fora finished item. So, the total cost of the vendor, per cycle, can be written as:

Costv ¼ Av þcQ

Pþhv2QT

21�

D

P

� �

þXms¼1

avs þ dsð ÞQus�s þhv1sQ

2

2


xþ

us�sP

� 1� �s �

D

x


Q

P

� � �: ð3Þ

Using T or Q/D as vendor’s cycle time, its annual cost can be written as:

TCv ¼AvD

QþcD

Pþhv2Q

21�

D

P

� �þD

Xms¼1

avs þ dsð Þus�s þhv1sQ

2


xþ

us�sP

� 1� �s �

D

x

� ��

þXms¼1

hv1sQus 1� �sð Þ 1�D

P

� �ð4Þ

Now, for a lot-for-lot (LFL) case, the suppliers have order costs, As, but no carrying costs. That is, they supplyall their raw material at the beginning of each cycle to the vendor. Thus, the annual cost for a supplier s is:

Costs ¼ As:

So, the annual cost of all the suppliers would be:

TCS ¼DPm

s¼1 As

Q: ð5Þ

Thus, the total annual cost of the whole supply chain for an equal cycle time mechanism, would be:

TC ¼D Av þ

Pms¼1 As

� Q

þcD

Pþhv2Q

21�

D

P

� �

þDXms¼1


2


x

�þ

us�sP

� 1� �s �

D

x

� �� þXms¼1


P

� �: ð6Þ

Taking the first derivative of this total annual cost and solving for dTC/dQ¼ 0 gives the following optimalproduction quantity:

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2D Av þ

Pms¼1 As

� hv2 1� D

P

� þD

Pms¼1 hv1s u2s�

2s

1þ�sx

� þ us�s

P 1� �s �Dx

� � �þ 2

Pms¼1 hv1sus 1� �sð Þ 1� D

P

� s

: ð7Þ

3.2 Integer-multiplier cycle time mechanism

Many researchers have found that the equal cycle time mechanism that follows a JIT schedule is not an optimalsolution to the problem considered. Thus, it is now assumed that the cycle time for the suppliers and the vendor is

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not the same but all the suppliers still follow the same cycle time. The suppliers’ cycle time is an integer-multiplier

of the basic cycle time T used by the vendor. As the vendor in this case follows the basic cycle time T, its annual cost

is given by Equation (4). On the other hand, for a supplier, during the non-production time, the inventory drops

every T years by TDs¼TQus. So, its inventory level in the non-production portion of the cycle is

(Ks� 1)TDs, (Ks� 2)TDs, . . . ,TDs, and zero as shown in Figure 2.Therefore, the inventory level for a supplier for its non-production period is KsT½ðKs � 1ÞQushs�=2. The total cost

for a supplier s, in a cycle, can be written as:

Costs ¼ As þ Ks Ks � 1ð ÞhsQusT

2,

where As and hs are, respectively, supplier’s s order cost and unit holding costs. Using a supplier’s cycle time as

KsT, its annual cost would be:

Cs ¼As

KsTþ

Ks � 1ð ÞhsQus2

:

Thus, the total annual cost of all the suppliers for the integer-multiplier mechanism, would be:

TCS ¼

Pms¼1

As

Ks

TþQPm

s¼1 Ks � 1ð Þhsus2

¼DPm

s¼1As

Ks

QþQPm

s¼1 Ks � 1ð Þhsus2

: ð8Þ

It should be noted that the above expression reduces to Equation (5) when Ks¼ 1. That is, the integer-multiplier

mechanism reduces to the LFL policy for an equal cycle time mechanism when Ks¼ 1. Using Equation (4), the

annual cost of the whole supply chain in the case of an integer-multiplier cycle time mechanism would be:

TC ¼D Av þ

Pms¼1

As

Ks

� Q

þcD

PþQ

2hv2 1�

D

P

� �þXms¼1

Ks � 1ð Þhsus

" #þXms¼1


P

� �

þDXms¼1


2


x

�þ

us�sP

� 1� �s �

D

x

� ��#: ð9Þ

Taking the first derivative of this total annual cost and solving for dTC/dQ¼ 0 gives the following optimal

production quantity:

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2D Av þ

Pms¼1

As

Ks

� hv2 1� D

P

� þPm

s¼1 Ks � 1ð Þhsus þDPm

s¼1 hv1s u2s�2s

1þ�sx

� þ us�s

P 1� �s �Dx

� � �þ 2


P

� vuuut :

ð10Þ

(Ks–1)Qus

(Ks–2)Qus

(Ks–3)Qus

(Ks–1)Qus

KsT

Figure 2. Supplier’s inventory level when integer-multiplier Ks¼ 4.


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3.3 Approximation for Ks

In this section, an analytical approach will be used to determine the optimal multipliers for mechanism 2,discussed in Section 3.2. Consider the total cost in Equation (9) for mechanism 2 that is the integer-multiplier Ks

mechanism. Assume for simplicity that Ks is a real number, then an approximate multiplier for supplier s can bedetermined by setting the first derivative of TC in Equation (9) with respect to Ks equal to zero and solving for Ks

to obtain:

Ks ffi1

Q

ffiffiffiffiffiffiffiffiffiffiffiffi2AsD

hsus

s: ð11Þ

Substituting Equation (11) in Equation (9) and simplifying results in:

TC ¼AvD

QþXms¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DAshsus

pþcD

PþQ

2hv2 1�

D

P

� ��Xms¼1

ushs

" #þXms¼1


P

� �

þDXms¼1


2


x

�þ

us�sP

� 1� �s �

D

x

� ��#:

An approximate production quantity, which is independent of the multipliers, would then be:

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AvD

hv2 1� DP

� �Pm

s¼1 ushs þDPm

s¼1 hv1s u2s�2s

1þ�sx

� þ us�s

P 1� �s �Dx

� � �þ 2


P

� s

: ð12Þ

To validate this approximation, 1000 examples were tried. That is, the exact and approximate productionquantities were computed through Equation (10) and Equation (12), respectively. The difference between the costscalculated with these production quantities, was almost zero in the 1000 cases. Thus, a solution procedure for thelast two mechanisms would be as follows:

Step 1: Estimate an approximate production quantity, using Equation (12).

Step 2: Estimate an approximate multiplier for supplier s using Equation (11).

Step 3: Determine integer values of the multipliers as Ksb c and Ksd e

Step 4: Determine an exact production quantity for each combination of the multipliers from Step 3, usingEquation (10).

Step 5: Determine an annual cost using Equation (9), for each combination from Step 3, using Q from Step 4.

Step 6: Determine an optimal annual cost of the supply chain as the minimum of the costs from Step 4. This willindicate the optimal production quantity and the optimal set of multipliers.

4. Model extensions

In this section, a number of extensions as outlined in Section 1 will be discussed for the model in Section 3.

4.1 Type I and type II errors in screening

The screening process in most of the supply chain literature is assumed to be error-free, for example Huang (2002)and Goyal et al. (2003); but it is quite realistic to account for type I and type II errors committed by inspectors inthis process with probabilities e1 and e2, respectively. In this section, it is assumed that the inspectors at the vendor’send commit errors while screening the suppliers’ items. That is, they will classify some non-defective items asdefectives while some defective items as non-defectives. In other words, they will attribute a percentage of defectives

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to each supplier, different from the actual one. Thus, the defective items of type s classified by the inspection process

would be:

Q0s ¼ Qus 1� �sð Þe1 þQus�s 1� e2ð Þ

¼ Qus 1� �sð Þe1 þ �s 1� e2ð Þ� �

¼ QusMs:

Thus, the fraction accommodating the leftovers of type s in a cycle will be given as:

�se ¼ 1� Mmax �Msð Þ,

where Mmax ¼ maxfMs, s ¼ 1, 2, . . . ,mg, and Equation (1) can be written as:

Qs ¼ Qus 1� Mmax �Msð Þ� �

¼ Qus�se: ð13Þ

So, the vendor’s total cost of the raw materials can now be written as:

Costvr ¼Xms¼1

avs þ dsð ÞQus�se þhv1sQ

2

2

u2s�2s 1þMsð Þ

xþ

us�seP

� 1�Ms �

D

x

� �� þ hv1sQus 1� �seð Þ T�

Q

P

� � �:

ð14Þ

The defective raw material misclassified by an inspector ends up making a defective product. This is assumed to

cost the vendor an extra cfe2QPm

s¼1 �s. This may be taken as a goodwill cost or warranty cost. The loss due

to misclassifying non-defective raw material is neglected for simplicity here. The rest of the model remains the same

as in Sections 3.1, 3.2 and 3.3.

4.2 Learning in vendor’s production process

In this section, it is assumed that the vendor’s production process follows Wright’s (1936) learning curve. That is, the

vendor produces the final product at an increasing production rate which is consumed at a constant rate. Let us

assume that Tpi, Tdi and Ti are the production time, depletion time and the cycle time, respectively, in any cycle, as

shown in Figure 3. The process produces a fixed quantity Q and builds up a maximum inventory Zi, in each cycle i.

The level of inventory in each cycle can be expressed as a function of time as:

�iðtÞ ¼QðtÞ �Dt, 05 t5Tpi

DTi �Dt, Tpi 5 t5Ti:

�ð15Þ

Zi

TdiTpi

Ti

Inventory

level

Time

Figure 3. Vendor’s inventory of the final product with learning in ith cycle.


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Let us now assume that b is the learning exponent, while 0� bi5 1 is the learning exponent in cycle i of

production. Faster learning is associated with higher values of b. The production time in a cycle i is

written as:

Tpi ¼

Z iQ

ði�1ÞQ

T1x�bi dx

or

Tpi ¼T1Q

1�bi i1�bi � ði� 1Þ1�bi� �

1� bið Þ, ð16Þ

where T1 ¼ 1=P is the time to assemble (produce) the first unit on the learning curve. To understand the changing

learning rate, assume that the vendor produces x units in one cycle and y units in the next. If T1 and T2 are times

to produce the first unit in the two cycles, the time to produce the yth unit can be written as:

Ty ¼ T1ðxþ yÞ�b ð17Þ

and Ty ¼ T2y�b2 or:

Ty ¼ T1ðxþ 1Þ�by�b2 : ð18Þ

Equating Equations (17) and (18), it can be written as:

b2 ¼b Logðxþ yÞ � Logðxþ 1Þ½ �

Logð yÞ:

In the case of producing a fixed quantity Q in each cycle, the new learning rate can be written as:

bi ¼b Log iQð Þ � Log i� 1ð ÞQþ 1

� �� LogðQÞ

: ð19Þ

Now, the average inventory of finished products in a cycle i can be written as:

Z Ti

0

�iðtÞ dt ¼

Z Tpi

0

Q�Dtð Þ dtþZiTdi

2:

After simplification, it can be written as:

Z Ti

0

�iðtÞ dt ¼Q2

2D�T1Q

2�bi i1�bi � ði� 1Þ2�bi� �1� bið Þ 2� bið Þ

: ð20Þ

So, the vendor’s total cost of the finished products in cycle i would be:

Costvf ¼ Av þcT1Q

1�bi i1�bi � ði� 1Þ1�bi� �1� bið Þ

þ hv2Q2

2D�T1Q

2�bi i1�bi � ði� 1Þ2�bi� �1� bið Þ 2� bið Þ

" #:

Inventory build-up behaves linearly when learning in production is fast. Therefore, using Equation (2), the

vendor’s total cost of the raw material, in cycle i would be:

Costvr ¼Xms¼1

avs þ dsð ÞQus�s þ hv1sQus 1� �sð Þ T� QP

� þ

hv1sQ2

2


xþQ�bius�sT1 i1�bi � ði� 1Þ2�bi

� �1� bið Þ

1� �s �D

x

� �( )2664

3775

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So, the vendor’s total cost in a cycle would be:

Costv¼AvþXms¼1

avsþdsð ÞQus�s

þhv1sQ

2

2

u2s�2s 1þ�sð Þ

xþQ�bius�sT1 i1�bi �ði�1Þ2�bi

� �1�bið Þ

1��s�D

x

� �( )hv1sQus 1��sð Þ T�

Q

P

� �264

375

þcT1Q

1�bi i1�bi �ði�1Þ1�bi� �1�bið Þ

þhv2Q2

2D�T1Q

2�bi i1�bi �ði�1Þ2�bi� �1�bið Þ 2�bið Þ

" #

and the vendor’s annual cost per cycle would be:

TCv ¼AvD

QþD

Xms¼1

hv1sQ

2



� �1� bið Þ

1� �s�D

x

� �( )þ avsþ dsð Þus�s

" #

þXms¼1

hv1sQus 1��sð Þ 1�D

P

� �þcT1DQ�bi i1�bi � ði� 1Þ1�bi

� �1� bið Þ

þ hv2Q

2�T1DQ1�bi i1�bi � ði� 1Þ2�bi

� �1� bið Þ 2� bið Þ

" #: ð21Þ

Using Equation (5), the total annual cost of the supply chain for the equal cycle time can be written as:

TC¼Avþ

Pms¼1As

� D

QþD

Xms¼1

hv1sQ

2



� �1� bið Þ

1� �s�D

x

� �( )

þ avsþ dsð Þus�s

2664

3775

þXms¼1

hv1sQus 1��sð Þ 1�D

P


� �1� bið Þ

þ hv2Q



" #: ð22Þ

An iterative procedure will be carried out to determine the level of learning at which there is no improvement

in the cost. This would indicate the optimal production quantity and the learning rate for the equal cycle time.

Mathematica 6 is used to solve through these expressions. It takes no time to plug these complex expressions and

obtain a solution with this computational software program.Using Equations (8) and (22), the annual cost of the supply chain in a cycle, for the integer-multiplier

mechanism, would be:

TC ¼Av þ

Pms¼1

As

Ks

� D

QþD

Xms¼1

hv1sQ

2



� �1� bið Þ

1� �s �D

x

� �( )

þ avs þ dsð Þus�s

2664

3775

þXms¼1


P


� �1� bið Þ

þ hv2Q



" #þ

QPms¼1

Ks � 1ð Þhsus

2: ð23Þ

For an approximate value of the multipliers in Equation (23), substituting Ks from Equation (11) in Equation

(23), we obtain:

TC ¼AvD

QþXms¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Asushs

p�Qushs2

� �

þDXms¼1


2



� �1� bið Þ

1� �s �D

x

� �( )" #


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þXms¼1


P


� �1� bið Þ

þ hv2Q



" #þ

QPms¼1

2Ms � 1�

hsus

2ð24Þ

Again, an approximate value of the production quantity and the learning rate would be computed throughiterating Equation (24). This will be used to calculate the real-numbered values of multipliers by employingEquation (14). An exact value of the optimal multipliers will be searched by calculating the cost in Equation (9). Theinteger-multipliers would be the values Ksb c, Ksd e, respectively. The minimum for the second policy will be foundthrough Mathematica 6, by plugging these integer values in Equation (23).

4.3 Learning in suppliers’ quality

In this section, it is assumed that the percentage of defectives per lot, from each supplier decreases followinga learning curve. This improved quality may be attributed to the human learning in production and/or inspectionat suppliers’ end. Jaber et al. (2008) discovered this behaviour in the items of an automotive industry. They showedthat the data follows a logistic learning curve of the form:

�ðnÞ ¼a

gþ ebn, ð25Þ

where a and g are the fit parameters while b is the learning rate and n is the number of shipments. Substituting thisin Equation (9), it becomes:

TC ¼D Av þ

Pms¼1

As

Ks

� Q

þcD

PþQ

2hv2 1�

D

P

� �þXms¼1

Ks � 1ð Þhsus

" #þXms¼1


P

� �

þDXms¼1


2

u2s�2s 1þ �sðiÞð Þ

x

�þ

us�sP

� 1� �sðiÞ �

D

x

� ��#ð26Þ

Thus, to find the level suppliers’ quality at which there is no further improvement in the above cost, it will beiterated through a number of shipments. This level of quality (�s) will be used to approximate Equation (12) and theprocedure to find the multipliers will remain the same as in Section 3.2.

5. Numerical analysis

Consider a two-level supply chain with two suppliers and a vendor. The vendor produces a single product andprocures its parts from overseas suppliers. Most of the data is obtained from Khouja (2003) and Salameh and Jaber(2000). The vendor’s production rate is taken to be three times its demand while its unit holding cost of the finalproduct is taken to be much more than that of the raw material. This accounts for the value added during theproduction process. For simplicity of the analysis, it is assumed that all the suppliers have the same setup cost.

The learning curve for the two suppliers is taken to be:

�1ðnÞ ¼40

999þ e0:8nand �2ðnÞ ¼

80

999þ e0:8n,

respectively.

5.1 Effect of human factors

The data and results for the different mechanisms under a number of human factors discussed in this paper are givenin Table 1. A base case for this data is solved to have a benchmark where there are no defective items from thesuppliers, no learning and there is no inspection error.

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It should be noted that for this base case, as one moves from mechanism 1 (equal cycle time) to mechanism 2(integer-multiplier cycle time), both the production quantity per cycle and the annual cost decrease. The rationalefor this is that an increased number of shipments force both the vendor and the suppliers to carry smallerinventories, thus decreasing the overall inventory cost. This indicates that there exists an optimal set of multipliersfor the given data and the overall performance of the supply chain starts deteriorating as we move away from thatset of multipliers. The outcome of this example is that once the multiplier (Ks) goes beyond a certain number, theannual cost starts increasing again. One should observe that mechanism 1 is a special case of mechanism 2, forKs¼ 1.

Khouja (2003) proposed another coordination mechanism, integer-power-of-two or 2Ms, whereKs¼ 2Ms

¼ 1, 2, 4, 8, etc, when Ms¼ 0, 1, 2, 3, etc. Ten thousand data sets for the model input parameters (i.e., As,avs, us, Av, hv1s, hv2, hs, etc) were randomly generated and used to optimise a corresponding number of numericalexamples for the model described in Section 3.2 first by optimising for Ks and then by replacing Ks in Equations (9)and (10) by 2Ms. The results showed that in three out of 10,000 examples, the 2Ms policy was negligibly better, whichis why this policy was not considered in this paper.

This trend remained the same for all the cases (human factors) discussed in the paper, as shown in Table 1. Thatis, the case with: (i) defective items from the suppliers, (ii) vendor’s inspection error, (iii) vendor’s learning inproduction; and (iv) suppliers’ learning in quality. As we include the defectives in our study, it is seen that there isa noticeable drop in the annual cost while the production quantity has a minor change. This indicates that thescreened items at the vendor’s end decrease its carrying cost.

5.2 Sensitivity analysis

An interesting outcome of this example suggests that including the defective items results in the reduction of theannual cost of the supply chain. The rationale for this unusual behaviour is that the leftovers considered in ourexample tend to decrease the order size and thus the annual cost, in case there are some defective items from thesuppliers.

Table 1. Input data and results of the numerical examples.

Av A1 A2 av1 av2 h1 h2 u1 u2 d hv1

200 400 400 1 1 0.5 1 10 3 0.1 2

hv2 �1 �2 x c D P e1 e2 b cf

15 0.008 0.032 175,200 1,000,000 10,000 30,000 0.008 0.0125 0.1 50

Base case (no defective items)

Q TC

Mechanism 1 808 501078Mechanism 2 Ks 3, 4 418 497070

Defective items Inspection error

Q TC Q TC

Mechanism 1 809 498,418 Mechanism 1 809 499,034Mechanism 2 Ks 3, 4 418 494,420 Mechanism 2 Ks 3, 4 418 495,030

Learning in production (vendor) Learning in quality (supplier)

Q TC Q TC

Mechanism 1 924 308,951 Mechanism 1 808 499,248Mechanism 2 Ks 3, 3 462 160,547 Mechanism 2 Ks 3, 4 418 496,902


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To further illustrate the effect of the defective items on the annual cost, the variation of the cost for the firstmechanism, with respect to the percentage of defectives from one supplier is shown in Figure 4. To avoid anyunusual behaviour, this analysis was carried out by avoiding the leftovers. The percentage of defectives from theother supplier was taken to be zero. It can be seen from Figure 4 that the defective items from supplier 1 have someimpact on the annual cost of the supply chain while those from supplier 2 have no effect on this cost. This isattributed to the number of parts from the two suppliers required for the final product. That is, the more the numberof parts, the more is the holding cost of the defective items and so is the annual cost of whole supply chain. Includingthe leftovers would change the shape of these curves. That is, the more the defectives, the more are the leftovers, andless is the quantity and the frequency of orders would be. This tends to decrease the annual cost of the supply chainwhen the fraction of defective items crosses some threshold value. Although this may sound as if having defectiveitems is advantageous, it is not so when it is put in a profit context.

Introducing the inspection error to the model brings in a slight increase in the annual cost. This accounts for thenon-defective parts misclassified. To illustrate the effect of the inspection errors, the variation of the annual costwith these errors, for the first mechanism is drawn in Figure 5. It can be seen that the type II error has a pronouncedeffect as compared to that of the type I error. That is, the defective items that are misclassified cause a major impacton the overall cost of the supply chain. However, if the loss due to misclassification of perfect items is included,it may change the magnitude of the effect of the type I error.

Learning at the vendor’s end brings in the most savings when compared to the base case (about 38%). Therationale for this is that learning decreases the production time and results in a major cut down in the overall cost forthe whole supply chain.

On the other hand, learning in the suppliers’ quality brings the supply chain very close to the state where thereare no defectives from the suppliers. That is why the cost of the chain is very close to that in the base case.

6. Summary and conclusions

In this paper, a two-level, multi-supplier, single-vendor supply chain was formulated. A vendor is supposed to askfor a number of components from different suppliers, which are needed to make a single product. Suppliers arebelieved to be providing a certain fixed percentage of defectives in their supplies. As it is costlier to send the defectiveitems back to the suppliers, the vendor institutes a 100% inspection process and sells the defectives in the localmarket at a discounted price.

Two mechanisms, as in Khouja (2003) were studied for the coordination between suppliers and the vendor. Thefirst mechanism is governed by an equal cycle time for all the stakeholders of the supply chain. In the second

Figure 4. Effect of inspection errors on the cost of supply chain.

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mechanism, each supplier’s cycle time is taken to be an integer-multiplier of the vendor’s cycle time. It was observedthat mechanism 1 is a special case of mechanism 2, for certain values of the multiplier. An approximate solutionmethod was provided to attain the multipliers in the second mechanism. This approximation was validated throughnumerical examples and was found to work well. The third mechanism in Khouja (2003) was ignored in this researchas it rarely results in better results than those of the second policy. The numerical examples presented showed thatthe integer-multipliers mechanism behaves better than the equal cycle time mechanism. That is, the suppliers aresupposed to follow a relaxed and practical approach of the integer-multipliers cycle time rather than forcingthemselves to follow an equal cycle time. The rationale is that after a certain level of multiples of the cycle time, thetotal cost of the supply chain starts rising up. Besides, the integer-multiplier presents a practical scenario forcoordination in a supply chain.

A number of human factors were brought into the picture in this paper. First of all, a scenario was considered inwhich the inspectors at the vendor’s end make misclassifications. This factor was shown to reduce the carrying costof the vendor and thus the overall annual cost of the supply chain. Next, the production process of the vendor wasassumed to follow learning as workers tend to perform the same job at a faster pace. This brought in a substantialdrop in the annual cost of the supply chain. Lastly, the quality of the suppliers’ items was assumed to follow alogistic learning curve. This effect resulted in a situation as if there are no defectives from the suppliers.

There are several aspects of this research that could be extended. First, the model could be extended byincorporating a stochastic percentage of defectives in the incoming quality of raw materials. Second, the scope of themodel could be expanded to investigate learning in inspection errors. Third, the impact of transportation cost on thesize and number of deliveries per cycle can be studied. Lastly, one could also study the effects of a probabilisticdemand from the vendor in response to the market’s behaviour. Each of the above model adaptations would furtherenhance the supply chain coordination problem from different perspectives.

Acknowledgements

M. Khan and M.Y. Jaber thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for the financialsupport. The reviewers’ comments have also been of great help in improving the quality and utility of the paper.

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Figure 5. Effect of inspection errors on the cost of supply chain.


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