International Journal of Production Research
ISSN: 0020-7543 (Print) 1366-588X (Online) Journal homepage: http://www.tandfonline.com/loi/tprs20
Joint optimisation of tracking capability and price
in a supply chain with endogenous pricing
James B. Dai, Lei Fan, Neville K.S. Lee & Jianbin Li
To cite this article: James B. Dai, Lei Fan, Neville K.S. Lee & Jianbin Li (2017): Joint optimisation
of tracking capability and price in a supply chain with endogenous pricing, International Journal of
Production Research, DOI: 10.1080/00207543.2017.1321800
To link to this article: http://dx.doi.org/10.1080/00207543.2017.1321800
Published online: 05 May 2017.
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Date: 08 May 2017, At: 01:06
International Journal of Production Research, 2017
https://doi.org/10.1080/00207543.2017.1321800
Joint optimisation of tracking capability and price in a supply chain with endogenous pricing
James B. Daia , Lei Fana , Neville K.S. Leeb and Jianbin Lic∗
a Economics and Management School, Wuhan University, Wuhan, P.R. China; b Department of Industrial Engineering and Logistics
Management, Hong Kong University of Science and Technology, Kowloon, Hong Kong; c School of Management, Huazhong University
of Science and Technology, Wuhan, P.R. China
(Received 28 March 2016; accepted 2 April 2017)
Tracking systems have been widely used to resolve the issues of product recall and food safety. Thus far, few researches
have been done on designing the tracking capability from the perspective of supply chain. In this paper, using the traceable
unit size at the manufacturer level to measure the tracking capability, we propose a non-convex non-linear programming
to jointly optimise the tracking capability and price considering the tracking cost and recall cost in a supply chain with
endogenous pricing. Results show that, in both centralised and decentralised supply chains, there is a unique tracking
capability and retailing/wholesale price with closed-form solutions to optimise the supply chain profit. When the cost ratio
(unit tracking cost/unit recall cost) is sufficiently large and small, the optimal tracking strategy is barcode tracking and unit
tracking, respectively, and otherwise, the optimal tracking strategy is batch tracking with an economic traceable unit size
which depends on the cost ratio, quality inspection threshold, supply defection rate and the supplier’s tracking capability.
Furthermore, in the context of large and small cost ratio, we find that improving tracking capability will enlarge and mitigate
the effect of double marginalisation, respectively. In particular, we find that the strict tracking regulation policy is more robust
than the subsidy policy to improve the supply chain tracking capability.
Keywords: design; optimisation; pricing theory; tracking capability; product recall
1. Introduction
As companies have increased global supply chains at the cost of product reliability to retain competitiveness, product recall
is becoming a fairly challenging issue in supply chain management, particularly in the food and pharmaceutical supply
chains. In 1997, Hudson Foods recalled 25 million pounds of beef products due to bacterial contamination, which is the
largest ground beef recall in US history (Loader and Hobbs 1999). In the early 2000s, the drug manufacturer, Merck (MRK)
recalled arthritis medication Vioxx due to increased risk of heart attacks, which cost Merck $4.85 billion in settled claims
and lawsuits. On 30 April 2010, a subsidiary of Johnson & Johnson, McNeil Consumer Healthcare, announced recalls of
43 over-the-counter children’s medicines in 12 countries (ALJAZEERA 2015). In 2014, the US Consumer Product Safety
Commission announced 313 recalls of toys, furniture, fitness equipments and baby food makers. From 2007 to 2014, Mattel
announced 33 recalls of toys and games due to high lead levels and possibility of causing choking and other hazards. These
aforementioned recalls not only damage the company’s reputation, but also incur significant financial loss, such as the cost
of replacing and fixing defective products.
The essential of product recall is to responsively identify the risky origin and then execute a complete recall of products
which cannot be clearly identified as safe. Thus far, the traceability system is considered as one of the most promising solutions
to alleviate or resolve recall cost (Huang et al. 2012). The underlying reason is that the traceability system has the following
potential benefits. First, the traceability system makes it possible to retrieve pedigree information to quickly identify the risky
origin or to allocate the recall liability in a fair way. Second, given a risk identified by national authorities or food businesses,
the traceability system can be used to trace the product upward the supply chain in order to isolate the pollution and prevent
contaminated products from reaching consumers. Third, the traceability system can reduce the recall cost by isolating the
recall of safe products. Fourth, improved traceability could protect the reputation of industries by limiting the size of recalls
(Pouliot and Sumner 2013). Furthermore, it is possible to improve supply chain efficiency through integration of traceability
with operations management functions. In this context, supply chain traceability has become a legal obligation within the
EU since 1 January 2005, and it has attracted considerable attention in other advanced economies as well, such as the US
consumer product safety law(Public law No.110-314), and the online food traceability system SEICA in Japan (Lee and
Park 2008). Thus far, China has initiated a traceability regulation as well, for instance, the announcement of new food safety
∗ Corresponding author. Email: jbli@hust.edu.cn
© 2017 Informa UK Limited, trading as Taylor & Francis Group
2
J.B. Dai et al.
law in 2015 which requires traceability throughout the entire supply chain, and the open of national food (product) safety
traceability platform. Therefore, it is interesting to investigate the design of traceability system in supply chains, particularly
the optimisation of tracking capability, which mainly determines the performance of traceability systems.
Supply chain traceability is defined as the ability to trace the history, application or location of an entity (traceable
unit) by means of recorded identifications throughout the supply chain (Nguyen 2004). Thus far, existing literatures focused
mainly on the traceability system architecture design, for instance, the design of modules and structures, and the associated
identification, coding and data access systems for various industries, such as cereal and oil (Liu et al. 2009), pork (Zhao,
Teng, and Wang 2009), poultry (Cebeci et al. 2009) and aircraft (Ngai et al. 2007). To facilitate data acquisition, RFID is
used for tracking processes and product recall in a meat supply chain (Hu et al. 2009), and web-based systems for data
processing are used in food supply chains (Ruiz-garcia, Steinberger, and Rothmund 2010). A design approach that adapts
the Axiomatic design method for supply chain traceability system with aligned interests is proposed by Dai, Ge, and Zhou
(2015b). However, these works do not consider economic analysis in evaluating the traceability system.
Economic analysis, as one of the key aspects in evaluating the performance of a traceability system, is important in
designing tracking capability. Dessureault (2006) assessed the benefits and costs of a traceability system based on an empirical
survey of Canadian dairy processing firms. Lindley (2007) investigated the trade-off between recall cost and tracking cost
in meat plants by a simulation model. Pouliot (2008) developed an analytical model to explore the relationship between
traceability and profit based on the willingness to pay for traceability. Fritz and Schiefer (2009) developed a decision table
with relative benefits achieved through reduction in recall costs as well as the costs associated with different tracking
systems. Kumar and Schmitz (2011) analysed the management of recalls in a consumer products supply chain, as well as the
reasons, costs and measures to prevent recalls by a Six Sigma DMAIC methodology. Segura Velandia, Kaur, and Whittow
(2016) investigated the feasibility of implementing an RFID system for manufacturing and assembly of crankshafts. Based
on comparison of various tracking technologies, Appelhanz et al. (2016) proposed a traceability information system for
capturing, processing and provision of product information in the industry of wood furniture, and employed a cost-benefit
model to investigate the system feasibility. Although these aforementioned works attempt to investigate the effect of tracking
capability on reducing recall cost, they do not specify the explicit relationship between tracking capability and recall cost
or supply chain performance. In this context, practitioners have difficulty in deciding the tracking capability to adopt a
traceability system. Therefore, the main objective of this paper is to quantify the relationship between tracking capability
and recall cost, and then optimise the tracking capability and price by maximising the supply chain performance.
Tracking capability mainly determines the performance of a traceability system (Dabbenea and Gayb 2011), and the
optimisation of tracking capability has attracted intensive research. Dupuy, Botta-Genoulaz, and Guinet (2005) proposed a
batch dispersion model based on a mixed-integer linear programming (MILP) to optimise the batch size and batch mixing
in a food industry. Using the concept of chain dispersion, Rong and Grunow (2010) developed a production and distribution
planning model for food supply chains using MILP with heuristic solutions for decision makers with various risk attitudes.
Thakur, Wang, and Hurburgh (2010) investigated a bulk grain blending problem of balancing total cost and traceability in
bulk grain handling, and proposed a multi-objective mixed-integer programming (MIP) model to minimise traceability effort
by minimising the food safety risk caused by lot aggregation at a grain elevator. Wang, Li, and Brien (2009) optimised the
production batch size and batch dispersion policy for perishable food. Piramuthu, Farahani, and Grunow (2013) studied the
recall dynamics in a three-stage perishable food supply network through three different visibility levels in the presence of
contamination based on the accuracy with which the contamination source is identified. They focused on studying several
different levels of granularity at which contamination could be traced back to its origin and the stage at which recall should
occur in a supply chain consisting of suppliers, manufacturers and retailers. Fan et al. (2015) investigated the impact of RFID
technology on supply chain decisions. Saak (2016) studied the questions of why and when a supply chain should invest in a
traceability system, and identified the context when perfect traceability is not optimal. In summary, most existing literatures
optimised the tracking capability mainly based on the dispersion without considering the benefit of traceability in reducing
recall cost and the cost of improving traceability. More importantly, they do not quantify the relationship between the recall
cost and tracking capability, and the relationship between the contamination and recall cost.
The research closet to our study is Dai, Tseng, and Zipkin (2015), which designed the tracking capability in terms of
item level, batch level, and barcode level based on given technology cost, recall cost and retailing price in a two-stage
supply chain consisting of a manufacturer and two suppliers. Furthermore, they proposed an interest-sharing mechanism
in designing supply chain traceability systems for better interest alignment for different supply chain members based on a
proposed liability calculation model which quantifies the expected total liability under different design parameters. However,
in practice, the retailing price could be endogenous, and recall cost depends on the amount of products put into market
by the manufacturer, the traceable unit size of finished products and raw materials and the quality inspection requirements
of raw materials. More importantly, they only focused on the tracking capability of item level, batch level and barcode
level, rather than a level of granularity that is in-between item and barcode levels. Aiello, Enea, and Muriana (2015) used
International Journal of Production Research
3
numerical analysis to identify the economical traceability unit size which optimises the supply chain profit based on the
expected value of the implementation of traceability systems for perishable products like fruits and vegetables. However,
they did not consider the contamination from the supplier and the interactions of tracking capability optimisation and supply
chain decisions. This motivates us to study the joint optimisation of tracking capability in terms of traceable unit size and
price considering the tracking cost and recall cost in a supply chain with endogenous pricing. Our work distinguishes itself
from existing literatures in the following aspects: (i) both tracking capability and price are jointly optimised in a supply
chain; (ii) tracking and pricing strategies considering unit tracking cost, unit recall cost, suppliers’ tracking capability and the
manufacturer’s quality inspection threshold are identified; (iii) rather by an exogenous recall probability, an explicit recall
cost model incorporating both the supplier’s and manufacturer’s tracking capability is formulated by combination theory;
(iv) the interaction between the tracking capability optimisation and supply chain decisions is investigated, such as the effect
of problem parameters on the effect of double margination.
The remainder of this article is organised as follows. Section 2 describes the formulation of the joint optimisation
problem in both centralised and decentralised supply chains. Section 3 obtains the explicit solution of the aforementioned
joint optimisation problem, and presents the optimal tracking and pricing strategies. Section 4 analyses the effect of problem
parameters on the optimal tracking capability, and the impact of tracking capability and problem parameters on pricing
strategies and supply chain profit. Finally, we conclude this paper in Section 5.
2. Problem formulation
Food safety and product recall cost mainly depend on the supply chain tracking capability, and one of the main measures
characterising the tracking capability is the traceable unit size, which refers to a batch of traceable products (Aiello, Enea,
and Muriana 2015). Thus far, there are two commonly used traceable units: unit product by RFID tracking and batch of
products by barcode tracking. The traceable unit based on a unit product is commonly used in livestock industry, while the
one based on batch of products is commonly used in dairy industry, agriculture, electronics and automotive industry. In this
paper, we consider the tracking capability in terms of the traceable unit based on batch of products. It is easy to understand
that the smaller the traceable unit size, the better the tracking capability. However, higher tracking capability results in a
larger number of traceable units, which induces a higher tracking cost. Therefore, it is fairly interesting to investigate the
optimal traceable unit size or tracking capability to balance tracking cost and recall cost to maximise the supply chain profit,
or to minimise the negative impact of defects on customers as well. Naturally, supply chain members are reluctant to adopt
tracking system when there is no regulation on supply chain traceability or the relationship between recall liability and supply
chain traceability is unclear. For instance, the 2008 recall of peanut butter in the USA due to the presence of salmonella. It
was the largest product recall ever in the history of the country, involving more than 200 food manufacturers downstream
in the supply chain resulting in the recall of more than 2100 products (Terreri 2009). Another example is that , in the US in
1997, Hudson Foods, a supplier of burgers to Burger King, recalled 25 million pounds of hamburger meat due to Escherichia
coli contamination. The company was not able to determine the origin of the outbreak (suspected to be one of its suppliers)
and eventually went out of business (Loader and Hobbs 1999).
In this study, an automated traceability system based on batch of products is used in a supply chain consisting of a supplier,
a manufacturer, a retailer and customers. The supply chain is assumed to face product recalls due to defective components from
contaminated suppliers only, but no defection from the manufacturer. We consider a risk-neutral monopolist manufacturer
ordering raw materials from a single supplier to be used in its production, and then sell its products through a single retailer
to end customers. Some raw materials are exposed to risks of defects, and the manufacturer is using quality inspection to
reduce the supply defection rate of raw materials by rejecting some traceable units with many defects. Other raw materials
in conforming traceable units will be shipped from the supplier to the manufacturer. Subsequently, the manufacturer equally
consume a traceable unit of raw materials into several traceable units of finished products. Then products are delivered to
the retailer according to orders made by the retailer, and finally they are shipped to end customers. Therefore, traceable unit
size of raw materials and products are different. We assume all defects in the finished products contaminated by the defective
raw materials will be detected by end customers. Once there exist defects in end customers, the manufacture is responsible
for recalling the entire traceable unit which contains defective products. The configuration of the investigated supply chain
is presented by Figure 1.
We assume that there exists x items of defects in each traceable unit with a size of n units of raw materials, and x follows a
Poisson distribution as the defection rate is fairly small in practice. When the number of defects in a traceable unit is larger than
t, raw materials in this traceable unit will be rejected, and we denote Z the probability of rejecting the incoming traceable
units of raw materials. Denote the traceable unit size of finished products by m, and for easy of calculation, we assume
n = zm, where z is a positive integer number. This indicates that a batch of raw materials will be used to produce multiple
batches of finished products, and this is very common in many industries, such as the food industry and pharmaceutical
4
J.B. Dai et al.
Figure 1. Flow diagram with quality inspection and product recall.
industry. Furthermore, we denote Cr , Ct and Ctr the unit recall cost of finished products, inspection cost of a traceable unit
and tracking cost of a traceable unit, respectively. We treat q as the sales volume and p to be the sales price of product.
We assume that with an output of q ∈ [0, 2a], the market clears at price a − q/2. This formulation are widely used in the
existing literatures like (Hu and Kostamis 2015). Here, a is a proxy for the market size. Throughout this study we slightly
abuse terminology and refer to a as just ‘market size’. In addition, we refer ws and wm to be unit wholesale price of raw
materials and finished products, respectively.
In the following, we will formulate the joint optimisation problem in both centralised and decentralised supply chains.
The motivations to investigate both supply chains composes the following aspects: (i) both supply chains are common in
practice, and we aim to provide joint optimisation strategies of both tracking capability and price for both supply chains;
(ii) to compare the optimisation strategies in both supply chains in order to explore the impact of integrating supply chain;
(iii) to investigate the effect of problem parameters and tracking capability on the effect of double marginalisation.
2.1 Centralised supply chain
In the centralised supply chain, the centralised firm purchases raw materials from the supplier to produce products, and then
sell them to end customers. The revenue of the firm is given by pq, where p is the retailing price, and q is the quantity of
products produced and sold. The cost of the firm composes four parts: the purchasing cost of raw materials ws Q(1 − Z ),
where Q(1 − Z ) is the quantity of raw materials that have passed the quality inspection test with a batch rejection probability
of Z ; the quality inspection cost Ct Qn , where Ct is the quality inspection cost per traceable unit of raw materials, n is the batch
size of raw materials and Q is the total quantity of raw materials ordered by the firm; the total tracking cost is Ctr mq , where Ctr
is the unit tracking cost, m is the traceable unit size of finished products and mq is the average number of traceable units need
to be tracked; the recall cost is Cr q f (m, n, λ, t), where f (m, n, λ, t) is the expected product recall rate contaminated by the
defects in raw materials which has passed the quality inspection test. Therefore, the expected profit function of the centralised
firm is given by pq − ws Q(1 − Z ) − Ct Qn − Ctr mq − Cr q f (m, n, λ, t). Since we only consider a one-period problem, by
assuming a raw material savage value of 0, obviously the centralised firm should make use of all raw materials in producing
products to satisfy the demand, which indicates the available quantity of raw materials passed the quality inspection Q(1− Z )
should be equal to demand of finished products q. Therefore, in this problem, the centralised firm optimises her profit by
setting a proper selling price p and tracking capability m, particularly to tradeoff the tracking cost and recall cost. Below is
the formulation of joint optimisation of tracking capability and price in the centralised supply chain:
q
q
− Ctr − Cr q f (m, n, λ, t)
max E(π ) = pq − ws q − Ct
(1)
p,m
n(1 − Z )
m
q
s.t. p − a + = 0,
2
n − m ≥ 0,
0 ≤ Z ≤ 1,
p ≥ 0, m ≥ 0, q ≥ 0, Ct ≥ 0, Ctr ≥ 0, Cr ≥ 0.
(2)
In the centralised firm, since the price is endogenous, q is related to p, and therefore, the optimal price depends on the tracking
capability at both the supplier level and the manufacturer level, and all other problem parameters. However, considering
the optimisation of tracking capability m, the essential of the problem is to tradeoff the tracking cost and recall cost. The
first constraint states the relationship between the quantity of products sold and the market price, which is widely used
in literatures, such as Hu and Kostamis (2015). The second constraint ensures that the batch size of finished products is
smaller than that of raw materials, meaning that a batch of raw materials will be used to produce multiple batches of finished
International Journal of Production Research
5
products, and this is common in most manufacturing companies, particularly in the food and pharmaceutical industry. The
third constraint ensures that materials from selected suppliers can only be partially rejected by the manufacturer’s quality
inspection policy, and otherwise, the supplier will not be selected. All other constraints stem from the definition of problem
parameters.
2.2 Decentralised supply chain
In the decentralised supply chain, the sequence of decision-making is that: the manufacturer first optimises her profit by setting
a proper order quantity Q, tracking capability m and wholesale price wm , and then the retailer based on the wholesale price
wm decides the optimal retailing price to achieve the maximal profit. To resolve the joint optimisation of tracking capability
and price, the retailer problem should be optimised before the manufacturer problem. Considering the retailer problem, the
profit of the retailer is given by ( p − wm )q, where p − a − q/2 = 0, it is easy to show that the optimal retailing price is given
m
by a+w
2 , and therefore, the optimal order quantity of the retailer is a − wm . Denote the output quantity of the manufacturer
by qm , similar to that in the centralised supply chain, the available quantity of raw materials passed the quality inspection
Q(1− Z ) should be equal to the output quantity of the manufacturer qm . The revenue of the manufacturer is determined by the
wholesale price wm , the output of the manufacturer qm and the optimal order quantity of the retailer a − wm . Therefore, the
qm
qm
profit function of the manufacturer is given by wm min(a −wm , qm )−ws qm −Ct n(1−Z
) −C tr m −Cr qm f (m, n, λ, t). When
qm ≥ a − wm , obviously, the profit function of the manufacturer is a decreasing function of qm , and therefore, qm ≤ a − wm .
Without loss of generality, when the output quantity qm ≤ a − wm , the manufacturer can decide a sufficient large wholesale
Ct
Ctr
price wm to ensure wm − ws − n(1−Z
) − m − Cr f (m, n, λ, t) > 0, which indicates that the profit of the manufacturer is
an increasing function of qm , and therefore, the optimal output quantity should be set as qm = a − wm . Therefore, similar
to the formulation in the centralised supply chain, the formulation of joint optimisation problem in the decentralised supply
chain is given by:
max E(πm ) = (wm − ws )qm − Ct
wm ,m
s.t.
qm − a + wm = 0,
qm
qm
− Ctr
− Cr qm f (m, n, λ, t)
n(1 − Z )
m
(3)
wm − ws ≥ 0,
n − m ≥ 0,
0 ≤ Z ≤ 1,
ws ≥ 0, m ≥ 0, qm ≥ 0, Ct ≥ 0, Ctr ≥ 0, Cr ≥ 0.
(4)
Similarly to that in the centralised supply chain, the optimisation of the manufacturer’s wholesale price in the decentralised
supply chain depends on the tracking capability of both the supplier and the manufacturer, and all other problem parameters
as well. The essential of the tracking capability optimisation is to tradeoff the tracking cost and recall cost. Therefore, we
expect that both centralised and decentralised supply chains will have the same optimal tracking capability. Moreover, the
recall cost of the supply chain depends on not only the manufacturer’s tracking capability, but also the supplier’s tracking
capability. However, in practice, most of the time it is the manufacturer who is responsible for product recall, and this explains
why the risky origin of most product recall events come from the supplier, for instance, the recall of meat by Hudson Foods
in 1997, the recall of toys by Mattel from 2007 to 2014. The first constraint in the problem states the relationship between the
quantity of products sold and the manufacturer’s wholesale price, which is based on the optimisation of the retailer problem
using the Stackelberg game, and all other constraints are exactly the same as that in the centralised supply chain.
3. Problem analysis
Instead of the MILP model proposed by Dupuy, Botta-Genoulaz, and Guinet (2005) and numerical analysis methodology
proposed by Dai, Tseng, and Zipkin (2015) and Aiello, Enea, and Muriana (2015) to identify the economic traceable unit
size, we propose a non-linear programming model to jointly optimise the tracking capability and price. The significance
of the proposed model lies in three aspects. First, rather than an exogenous recall probability, an explicit recall cost model
incorporating tracking capability of both the supplier and the manufacturer is formulated by combination theory. Second,
the proposed model can investigate the effect of supply chain tracking capability, but not only the tracking capability of the
manufacturer, on supply chain decisions. Third, even the nonlinear programming model we proposed is non-convex, we have
obtained explicit solutions for optimal tracking and pricing strategies under various situations, which is much easier to use
than numerical analysis results.
6
J.B. Dai et al.
3.1 Expected product recall rate
In this study, we only consider the product recall induced by the contamination from the supplier, meaning that defective raw
materials used in the production will contaminate the finish products and induce product recall (Dai, Tseng, and Zipkin 2015).
Therefore, quality inspection of raw materials is commonly used to reduce the product recall rate. However, in practice, most
of the time it is not economical to use full inspection on incoming raw materials, furthermore, full inspection still cannot
guarantee zero defection in the case of perishable raw materials. Therefore, sampling inspection is widely used to control the
defection of incoming raw materials. According to Aiello, Enea, and Muriana (2015), we assume the defects of raw materials
in each traceable unit follows a Poisson process and denote the average defection rate by λ, the probability of an incoming
traceable unit of raw materials with k defects before inspection is given by:
λk e−λ
, k = 0, 1, . . . , n.
k!
Pb (k) =
(5)
Considering a sampling inspection policy in which a traceable unit of raw materials with more than t detected defects will
be rejected, the probability of k defects in a traceable unit of raw materials that has passed the quality inspection is given by:
λk /k!
, k = 0, 1, 2, . . . , t.
j
j=0 λ /j!
Pa (k) = t
(6)
As each traceable unit of raw materials will be used to produce z = n/m traceable units of finished products, consider a
traceable unit of raw materials with k defects used in the production, k defects will be randomly flowed into z traceable units
of finished products and then contaminate finished products. If there exists defects in a traceable unit of finished products, the
entire traceable unit of finished products will be recalled by a conservative recall plan which is widely used in practice (Dai,
Tseng, and Zipkin 2015). Therefore, given a batch of
raw materials used with k defects, the number of traceable units recalled
equals to number of positive solutions in equation zj=1 x j = k, where x j is a non-negative integer number. Considering
this random allocation problem, denote the number of traceable units recalled by a random variable y, the probability of
y = j traceable units recalled is given by:
z k−1
j
j−1
Pr ( j) = z+k−1 , j = 1, 2, . . . , z.
(7)
k
Therefore, the expected number of traceable units recalled conditioning on k defects in a traceable unit of raw materials is
given by:
k−1
z k−1
z
z zj=1 z−1
j=1 j j j−1
j−1 j−1
E(y|k) =
=
.
(8)
z+k−1
z+k−1
k
z−1 k−1
k
Based on Vandermondes Convolution Formula: j=1 j−1 j−1 = z+k−2
k−1 . Therefore, the expected number of traceable
units recalled conditioning on using raw materials which has k defects in each traceable unit can be reformulated into:
z z+k−2
kn
k−1
.
(9)
E(y|k) = z+k−1 =
(k − 1)m + n
z
k
Therefore, in the sampling inspection threshold of t defects for the incoming raw materials
with a defection rate of λ, the
expected number of traceable units recalled per batch of raw materials is determined by tk=1 Pa (k)E(y|k), and therefore,
the expected product recall rate per unit of raw materials is given by:
f (m, n, λ, t) =
t
k=1
km
λk /k!
.
t
(k − 1)m + n j=0 λ j /j!
(10)
According to the aforementioned formula, the product recall cost not only depends on the tracking capability of the
manufacturer, but also the tracking capability of the supplier. Therefore, it is interesting to investigate the tracking capability
optimisation from the supply chain perspective.
3.2 Joint optimisation of the tracking capability and price
Using the formulation of the expected product recall rate, we show that the second derivative of the objective function can
be either positive or negative, which indicates that the joint optimisation of tracking capability and price in either centralised
International Journal of Production Research
7
or decentralised supply chain is a non-convex program. In this subsection, we will depict how the joint optimisation problem
be addressed to obtain the closed-form solution. Below is a theorem that states the optimal tracking capability and price in
the centralised supply chain. Unless otherwise noted, all proofs are in Appendix.
Theorem 1 In the centralised supply chain dominated by the manufacturer, the optimal tracking capability and price
depend on the unit tracking cost, unit recall cost, quality inspection threshold and the supplier’s tracking capability:
Case 1 When CCtrr > δ(t, n), there exists a unique optimal tracking capability with a traceable unit size of n and the
associated optimal price pc∗ to optimise the supply chain profit is given by:
t
Ct
λk /k!
1
Ctr
∗
+
+ Cr
pc =
.
(11)
a + ws +
t
j
2
n
n(1 − Z )
j=0 λ /j!
k=1
Case 2 When CCtrr ≤ ϕ(t, n), there exists a unique optimal tracking capability with a traceable unit size of 1 and the
associated optimal price pc∗ to optimise the supply chain profit is given by:
t
λk /k!
k
1
Ct
∗
pc =
+ Cr
.
(12)
a + ws + Ctr +
2
n(1 − Z )
k − 1 + n tj=0 λ j /j!
k=1
Case 3 When ϕ(t, n) < CCtrr ≤ δ(t, n), there exists a unique optimal tracking capability with a traceable unit size of m ∗c
and price pc∗ to optimise the supply chain profit, and they are given by the following equation group:
Ctr
kn
λk /k!
−
C
= 0.
r
t
2
j
∗
(m ∗c )2
j=0 λ /j!
k=1 (k − 1)m c + n
t
∗
k /k!
λ
km
1
C
C
tr
t
c
+ Cr
.
pc∗ =
a + ws + ∗ +
2
mc
n(1 − Z )
(k − 1)m ∗c + n tj=0 λ j /j!
t
(13)
k=1
where ϕ(t, n) =
t
λk /k!
kn
k=1 (k−1+n)2 t λ j /j!
j=0
and δ(t, n) =
t
λk /k!
n
k=1 k t λ j /j! .
j=0
Theorem 1 presents closed-form expressions for the optimal tracking capability and price in terms of problem parameters
in a centralised supply chain, and a threshold policy is obtained. When the cost ratio (unit tracking cost/ unit recall cost) is
large enough, meaning that it is larger than a threshold which depends on the quality inspection threshold and the traceable
unit size of incoming raw materials, the optimal tracking strategy is to use unit tracking, such as RFID tracking technology.
When the cost ratio is small enough, the optimal tracking strategy is to use batch tracking with a traceable unit size of n, which
implies a traceable unit of finish products are just produced by a batch of raw materials. This explains why firms have no
incentive to adopt traceability systems when there is no regulation or in the early stage of tracking technology development.
When the cost ratio is moderate, batch tracking with an economic traceable unit size of m ∗c is a dominated strategy, such
as barcode tracking technology. Based on the above theorem, the supply chain can be motivated to choose unit tracking by
imposing a very large recall cost or significantly reduce the tracking cost. Furthermore, we find that the optimal retailing
price depends on the optimal tracking capability, and their relationship will be investigated in the next section.
In the decentralised supply chain, the problem can be formulated by a Stackberg game. The sequence of events is that:
(1) The manufacturer sets an optimal tracking capability and wholesale price to maximise her profit; (2) based on the wholesale
price, the retailer places an order of q products to the manufacturer to maximise his profit. The solution of this game can be
obtained by backward induction. Below is a theorem that states the optimal tracking capability, wholesale price and retailing
price in the decentralised supply chain.
Theorem 2 In the decentralised supply chain, the optimal tracking capability and price depend on the unit tracking cost,
unit recall cost, quality inspection threshold and the supplier’s tracking capability:
Case 1 When CCtrr > δ(t, n), there exists a unique optimal tracking capability with a traceable unit size of n and the
∗ and retailing price p ∗ to maximise the supply chain profit, and they are given by:
associated optimal wholesale price wm
d
t
Ct
λk /k!
1
Ctr
∗
+
+ Cr
wm =
.
a + ws +
t
j
2
n
n(1 − Z )
j=0 λ /j!
k=1
t
k /k!
C
λ
1
C
tr
t
+
+ Cr
pd∗ =
.
(14)
3a + ws +
t
j
4
n
n(1 − Z )
j=0 λ /j!
k=1
8
J.B. Dai et al.
Case 2 When CCtrr ≤ ϕ(t, n), there exists a unique optimal tracking capability with a traceable unit size of 1 and the
∗ and retailing price p ∗ to maximise the supply chain profit, and they are given by:
associated optimal wholesale price wm
d
t
λk /k!
k
1
Ct
∗
+ Cr
wm =
.
a + ws + Ctr +
2
n(1 − Z )
k − 1 + n tj=0 λ j /j!
k=1
t
λk /k!
k
1
C
t
∗
+ Cr
pd =
.
(15)
3a + ws + Ctr +
4
n(1 − Z )
k − 1 + n tj=0 λ j /j!
k=1
Case 3 When ϕ(t, n) < CCtrr ≤ δ(t, n), there exists a unique optimal tracking capability with an economic traceable unit
∗ and retailing price p ∗ to maximise the supply chain profit, and they are
size of m ∗d , associated optimal wholesale price wm
d
given by the following equation group:
Ctr
kn
λk /k!
−
C
= 0.
r
(m ∗d )2
[(k − 1)m ∗d + n]2 tj=0 λ j /j!
k=1
t
∗
k /k!
km
λ
1
C
C
tr
t
d
∗
+ Cr
=
wm
.
a + ws + ∗ +
2
md
n(1 − Z )
(k − 1)m ∗d + n tj=0 λ j /j!
k=1
t
km ∗d
λk /k!
1
Ctr
Ct
∗
+ Cr
pd =
.
3a + ws + ∗ +
4
md
n(1 − Z )
(k − 1)m ∗d + n tj=0 λ j /j!
t
(16)
k=1
where ϕ(t, n) =
t
λk /k!
kn
k=1 (k−1+n)2 t λ j /j!
j=0
and δ(t, n) =
t
λk /k!
n
k=1 k t λ j /j! .
j=0
Theorem 2 presents closed-form expressions for the optimal tracking capability and the associated optimal wholesale
price and retailing price in terms of problem parameters in a decentralised supply chain, and a threshold policy is obtained
as well. When the cost ratio is larger than a threshold which depends on the quality inspection threshold and the traceable
unit size of incoming raw materials, the optimal tracking strategy is to use unit tracking, such as RFID tracking technology.
When the cost ratio is smaller than the other threshold, batch tracking with a traceable unit size of n is a dominate strategy.
Similarly, this explains that why firms have no incentive to adopt tracking systems when there is no regulation or when the
development of tracking technology is in the early stage. When the cost ratio is moderate, batch tracking with an economic
traceable unit size of m ∗c is a dominate strategy, such as barcode tracking technology. Under this setting, both the optimal
wholesale price and retailing price depends on the optimal tracking capability. When the tracking capability deviates from
the optimal tracking capability, the optimal wholesale price and retailing price increases, which indicates the optimisation of
tracking capability could reduce double marginalisation.
Theorem 3 In both centralised and decentralised supply chains, the optimal tracking capability can be further depicted
by the cost ratio (unit tracking cost/unit recall cost) and the quality inspection threshold:
Case 1 When t ≥ t0 and δ(n, n) ≥ CCtrr ≤ δ(1, n), or when CCtrr ≥ δ(1, n), there exists a unique optimal tracking capability
with a traceable unit size of n to optimise the supply chain profit.
Case 2 When 0 ≤ CCtrr ≤ ϕ(1, n), or when t ≥ t1 and ϕ(n, n) ≥ CCtrr ≤ ϕ(n, n), there exists a unique optimal tracking
capability with a traceable unit size of 1 to optimise the supply chain profit.
Case 3 Otherwise, there exists a unique optimal tracking capability with a economic traceable unit size of m ∗ to optimise
the supply chain profit, and it satisfies the following equation:
Ctr
kn
λk /k!
− Cr
= 0.
t
∗
2
∗
2
j
(m )
[(k − 1)m + n]
j=0 λ /j!
t
(17)
k=1
where ϕ(t, n) =
t
λk /k!
kn
k=1 (k−1+n)2 t λ j /j! ,
j=0
δ(t, n) =
t
λk /k!
Ctr
n
k=1 k t λ j /j! , Cr
j=0
= δ(t0 , n) and
Ctr
Cr
= ϕ(t1 , n).
Theorem 3 depicts the tracking strategies in terms of the cost ratio and the quality inspection threshold, and they are
shown in Figure 2. When the cost ratio is large enough, using a traceable unit size of n is a dominate strategy, on the other
hand, when the cost ratio is small enough, unit tracking by RFID technology is a dominate strategy. When cost ratio is in a
proper moderate range, batch tracking with an economic traceable unit size of m ∗ is preferred. In other settings, the optimal
tracking strategies can be depicted by a threshold policy which depends on the quality inspection threshold. When the unit
tracking cost is moderately small compared to unit recall cost, if the quality inspection threshold is larger than a certain value,
International Journal of Production Research
9
Figure 2. Impact of the ratio of unit tracking cost to unit recall cost and the quality inspection threshold on optimal tracking strategies.
unit tracking by RFID technology is a dominate strategy, and otherwise, batch tracking with an economic traceable unit size
of m ∗ is preferred. When the unit tracking cost is moderately large compared to unit recall cost, if the quality inspection
threshold is larger than a certain value, batch tracking with a traceable unit of n is a dominate strategy, and otherwise, batch
tracking with an economic traceable unit size of m ∗ is preferred.
Considering the additional willingness to pay for traceable products, according to Hu and Kostamis (2015) Saak (2016),
the price response function can be reformulated as:
p = a − q/2 + γ /m
(18)
where γ /m denotes the additional willingness to pay for a traceable product with a tracking capability of m, obviously,
smaller m contributes to better tracking capability, and therefore, results in a larger additional willingness to pay for each
product. Then the centralised supply chain profit is:
q
q
E(π ) = pq − ws q − Ct
− (Ctr + γ ) − Cr q f (m, n, λ, t)
(19)
n(1 − Z )
m
Denote Ctr = Ctr + γ , the supply chain profit considering the additional willingness to pay for traceable products will
be the exactly same as that we stated in Equation.(1) in the centralised supply chain. Similarly, we have the same result in
the decentralised supply chain. Therefore, the aforementioned optimal tracking and pricing strategies will still hold even
considering the additional willingness to pay for traceable products, except the explanation of the unit tracking cost is a little
different.
4. Solution analysis
In this section, we focus on analysing the properties of the optimal tracking capability, the optimal retailing price and the
optimal wholesale price. First, we compare the optimal tracking capability in both centralised and decentralised supply chains.
Second, we explore the impact of problem parameters on the optimal tracking capability and other supply chain decisions. At
last, we compare the supply chain decisions incorporating tracking capability in both centralised and decentralised systems.
Lemma 1
Both centralised and decentralised supply chains have the same optimal tracking capability.
Proof. According to Theorems 1 and 2, when unit tracking cost is large or small enough compared to unit recall cost, the
optimal tracking capability will be obtained by a traceable unit size of n and 1, respectively. Otherwise, the optimal tracking
capability in both centralised and decentralised supply chains satisfies the same equation which has a unique solution of m ∗ .
Therefore, the optimal tracking capability in both centralised and decentralised supply chains are equal (m ∗ = m ∗c = m ∗d ).
Lemma 1 compares the optimal tracking capability in both centralised and decentralised supply chains with endogenous
pricing and product recall. The essential of tracking capability optimisation in the joint optimisation problem we proposed is
to tradeoff the tracking cost and recall cost, which is the same in both centralised and decentralised supply chains. Therefore,
10
J.B. Dai et al.
both systems have the same optimal tracking capability, which implies that centralising supply chain cannot improve the
optimal tracking capability.
Lemma 2 In both centralised and decentralised supply chains, when the cost ratio (unit tracking cost/unit recall cost) is
in a moderate range, the optimal tracking capability increases when unit recall cost increases, unit tracking cost decreases,
or the quality inspection threshold increases. Otherwise, the optimal tracking capability keeps the same despite the change
of problem parameters.
Lemma 2 depicts the effect of problem parameters on determining the optimal tracking capability in both centralised and
decentralized supply chains. When the ratio of unit tracking cost to unit recall cost is in a moderate range, in order to improve
the supply chain tracking capability, governance department could issue a strict product recall law or impose a punishment
on the manufacturer with product recall to increase the unit product recall cost, or subsidise the manufacturer which adopts
tracking systems or cost effective tracking technology development to reduce the unit tracking cost. Many existing literatures
have investigated the recall cost sharing mechanism between the supplier and the manufacturer, and we find that recall cost
sharing will encourage the manufacturer to adopt weak tracking capability which brings food safety issues. Particularly, when
unit tracking cost is sufficiently large or small compared to unit recall cost, these aforementioned strategies cannot improve
the supply chain tracking capability.
Lemma 3 If the cost ratio (unit tracking cost/unit recall cost) is sufficiently large, as the tracking capability increases,
the optimal retailing price increases, while the optimal supply chain profit decreases. On the other hand, as the tracking
capability increases, the optimal retailing price decreases, while the optimal supply chain profit increases. If the cost ratio is
in a moderate range, when the tracking capability is higher than the optimal tracking capability, as the tracking capability
increases, the optimal retailing price increases, while the optimal supply chain profit decreases, otherwise, as the tracking
capability increases, the optimal retailing price decreases, while the optimal supply chain profit increases.
Proof. If the unit tracking cost is sufficiently large compared to unit recall cost, according to Theorems 1 and 2, in both
p)
centralised and decentralised supply chains, the first-order derivative of the supply chain profit is given by: ∂ P(m,
=
∂m
k /k!
k /k!
λ
∂
p
λ
t
t
Ctr
Ctr
kn
kn
2(a − p)[ m 2 − Cr k=1 [(k−1)m+n]2 t λ j /j! ] > 0, and therefore, ∂m = − 2m 2 + Cr k=1 2[(k−1)m+n]2 t λ j /j! < 0.
j=0
j=0
This indicates that as the tracking capability increases, the optimal retailing price increases, while the optimal supply chain
profit decreases.
If the unit tracking cost is sufficiently small compared to unit recall cost, according to Theorems 1 and 2, in both
p)
λk /k!
kn
t
centralised and decentralised supply chains, ∂ P(m,
= 2(a − p)[ Cmtr2 − Cr tk=1 [(k−1)m+n]
] < 0, and therefore,
2
j
∂m
j=0 λ /j!
k
∂p
λ /k!
t
Ctr
kn
k=1 2[(k−1)m+n]2 t λ j /j! > 0. This indicates that as the tracking capability increases, the optimal
∂m = − 2m 2 + Cr
j=0
retailing price decreases, while the optimal supply chain profit increases. If the ratio of the unit tracking cost to unit recall
cost is in a moderate range, according to Theorems 1 and 2, in both centralised and decentralised supply chains, there exists
p)
λk /k!
kn
t
an optimal tracking capability m ∗ . When m < m ∗ , ∂ P(m,
= 2(a − p)[ Cmtr2 − Cr tk=1 [(k−1)m+n]
] > 0, and
2
j
∂m
j=0 λ /j!
k
p)
λ /k!
∂p
Ctr
kn
t
= 2(a − p)[ Cmtr2 − Cr tk=1 [(k−1)m+n]
] < 0. Therefore, when m < m ∗ , ∂m
= − 2m
when m > m ∗ , ∂ P(m,
2
2 +
j
∂m
j=0 λ /j!
t
k
λ /k!
kn
t
< 0. This indicates that as the tracking capability increases, the optimal retailing price
Cr k=1 2[(k−1)m+n]
2
j
j=0 λ /j!
t
∂p
λk /k!
Ctr
kn
= − 2m
increases, while the optimal supply chain profit decreases. When m > m ∗ , ∂m
2 + Cr
k=1 2[(k−1)m+n]2 t λ j /j!
j=0
> 0. This indicates that as the tracking capability increases, the optimal retailing price decreases, while the optimal supply
chain profit increases.
Lemma 3 depicts the impact of tracking capability on the optimal retailing price and supply chain profit, and find that
improving tracking capability cannot guarantee the improvement or reduction of supply chain profit, however, it depends on
the value of unit tracking cost, unit recall cost and the current tracking capability. Considering a setting currently with no
tracking systems or no regulation on tracking capability, the unit tracking cost usually will be much larger than unit recall
cost, according to Lemma 3, improving tracking capability induces reduction of supply chain profit and increase of retailing
price, which hurts the supply chain members and end customers. This indicates that supply chain members are reluctant to
improve tracking capability. Currently, regulation and subsidy are two widely used strategies to encourage the adoption of
tracking systems to strut out the aforementioned dilemma. If the governance department issues a strict recall law to impose
a huge punishment on the manufacturer with product recalls, unit recall cost will be much higher than unit tracking cost,
according to Lemma 3, improving tracking capability will improve the supply chain profit, and therefore, the manufacturers
are encouraged to adopt tracking systems to improve the tracking capability up to unit tracking. If the governance department
International Journal of Production Research
11
issues a normal recall law to impose a moderate punishment on the manufacturer with product recalls, the cost ratio (unit
tracking cost/unit recall cost) will be in a moderate range. According to Lemma 3, manufacturers will first improve the tracking
capability to batch tracking with an economic traceability unit size. As more and more manufacturers are adopting tracking
systems, the unit tracking cost will be reduced, and this further motivates the manufacturer to improve the tracking capability.
If the tracking cost is extensively reduced with a proper punishment on product recall, it is possible for the manufacturer
to achieve unit tracking. If the governance department propose to use subsidy policy in adopting tracking systems, when a
large subsidy is used, unit tracking cost will be extensively reduced to the level which is much smaller than the unit recall
cost, according to Lemma 3, the manufacturers are encouraged to adopt tracking systems to improve the tracking capability
up to unit tracking. As more and more manufacturers are adopting tracking systems, the unit tracking cost will be reduced.
The subsidy policy can only be abolished when the real unit tracking cost is at least comparable to unit recall cost, and
therefore, manufacturers will keep their tracking capability at certain tracking level with an economic traceability unit size,
otherwise, manufacturers will go back to the original status with no tracking systems. When a moderate subsidy is used, the
ratio of unit tracking cost to unit recall cost will be in a moderate range, according to Lemma 3, manufacturers will first
improve the tracking capability to batch tracking with an economic traceability unit size. As more and more manufacturers
are adopting tracking systems, the unit tracking cost will be reduced. If it is reduced extensively, manufacturers will improve
their tracking capability to unit tracking, however, when the subsidy policy stops, manufacturers will keep their tracking
capability at certain batch tracking level with an economic traceability unit size. If unit tracking cost is reduced only in a
small amount, manufacturers will keep their tracking capability, however, when the subsidy policy stops, manufacturers will
abolish the tracking system.
Lemma 4 In both centralised and decentralised supply chains, with the increase of unit recall cost, unit tracking cost or
quality inspection threshold, the optimal retailing price and wholesale price increase, while the optimal supply chain profit
decreases.
Lemma 4 depicts the impact of problem parameters on the optimal retailing price and supply chain profit. This explains
that why firms are reluctant to adopt tracking systems, however, they can adopt a strict quality inspection policy to improve
their profit and the supply chain profit. From the perspective of governance department, they can use subsidy to encourage
manufacturers to adopt tracking systems to reduce the unit tracking cost, and then increases the supply chain profit which
decreases the retailing price and wholesale price. However, considering the regulation on tracking capability, it forces firms
to improve the tracking capability by increasing unit recall cost, the profit of the manufacturer and the supply chain actually
decreases.
Lemma 5 The centralised supply chain has a lower retailing price, but a higher profit than that of the decentralised supply
chain. If the cost ratio (unit tracking cost/unit recall cost) is sufficiently large, as tracking capability increases, the difference
of optimal retailing price and supply chain profit increases. On the other hand, as the tracking capability increases, the
difference of optimal retailing price and supply chain profit decreases. If the cost ratio is in a moderate range, when the
tracking capability is less than the optimal tracking capability which maximises the manufacturer’s profit, as the tracking
capability increases, the difference of optimal retailing price and supply chain profit decreases, otherwise, the difference of
optimal retailing price and supply chain profit increases.
∗ . Recall that p ∗ = (a +w ∗ )/2, thus p ∗ − p ∗ = (a +w ∗ )/2− p ∗ =
Proof. According to Theorems 1 and 2, we have: pc∗ = wm
m
c
m
c
d
d
∗
∗
∗
(a + pc )/2 − pc = (a − pc )/2. Since p = a − q/2 ≤ a holds for any p, we have a − pc∗ ≥ 0, which implies pd∗ − pc∗ ≥ 0.
∂ p∗
According to Lemma 3, if the unit tracking cost is sufficiently large compared to unit recall cost, ∂mc < 0. Therefore, if m
increases, pc∗ will decreases, and then pd∗ − pc∗ increases. This indicates that the optimal retailing price gap will be reduced
∂ p∗
by improving the tracking capability. If the unit tracking cost is sufficiently small compared to unit recall cost, ∂mc > 0.
Therefore, if m increases, pc∗ will increases, and then pd∗ − pc∗ decreases. This indicates that the optimal retailing price gap
will be enlarged by improving the tracking capability. If the ratio of the unit tracking cost to unit recall cost is in a moderate
range, according to Lemma 3, in both centralised and decentralised supply chains, there exists an optimal tracking capability
∂p
< 0. Therefore, if m increases, pc∗ will decrease, and then pd∗ − pc∗ increases. This indicates
m ∗ , such that, when m < m ∗ , ∂m
∂p
> 0. Therefore,
that the optimal retailing price gap will be reduced by improving the tracking capability. When m > m ∗ , ∂m
∗
∗
∗
if m increases, pc will increases, and then pd − pc decreases. This indicates that the optimal retailing price gap will be
enlarged by improving the tracking capability.
According to Theorem 1 and 2, the optimal supply chain profit of centralised and decentralised supply chains are given
∗ )2 , respectively. Since w ∗ = p ∗ , the optimal supply chain profit difference is
by: E c∗ (π ) = 2(a − pc∗ )2 and E d∗ (π ) = (a − wm
m
c
∂[E c∗ (π)−E d∗ (π)]
∂[E c∗ (π)−E d∗ (π)]
∂ p∗
given by E c∗ (π ) − E d∗ (π ) = (a − pc∗ )2 , and therefore,
=
−2(a
−
pc∗ ) ∂mc . Since (a − pc∗ ) > 0,
∂m
∂m
∗
∂p
always have the opposite sign with ∂mc .
12
J.B. Dai et al.
100
Quality inspection threshold(t=2)
Quality inspection threshold(t=10)
90
Optimal traceable unit size
80
70
60
50
40
30
20
10
0
−2
−1
0
1
2
3
The log ratio of unit tracking cost to unit recall cost
Figure 3. Impact of the cost ratio (unit tracking cost/unit recall cost) on the optimal tracking capability.
Lemma 5 depicts the effect of tracking capability on the difference of optimal retailing price and supply chain profit
between centralised and decentralised supply chain. The profit of centralised supply chain is larger than that in the decentralised
supply chain, which indicates that the wholesale contract considering tracking capability cannot coordinate the supply chain.
However, when the cost ratio is sufficiently large or in a moderate range with a tracking capability which is less than the
manufacturer’s optimal tracking capability, improving tracking capability could reduce the difference of optimal retailing
price and supply chain profit, which implies that improving the tracking capabilities could mitigate double marginalisation
and difference of supply chain profit. Otherwise, improving the tracking capabilities could enlarge double marginalisation
and difference of supply chain profit.
Lemma 6 In both centralised and decentralised supply chains, as the unit recall cost, unit tracking cost or quality inspection
threshold increases, the difference of optimal retailing price and supply chain profit decreases.
∂ pc∗
∂Cr
> 0. Recall that pd∗ − pc∗ = (a − pc∗ )/2, thus
∗
∂( pd∗ − pc∗ )
∂p
= − 12 ∂Ccr < 0. This indicates that the optimal retailing price difference decreases as unit recall cost increases. Since
∂Cr
∂[E c∗ (π)−E d∗ (π)]
∂[E c∗ (π)−E d∗ (π)]
∂ p∗
= −2(a − pc∗ ) ∂Ccr . Since (a − pc∗ ) > 0, thus
< 0.
E c∗ (π ) − E d∗ (π ) = (a − pc∗ )2 , and therefore,
∂Cr
∂Cr
Proof. Considering the unit recall cost, according to Lemma 4 and 5,
This indicates that the optimal supply chain profit difference decreases as unit
recall cost increases.
∂ p∗
Considering the unit tracking cost, according to Lemma 4 and 5, ∂Ctrc > 0. Recall that pd∗ − pc∗ = (a − pc∗ )/2,
thus
∂( pd∗ − pc∗ )
∂Ctr
∂ p∗
= − 12 ∂Ctrc < 0. This indicates that the optimal retailing price difference decreases as unit tracking cost
increases. Since E c∗ (π ) − E d∗ (π ) = (a − pc∗ )2 , and therefore,
∂[E c∗ (π)−E d∗ (π)]
∂Ctr
∂[E c∗ (π)−E d∗ (π)]
∂Ctr
∂ p∗
= −2(a − pc∗ ) ∂Ctrc . Since (a − pc∗ ) > 0, thus
< 0. This indicates that the optimal supply chain profit difference decreases as unit tracking cost increases.
Considering the quality inspection threshold, according to Lemma 4 and 5,
thus
∂( pd∗ − pc∗ )
∂t
= − 12
increases. Since
∂[E c∗ (π)−E d∗ (π)]
∂t
∂ pc∗
∂t
E c∗ (π )
∂ pc∗
∂t
> 0. Recall that pd∗ − pc∗ = (a − pc∗ )/2,
< 0. This indicates that the optimal price difference decreases as the quality inspection threshold
− E d∗ (π ) = (a − pc∗ )2 , and therefore,
∂[E c∗ (π)−E d∗ (π)]
∂t
= −2(a − pc∗ )
∂ pc∗
∂t .
Since (a − pc∗ ) > 0, thus
< 0. This indicates that the optimal supply chain profit difference decreases as the quality inspection threshold
increases.
Lemma 6 depicts the effect of problem parameters on the difference of optimal retailing price and supply chain profit.
Increasing the problem parameters will decrease the difference of optimal retailing price and supply chain profit, meaning
that increasing parameters could mitigate double marginalisation. Therefore, if the governance department is imposing
punishment on manufacturers with product recall, the unit recall cost will be increased, although double marginalisation
is mitigated, customers pay a higher price for each product. If the governance department is imposing subsidy policy on
adopting tracking systems, unit tracking cost will be reduced, although double marginalisation is enlarged, customers actually
pay a lower price. Therefore, in the scenario of subsidy policy, retailers are more encouraged to adopt other supply contracts
to mitigate double marginalisation or even to coordinate the supply chain.
International Journal of Production Research
190
Centralized supply chain (t=2)
Centralized supply chain (t=10)
Decentralized supply chain (t=2)
Decentralized supply chain (t=10)
180
170
Optimal retailing price
13
160
150
140
130
120
110
100
−2
−1
0
1
2
The log ratio of unit tracking cost to unit recall cost
Figure 4. Impact of the cost ratio (unit tracking cost/unit recall cost) on the optimal retailing price.
4
2
x 10
1.8
Centralized supply chain(t=2)
Centralized supply chain(t=10)
Decentralized supply chain(t=2)
Decentralized supply chain(t=10)
1.6
Supply chain profit
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−2
−1
0
1
2
The log ratio of unit tracking cost to unit recall cost
Figure 5. Impact of the cost ratio (unit tracking cost/unit recall cost) on the supply chain profit.
To further analyse the impact of key parameters on the optimal strategies obtained by the proposed model, we have
conducted a numerical experiment on the sensitivity analysis. Based on consultation with three industrial experts in product
recall, the setting of the experiment is chosen as: a = 200, n = 100, λ = 10, cr = 80, ct = 1, Z = 0.05 and ws = 0 for
easy calculation, and then we run the experiment at two different quality inspection levels of t = 2 and t = 10. Figure 3
presents the impact of the cost ratio (unit tracking cost/unit recall cost) on the optimal tracking capability at different quality
inspection threshold values. We find that when the cost ratio is in a moderate range, the optimal traceable unit size of batch
tracking increases as the cost ratio increases, and otherwise, small cost ratio results in unit tracking, large cost ratio results
in barcode tracking. Furthermore, larger quality inspection threshold results in larger range of cost ratio with unit tracking
and barcode tracking, which indicates firms using accommodative quality inspection policy are more likely to choose unit
tracking and barcode tracking.
Figure 4 presents the impact of the cost ratio (unit tracking cost/unit recall cost) on the optimal retailing price in both
centralised and decentralised supply chains at different quality inspection threshold values. We find the optimal retailing
price is an increasing function of unit recall cost and unit tracking cost. With the increment of the unit tracking cost, the
difference of retailing price in centralised and decentralised supply chains is reduced, which indicates that the effect of double
marginalisation is mitigated.
14
J.B. Dai et al.
Figure 5 presents the impact of the cost ratio (unit tracking cost/unit recall cost) on the supply chain profit in both
centralised and decentralised supply chains at different quality inspection threshold values. When the unit tracking cost
increases, the profit of both supply chains and the profit difference decrease. This explains that most firms are reluctant to
adopt traceability system when the tracking technology is unmatured, and at the same time, the advantage of integrating supply
chains is limited. Furthermore, increasing the quality inspection threshold will decrease the profit difference of both supply
chains, which indicates that firms using strict quality inspection policy are more likely to choose supply chain integration.
5. Conclusion
In this paper, using the traceable unit size at the manufacturer level to measure the tracking capability, we propose a nonconvex non-linear programming to jointly optimise the tracking capability and price considering the tracking cost and recall
cost in a three-stage supply chain with endogenous pricing, and then explore the interactions between tracking capability
and supply chain decisions. First, we establish the relationship between the recall cost and the unit tracking cost, the unit
recall cost, the manufacture’s quality inspection threshold, the supply deflection rate and the tracking capability of both the
supplier and the manufacturer. Second, we present the closed-form expressions of the unique optimal tracking capability
and retailing/wholesale price to optimise the supply chain profit for both centralised and decentralised supply chains. Third,
optimal tracking strategies considering key parameters are presented. Given the manufacturer’s quality inspection policy,
with the increase of the cost ratio (unit tracking cost/unit recall cost), the optimal tracking strategies first changes from unit
tracking to batch tracking with an economic traceable unit size, and then moves to barcode tracking with the same traceable
unit size of the supplier. When the cost ratio is moderately large and small, with the increase of the quality inspection
threshold, the optimal tracking capability moves from batch tracking to barcode tracking and unit tracking, respectively.
Considering the interactions between tracking capability and supply chain decisions, we find that in the context of large and
small cost ratio, improving tracking capability will enlarge and mitigate the effect of double marginalisation, respectively.
In practice, tracking regulation and tracking subsidy are widely used to improve the supply chain tracking capability, and
we find that the strict tracking regulation policy is more robust than the tracking subsidy policy to improve the supply chain
tracking capability.
With realisation of works in this paper, our research can be further extended in the following aspects: (1) jointly optimise
the supply chain tracking capability and price under shared recall cost; (2) rather than the wholesale price contract, jointly
optimise the supply chain tracking capability and price under other supply contracts; (3) consider the impact of competition
on designing supply chain tracking capability, for instance, competition of tracking capability between two manufacturers.
Therefore, it would be interesting to explore how the decision of one manufacture may affect the other’s tracking and pricing
strategies.
Acknowledgements
The authors would like to thank editors and referees for their valuable suggestions which can significantly improve our manuscript.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work was supported by the National Natural Science Foundation of China [grant number 71671133], [grant number 71301122], [grant
number 71571079]; Soft Science of Hubei Province of China [grant number 2016ADC074]; National Social Science Foundation of China
[grant number 15ZDA061]; Research Fund for Academic Team of Young Scholars at Wuhan University [grant number Whu2016013].
References
Aiello, G., M. Enea, and C. Muriana. 2015. “The Expected Value of the Traceability Information.” European Journal of Operation Research
244: 176–186.
ALJAZEERA. 2015. “US Firm Recalls Children’s Drugs”. Retrieved November 3. http://www.aljazeera.com/news/americas/2010/05/
2010518535960539.html
Appelhanz, S., V. S. Osburg, W. Toporowski, and M. Schumann. 2016. “Traceability System for Capturing, Processing and Providing
Consumer Relevant Information about Wood Products: System Solution and its Economic Feasibility.” Journal of Cleaner Production
110: 132–148.
International Journal of Production Research
15
Cebeci, Z., Y. Erdogan, T. Alemdar, L. Celik, M. Boga, Y. Uzun, H. D. Coban, M. Gorgulu, and F. Tosten. 2009. “Development of an
ICT-based Traceability System in Compound Feed Industry.” Proceedings of the 4th Aspects and Visions of Applied Economics and
Informatics, Debrecen, Hungary, 854–864.
Dabbenea, F., and G. Gayb. 2011. “Food Traceability Systems: Performance Evaluation and Optimization.” Computers and Electronics in
Agriculture 75: 139–146.
Dai, H., M. M. Tseng, and P. H. Zipkin. 2015. “Design of Traceability Systems for Product Recall.” International Journal of Production
Research 53: 511–531.
Dai, H., L. Ge, and W. Zhou. 2015b. “A Design Method for Supply Chain Traceability Systems with Aligned Interests.” International
Journal of Production Economics 170: 14–24.
Dessureault, S. 2006. “An Assessment of the Business Value of Traceability in the Canadian Diary Processing Industry." Thesis (master).
University of Guelph.
Dupuy, C., V. Botta-Genoulaz, and A. Guinet. 2005. “Batch Dispersion Model to Optimise Traceability in Food Industry.” Journal of Food
Engineering 70: 333–339.
Fan, T., F. Tao, S. Deng, and S. Li. 2015. “Impact of RFID Technology on Supply Chain Decisions with Inventory Inaccuracies.”
International Journal of Production Economics 159: 117–125.
Fritz, M., and G. Schiefer. 2009. “Tracking.” Tracing, and Business Process Interests in Food Commodities: A Multi-level Decision
Complexity, International Journal of Production Economics 117: 317–329.
Huang, G. Q., T. Tu, Y. F. Zhang, and H. D. Yang. 2012. “RFID-enabled Product-service System for Automotive Part and Accessory
Manufacturing Alliances.” International Journal of Production Research 50 (14): 3821–3840.
Hu, Z., Z. Jian, S. Ping, Z. Xiaoshuan, and M. Weisong. 2009. “Modeling Method of Traceability System Based on Information Flow in
Meat Food Supply Chain.” WSEAS Transactions on Information Science and Applications 7 (6): 1094–1103.
Hu, B., and D. Kostamis. 2015. “Managing Supply Disruptions When Sourcing from Reliable and Unreliable Suppliers.” Production and
Operations Management 24: 808–820.
Kumar, S., and S. Schmitz. 2011. “Managing Recalls in a Consumer Product Supply Chain - Root Cause Analysis and Measures to Mitigate
Risks.” International Journal of Production Research 49: 235–253.
Lee, D., and J. Park. 2008. “RFID-enabled Traceability in the Supply Chain.” Industrial Management & Data Systems 108 (6): 713–725.
Liu, S., H. Zheng, H. Meng, H. Hu, J. Wu, and C. Li. 2009. “Study on Full Supply Chain Quality and Safety Traceability Systems for Cereal
and Oil Products. In IFIP International Federation for Information Processing.” Computer and ComputingTechnologies in Agriculture
II 3: 2265–2273.
Loader, R., and E. J. Hobbs. 1999. “Strategic Responses to Food Safety Legislation.” Food Policy 24: 685–706.
Lindley, B. 2007. “Trade-off between Cost of Traceability within a Small and Large Commercial Meat Plant and Economic Benefits of
Reducing the Number of Recalls and Size of Recalls." Thesis (master). Iowa State University.
Ngai, E. W. T., T. C. E. Cheng, K. H. Lai, P. Y. F. Chai, Y. S. Choi, and R. K. Y. Sin. 2007. “Development of an RFID-based Traceability
System: Experiences and Lessons Learned from an Aircraft Engineering Company.” Production and Operation Management 16 (5):
554–568.
Nguyen, QV 2004. “Traceability System of Fish Products-legislation to Implementation in Selected Countries (Report).” National fisheries
Inspection and Veterinary Directorate.
Piramuthu, S., P. Farahani, and M. Grunow. 2013. “RFID-generated Traceability for Contaminated Product Recall in Perishable Food
Supply Networks.” European Journal of Operational Research 225: 253–262.
Pouliot, S. 2008. “Traceability and Food Safety: Liability, Reputation, and Willingness to Pay.” Thesis (PhD). University of California.
Pouliot, S., and D. A. Sumner. 2013. “Traceability, Recalls, Industry Reputation and Product Safety.” European Review of Agricultural
Economics 40 (1): 121–142.
Rong, A., and M. Grunow. 2010. “A Methodology for Controlling Dispersion in Food Production and Distribution.” OR Spectrum 32:
957–978.
Ruiz-garcia, L., G. Steinberger, and M. Rothmund. 2010. “A Model and Prototype Implementation for Tracking and Tracing Agricultural
Batch Products Along the Food Chain.” Food Control 21 (2): 112–121.
Saak, A. E. 2016. “Traceability and Reputation in Supply Chains.” International Journal of Production Economics 77: 149–162.
Segura Velandia, D. M., N. Kaur, W. G. Whittow, P. P. Conway and A. A. West. 2016. “Towards Industrial Internet of Things: Crankshaft
Monitoring.” Traceability and Tracking Using RFID, Robotics and Computer-Integrated Manufacturing 41: 66–77.
Terreri, A. 2009. “Preventing the next Product Recall.” Food Logistics 111: 2025.
Thakur, M., L. Wang, and C. R. Hurburgh. 2010. “A Multi-objective Optimization Approach to Balancing Cost and Traceability in Bulk
Grain Handling.” Journal of Food Engineering 101: 193–200.
Wang, X., D. Li, and C. O. Brien. 2009. “Optimisation of Traceability and Operations Planning: An Integratedmodel for Perishable Food
Production.” International Journal of Production Research 47: 2865–2886.
Zhao, D., C. Teng, and X. Wang. 2009. “Design of traceability system for pork safety production based on RFID.” Proceedingsof the 2nd
International Conference on Intelligent Computation Technology and Automation, Changsha, China, 562–565.
16
J.B. Dai et al.
Appendix 1
Proof of Theorem 1 Denote the expected supply chain profit in the centralised supply chain by P(m, p), taking the first-order and
second-order derivatives on P(m, p) with respect to m, respectively, we get:
⎤
⎡
t
k /k!
C
kn
λ
∂ P(m, p)
tr
⎦.
= 2(a − p) ⎣ 2 − Cr
∂m
m
[(k − 1)m + n]2 tj=0 λ j /j!
k=1
∂ 2 P(m, p)
−4(a − p)
=
ω(m).
∂m 2
m3
k
where ω(m) = Ctr − Cr tk=1 k(k−1)n 3 tλ /k!j . Obviously, ω(m) is a strict decreasing function of m. Recall that ω(1) =
(k−1+n/m)
j=0 λ /j!
t
λk /k!
k(k−1)n λk /k!
t
= Ctr −Cr ψ(t, n), ω(n) = Ctr −Cr tk=1 (k−1)n
= Ctr −Cr φ(t, n), where ψ(t, n) =
Ctr −Cr k=1
j
(k−1+n)3 tj=0 λ j /j!
k2
j=0 λ /j!
k
k
t
t
λ /k!
λ /k!
∂ P(m, p)
k(k−1)n
(k−1)n
t
kn
k=1 (k−1+n)3 t λ j /j! and φ(t, n) =
k=1 k 2 t λ j /j! . Since
k=1 (k−1+n)2
∂m |m=1 = 2(a − p)[C tr − Cr
j=0
j=0
k
t
2(a− p)
λk /k!
1
tλ /k!j ] = 2(a − p)[C tr − Cr ϕ(t, n)] and ∂ P(m, p) |m=n = 2(a − p)( Ctr
[Ctr − Cr δ(t, n)],
2 − Cr
k=1 kn t λ j /j! ) =
∂m
n
n2
λ
/j!
j=0
j=0
k
k
t
kn
λ /k!
and δ(t, n) = tk=1 nk tλ /k!j .
where ϕ(t, n) = k=1
(k−1+n)2 tj=0 λ j /j!
j=0 λ /j!
k
k
t
2
k(k−1)n
kn
kn
kn
tλ /k!j
< Ctr − Cr tk=1 k(k−1)n3 tλ /k!j ,
Since
2 −
3 =
3 > 0, thus C tr − Cr
2
k=1
(k−1+n)
(k−1+n)
(k−1+n)
(k−1+n)
j=0 λ
/j!
(k−1+n)
j=0 λ
/j!
kn
therefore Ctr −Cr ϕ(t, n) < ω(1). Recall that t > 1, thus
< (k−1)n
, therefore ϕ(t, n) < φ(t, n), namely ω(n) < Ctr −Cr ϕ(t, n).
(k−1+n)2
k2
k /k!
k
t
λ
(k−1)n
n
n
n
t λ /k!j , therefore C tr − Cr δ(t, n) <
Recall that k −
= 2 > 0, thus Ctr − Cr k=1 k t
< Ctr − Cr tk=1 (k−1)n
j
k2
k
k2
j=0 λ /j!
j=0 λ /j!
ω(n). This means that Ctr − Cr δ(t, n) < ω(n) < Ctr − Cr ϕ(t, n) < ω(1). Since ω(m) with respect to m is a strict decreasing function,
there exists 3 cases as follows: ω(1) < 0; ω(n) > 0; ω(1) > 0 and ω(n) < 0.
2
p)
Case 1 When ω(1) < 0, namely CCtr < ψ(t, n), ∂ P(m,
> 0. Since Ctr − Cr δ(t, n) < ω(n) < Ctr − Cr ϕ(t, n) < ω(1), we have
2
r
∂m
∂ P(m, p)
∂ P(m, p)
p) is a strict decreasing function of m, which indicates that there exists the only
∂m |m=1 < 0,
∂m |m=n < 0. Therefore, P(m,
optimal tracking capability with a traceable unit size m ∗c = 1 to maximise the supply chain profit.
2
p)
Case 2 When ω(n) > 0, namely CCtr < φ(t, n), ∂ P(m,
< 0. Since Ctr − Cr δ(t, n) < ω(n) < Ctr − Cr ϕ(t, n) < ω(1), we have
r
∂m 2
∂ P(m, p)
∂ P(m, p)
Ctr − Cr ϕ(t, n), which implies ∂m |m=1 > 0. If ∂m |m=n > 0, namely δ(t, n) < CCtr , there exists a unique optimal tracking
r
p)
Ctr
capability with a traceable unit size of m ∗c = n to maximise the supply chain profit. If ∂ P(m,
∂m |m=n < 0, namely φ(t, n) < Cr < δ(t, n),
p)
∗
there exists a unique optimal tracking capability with a traceable unit size of m ∗c which satisfies ∂ P(m,
∂m |m=m c = 0 to maximise the
supply chain profit.
2
p)
first is negative, and then increases to be positive. Since Ctr −
Case 3 When ω(1) > 0 and ω(n) < 0, meaning that ∂ P(m,
2
∂m
p)
∂ P(m, p)
Ctr
Ctr
Ctr
Cr δ(t, n) < ω(n), we have ∂ P(m,
∂m |m=n < 0. If
∂m |m=1 > 0, we have ψ(t, n) < Cr , ϕ(t, n) < Cr and Cr < φ(t, n), namely
ϕ(t, n) < CCtr < φ(t, n). Therefore, there exists a unique optimal tracking capability with a traceable unit size of m ∗c which satisfies
r
∂ P(m, p)
∂ P(m, p)
Ctr
Ctr
Ctr
∗
∂m |m=m c = 0 to maximise the supply chain profit. If
∂m |m=1 < 0, we have ψ(t, n) < Cr , ϕ(t, n) < Cr and Cr > φ(t, n),
namely ψ(t, n) < CCtr < ϕ(t, n). Therefore, there exists a unique optimal tracking capability with a traceable unit size of m ∗c = 1 to
r
maximise the supply chain profit.
Therefore, when CCtr > δ(t, n), there exists a unique optimal tracking capability with a traceable unit size of n, when CCtr < φ(t, n),
r
r
the optimal traceable unit size is m ∗c = n, when CCtr ≤ ϕ(t, n), there exists a unique optimal tracking capability with a traceable unit size
r
of 1, and when ϕ(t, n) < CCtr ≤ δ(t, n), there exists a unique optimal tracking capability with a unique traceable unit size of m ∗c which
r
satisfies the following first-order condition:
t
kn
λk /k!
Ctr
− Cr
=0
t
∗
∗
2
2
j
(m c )
[(k − 1)m c + n]
j=0 λ /j!
k=1
(A1)
Considering the optimal retailing price, since the supply chain profit P(m. p) is a concave function of p, the optimal retailing price is
obtained by the first-order condition:
⎤
⎡
t
∗
k /k!
λ
km
1
C
C
tr
t
c
⎦.
+ Cr
pc∗ = ⎣a + ws + ∗ +
2
mc
n(1 − Z )
(k − 1)m ∗c + n tj=0 λ j /j!
k=1
(A2)
International Journal of Production Research
17
Proof of Theorem 2 Denote the expected supply chain profit in the decentralised supply chain by F(m, wm ), taking the first-order and
second-order derivatives on F(m, wm ) with respect to m, respectively, we get:
⎡
⎤
t
λk /k!
∂ F(m, wm ) ⎣
km
Ctr
Ct
= a + ws +
− 2wm ⎦ = 0.
+
+ Cr
∂wm
m
n(1 − Z )
(k − 1)m + n tj=0 λ j /j!
k=1
∂ 2 F(m, wm )
−2(a − wm )
ω(m).
=
∂m
m3
k
λk /k!
∂ F(m,wm )
k(k−1)n
kn
t λ /k!j ] =
|m=1 = (a − wm )[Ctr − Cr tk=1
2
k=1 (k−1+n/m)3 t λ j /j! . Since
∂m
(k−1+n)
j=0
j=0 λ /j!
t
a−wm
n λk /k! ) = a−wm [C − C δ(t, n)], where ϕ(t, n) =
m)
|
=
(C
−
C
(a − wm )[Ctr − Cr ϕ(t, n)] and ∂ F(m,w
m=n
tr
r
tr
r
k=1 k t λ j /j!
∂m
n2
n2
where ω(m) = Ctr − Cr
t
j=0
t
λk /k!
λk /k!
kn
n
k=1 (k−1+n)2 t λ j /j! and δ(t, n) =
k=1 k t λ j /j! . Based on Theorem 1, there exists 3 cases as follows:
j=0
j=0
2
m)
> 0. Since Ctr − Cr δ(t, n) < ω(n) < Ctr − Cr ϕ(t, n) < ω(1), we have
Case 1 When ω(1) < 0, namely CCtr < ψ(t, n), ∂ F(m,w
r
∂m 2
∂ F(m,wm )
∂ F(m,wm )
|
<
0,
|
<
0.
Therefore,
F(m,
w
m=n
m ) is a strict decreasing function of m, which indicates that there exists the
m=1
∂m
∂m
only optimal tracking capability with a traceable unit size m ∗d = 1 to maximise the supply chain profit.
2
m)
Case 2 When ω(n) > 0, namely CCtr < φ(t, n), ∂ F(m,w
< 0. Since Ctr − Cr δ(t, n) < ω(n) < Ctr − Cr ϕ(t, n) < ω(1), we have
r
∂m 2
∂ F(m,wm )
∂ F(m,wm )
|
>
0.
If
|m=n > 0, namely δ(t, n) < CCtr , there exists a unique optimal tracking
Ctr − Cr ϕ(t, n), which implies
m=1
∂m
∂m
r
m)
capability with a traceable unit size of m ∗d = n to maximize the supply chain profit. If ∂ F(m,w
|
< 0, namely φ(t, n) < CCtr < δ(t, n),
m=n
∂m
r
∂ F(m,wm )
∗
there exists a unique optimal tracking capability with a traceable unit size of m d which satisfies
|m=m ∗ = 0 to maximise the
∂m
d
t
supply chain profit.
2
m)
first is negative, and then increases to be positive. Since Ctr −
Case 3 When ω(1) > 0 and ω(n) < 0, meaning that ∂ F(m,w
∂m 2
∂ F(m,wm )
∂ F(m,wm )
|m=n < 0. If
|m=1 > 0, we have ψ(t, n) < CCtr , ϕ(t, n) < CCtr and CCtr < φ(t, n),
Cr δ(t, n) < ω(n), we have
∂m
∂m
r
r
r
namely ϕ(t, n) < CCtr < φ(t, n). Therefore, there exists a unique optimal tracking capability with a traceable unit size of m ∗d which
r
∂ F(m,wm )
m)
∗ = 0 to maximise the supply chain profit. If
satisfies ∂ F(m,w
|
|m=1 < 0, we have ψ(t, n) < CCtr , ϕ(t, n) < CCtr and
m=m
∂m
∂m
r
d
r
Ctr
Ctr
Cr > φ(t, n), namely ψ(t, n) < Cr < ϕ(t, n). Therefore, there exists a unique optimal tracking capability with a traceable unit size of
∗
m d = 1 to maximise the supply chain profit.
Therefore, when CCtr > δ(t, n), there exists a unique optimal tracking capability with a traceable unit size of n, when CCtr < φ(t, n),
r
r
the optimal traceable unit size is m ∗d = n, when CCtr ≤ ϕ(t, n), there exists a unique optimal tracking capability with a traceable unit size
r
of 1, and when ϕ(t, n) < CCtr ≤ δ(t, n), there exists a unique optimal tracking capability with an economic traceable unit size of m ∗d which
r
satisfies the following first-order condition:
t
Ctr
kn
λk /k!
−
C
=0
r
t
∗
∗
j
(m d )2
[(k − 1)m d + n]2
j=0 λ /j!
k=1
(A3)
Considering the optimal wholesale price, since the supply chain profit F(m, wm ) is a concave function of wm , the optimal wholesale price
is obtained by the first-order condition:
∗ =
wm
t
km ∗d
λk /k!
1
Ctr
Ct
[a + ws + ∗ +
+ Cr
].
t
∗
2
md
n(1 − Z )
(k − 1)m d + n j=0 λ j /j!
k=1
(A4)
18
J.B. Dai et al.
Proof of Theorem 3
For both centralised and decentralised supply chain, recall that:
ϕ(t + 1, n) − ϕ(t, n)
kn
λk /k!
kn
λk /k!
−
t
t+1 j
2
2
j
(k − 1 + n)
(k − 1 + n)
j=0 λ /j!
k=1
k=1
j=0 λ /j!
t
t
kn
λk /k!
kn
λk /k!
n(t + 1) λt+1 /(t + 1)! +
−
=
t
t+1 j
t+1 j
2
2
2
j
(k − 1 + n)
(t + n)
(k − 1 + n)
j=0 λ /j!
k=1
k=1
j=0 λ /j!
j=0 λ /j!
=
t+1
t
t
kn
λk
n(t + 1) λt+1 /(t + 1)!
−λt+1 /(t + 1)!
+
= t+1
t+1 j
t
j
j
(k − 1 + n)2 k!
(t + n)2
j=0 λ /j! k=1
j=0 λ /j!
j=0 λ /j!
t
kn
λk /k!
λt+1 /(t + 1)! n(t + 1) ].
= t+1
−
[
t
j
j
(t + n)2
(k − 1 + n)2
j=0 λ /j!
k=1
j=0 λ /j!
(A5)
xn
kn
when n ≥ t + 1 ≥ 3, recall that function
is a strict increasing function of x, we have
< (t+1)n2 , therefore,
(x−1+n)2
(k−1+n)2
(t+n)
t
λk /k!
λk /k!
n(t+1) t
n(t+1)
kn
<
<
,
thus
ϕ(t
+
1,
n)
−
ϕ(t,
n)
>
0.
This
means
that
ϕ(t,
n) with respect
t
2
2
2
j
j
k=1
k=1 t
(k−1+n)
j=0 λ
(t+n)
/j!
to t is a strict increasing function. Recall that
j=0 λ
/j!
(t+n)
δ(t + 1, n) − δ(t, n)
=
t+1
t
λk /k!
λk /k!
n
n
−
t
t+1 j
j
k
k
j=0 λ /j!
k=1
k=1
j=0 λ /j!
=
t
t
λk /k!
λk /k!
n
n λt+1 /(t + 1)! n
+
−
t
t+1 j
t+1 j
j
k
(t + 1)
k
j=0 λ /j!
k=1
k=1
j=0 λ /j!
j=0 λ /j!
n λk
n λt+1 /(t + 1)!
−λt+1 /(t + 1)!
+
= t+1
t+1 j
t
j
j
k k!
t +1
j=0 λ /j! k=1
j=0 λ /j!
j=0 λ /j!
⎤
⎡
t
1
λk /k! ⎦
nλt+1 /(t + 1)! ⎣ 1
−
= t+1
t
j
j
t +1
k
j=0 λ /j!
k=1
j=0 λ /j!
⎡
⎤
t
t
j
k
λ
λ
1
nλt+1 /(t + 1)!
⎣
⎦.
− (t + 1)
=
j!
k k!
(t + 1) t+1 λ j /j! t
λ j /j!
t
j=0
j=0
j=0
(A6)
k=1
t
t−1 λ2
1 λt
λj
1 λk
j=0 j! − (t + 1) k=1 k k! = 1 − tλ − 2 2! − · · · − t t! < 0, therefore δ(t + 1, n) − δ(t, n) < 0. This
means that δ(t, n) with respect to t is a decreasing function. This means that when δ(n, n) < CCtr < δ(1, n), ∃t0 s.t. CCtr = δ(t0 , n) and
r
r
when ϕ(1, n) < CCtr < ϕ(n, n), ∃t1 s.t. CCtr = ϕ(t1 , n). Since ϕ(1, n) < ϕ(n, n) < δ(n, n) < δ(1, n), based on Theorem 1, there exists
r
r
Assume 1 − tλ < 0, thus
t
five cases as follows:
p)
Case 1 When CCtr < ϕ(1, n), since Ctr − Cr δ(t, n) < ω(n) < Ctr − Cr ϕ(t, n) < ω(1), for all t, we have ∂ P(m,
∂m |m=1 < 0 and
r
∂ P(m, p)
∂m |m=n < 0. According to Theorem 1, therefore, P(m, p) is a strict decreasing function of m, which indicates that there exists the
only optimal tracking capability with a traceable unit size m ∗c = 1 to maximise the supply chain profit.
p)
Case 2 When ϕ(1, n) < CCtr < ϕ(n, n), since Ctr − Cr δ(t, n) < ω(n) < Ctr − Cr ϕ(t, n) < ω(1), if t < t1 , we have ∂ P(m,
∂m |m=1 > 0
r
p)
∗
and ∂ P(m,
∂m |m=n < 0. According to Theorem 1, there exists a unique optimal tracking capability with a traceable unit size of m c which
∂ P(m, p)
∂ P(m, p)
∂ P(m, p)
satisfies ∂m |m=m ∗c = 0 to maximise the supply chain profit. If t > t1 , we have ∂m |m=1 < 0 and ∂m |m=n < 0. P(m, p)
is a strict decreasing function of m, which indicates that there exists the only optimal tracking capability with a traceable unit size m ∗c = 1
to maximise the supply chain profit.
p)
Case 3 When ϕ(n, n) < CCtr < δ(n, n), since Ctr − Cr δ(t, n) < ω(n) < Ctr − Cr ϕ(t, n) < ω(1), for all t, we have ∂ P(m,
∂m |m=1 > 0
r
p)
∗
and ∂ P(m,
∂m |m=n < 0. According to Theorem 1, there exists a unique optimal tracking capability with a traceable unit size of m c which
∂ P(m, p)
satisfies ∂m |m=m ∗c = 0 to maximise the supply chain profit.
p)
Case 4 When δ(n, n) < CCtr < δ(1, n), since Ctr − Cr δ(t, n) < ω(n) < Ctr − Cr ϕ(t, n) < ω(1), if t < t0 , we have ∂ P(m,
∂m |m=1 > 0
r
p)
∗
and ∂ P(m,
∂m |m=n < 0. According to Theorem 1, there exists a unique optimal tracking capability with a traceable unit size of m c which
∂ P(m, p)
∂ P(m, p)
∂ P(m, p)
satisfies ∂m |m=m ∗c = 0 to maximise the supply chain profit. If t > t0 , we have ∂m |m=1 > 0 and ∂m |m=n > 0, there
exists a unique optimal tracking capability with a traceable unit size of m ∗c = n to maximise the supply chain profit.
International Journal of Production Research
19
p)
Case 5 When δ(1, n) < CCtr , since Ctr − Cr δ(t, n) < ω(n) < Ctr − Cr ϕ(t, n) < ω(1), for all t, we have ∂ P(m,
∂m |m=1 > 0 and
r
∂ P(m, p)
∗
∂m |m=n > 0. According to Theorem 1, there exists a unique optimal tracking capability with a traceable unit size of m c = n to
maximise the supply chain profit.
Similarly, in decentralised supply chain system, aforementioned results always hold.
Proof of Lemma 2 According to Theorems 1 and 2, when the unit tracking cost is large or small enough compared to the unit recall cost,
the optimal tracking capability will be obtained by a traceable unit size of n and 1, respectively, and therefore, it is independent on problem
parameters. However, when the ratio of unit tracking cost to unit recall cost is in a moderate range, the optimal tracking capability m ∗ is
obtained by the first-order condition. We will explore the impact of problem parameters on the optimal tracking capability one by one:
(1) Considering the unit recall cost, taking the first-order derivative on Cr , yields:
f m ∗ (m ∗ , n, λ, t) +
∗
2Ctr ∂m ∗
(m ∗ , n, λ, t) ∂m = 0.
+
C
f
∗
r
m
∂Cr
(m ∗ )3 ∂Cr
k
Plugging in the first-order condition C∗tr 2 − Cr f m ∗ (m ∗ , n, λ, t) tλ /k!j
(m )
f m ∗ (m ∗ , n, λ, t) +
where f m ∗ (m ∗ , n, λ, t) = tk=1
Therefore,
j=0 λ
/j!
(A7)
= 0, yields:
Cr
∂m ∗
[2 f m ∗ (m ∗ , n, λ, t) + m ∗ f m ∗ (m ∗ , n, λ, t)]
= 0.
∗
m
∂Cr
(A8)
λk /k!
λk /k!
kn
, and f m ∗ (m ∗ , n, λ, t) = − tk=1 2k(k−1)n
.
[(k−1)m ∗c +n]2 tj=0 λ j /j!
[(k−1)m ∗c +n]3 tj=0 λ j /j!
2 f m ∗ (m ∗ , n, λ, t) + m ∗ f m ∗ (m ∗ , n, λ, t)
t
kn
λk /k!
2k(k − 1)n
λk /k!
∗
−
m
t
t
∗
2
∗
3
j
j
[(k − 1)m + n]
[(k − 1)m + n]
j=0 λ /j!
j=0 λ /j!
k=1
k=1
t
2kn 2
λk /k!
=
> 0.
t
∗
3
j
[(k − 1)m c + n]
j=0 λ /j!
k=1
=
t
(A9)
∗
Recall that f m ∗ (m ∗ , n, λ, t) > 0, and therefore ∂m
∂Cr < 0. This indicates that when the unit recall cost increases, the optimal
traceable unit size decreases, and the optimal tracking capability increases.
(2) Considering the unit tracking cost, taking the first-order derivative on Ctr , yields:
∗
1
2Ctr ∂m ∗
(m ∗ , n, λ, t) ∂m = 0.
−
−
C
f
∗
r
m
∂Ctr
(m ∗ )2
(m ∗ )3 ∂Ctr
k
Plugging in the first-order condition C∗tr 2 − Cr f m ∗ (m ∗ , n, λ, t) tλ /k!j
(m )
j=0 λ
/j!
(A10)
= 0, yields:
1
Cr
∂m ∗
− ∗ [2 f m ∗ (m ∗ , n, λ, t) + m ∗ f m ∗ (m ∗ , n, λ, t)]
= 0.
∗
2
m
∂Ctr
(m )
(A11)
∗
Cr
∂m > 0. This indicates that when the
∗
∗ ∗
Since 1∗ 2 > 0, and m
∗ [2 f m ∗ (m , n, λ, t) + m f m ∗ (m , n, λ, t)] > 0, and therefore, ∂C
tr
(m )
unit tracking cost increases, the optimal traceable unit size increases, and the optimal tracking capability decreases.
(3) Considering the quality inspection threshold of t and t + 1, according to the first-order condition, we have:
t
(m ∗ (t))2 kn
λk /k!
Ctr
=
.
Cr
[(k − 1)m ∗ (t) + n]2 tj=0 λ j /j!
k=1
t+1
(m ∗ (t + 1))2 kn
λk /k!
Ctr
=
t+1 j .
∗
2
Cr
[(k − 1)m (t + 1) + n]
λ /j!
k=1
(A12)
j=0
Therefore
t+1
t
(m ∗ (t + 1))2 kn
λk /k!
(m ∗ (t))2 kn
λk /k!
.
=
[(k − 1)m ∗ (t + 1) + n]2 t+1 λ j /j!
[(k − 1)m ∗ (t) + n]2 tj=0 λ j /j!
k=1
k=1
j=0
j
Since tj=0 λ j /j! < t+1
j=0 λ /j!, yields,
t+1
t
(m ∗ (t + 1))2 knλk /k!
(m ∗ (t))2 knλk /k!
<
.
[(k − 1)m ∗ (t + 1) + n]2
[(k − 1)m ∗ (t) + n]2
k=1
k=1
(A13)
(A14)
20
J.B. Dai et al.
Subtracting two terms, yields,
t
k=1
λk
(m ∗ (t))2
(t + 1)(m ∗ (t + 1))2 λt+1
(m ∗ (t + 1))2
<0
−
)
+
(
(k − 1)! [(k − 1)m ∗ (t + 1) + n]2
[(k − 1)m ∗ (t) + n]2
[(k − 1)m ∗ (t + 1) + n]2 (t + 1)!
(A15)
2
x
Recall that function
is a strict increasing function of x, and therefore, m ∗ (t) > m ∗ (t + 1) must holds. This indicates
[(k−1)x+n]2
that when the quality inspection threshold increases, the optimal traceable unit size decreases, and the optimal tracking capability
increases.
∂ p∗
∂ p∗
∂w∗
Considering the unit recall cost, according to Theorems 1 and 2, if CCtr > δ(t, n), we have: ∂Cc = ∂Cd = ∂Cm =
r
r
r
r
∗
∂ pd∗
∂ pc∗
∂wm
λk /k!
Ctr
Ctr
1 t
k
>
0.
If
≤
ϕ(t,
n),
we
have:
=
=
=
>
0.
If
ϕ(t,
n)
<
≤
δ(t,
n),
k=1 t λ j /j!
k=1 k−1+n t λ j /j!
Cr
∂Cr
∂Cr
∂Cr
Cr
2
2
j=0
j=0
∗
∗
∗
k
k
∗
∂p
∂p
∂w
λ /k!
λ /k!
kn
km ∗
) ∂m + 12 tk=1 (k−1)m
. Recall the
we have: ∂Cc = ∂Cd = ∂Cm = (− Ctr∗ 2 + Cr tk=1
∗ +n t
j
r
r
r
2[(k−1)m ∗ +n]2 tj=0 λ j /j! ∂Cr
2m
j=0 λ /j!
∗
∗
∗
k
∂p
∂p
∂w
t
kn
tr − C
t λ /k!j
= 0, and thus ∂Cc = ∂Cd = ∂Cm > 0. This indicates that the optimal
first order condition C∗2
r
∗
2
k=1
Proof of Lemma 4
λk /k!
1 t
[(k−1)m +n]
m
j=0 λ
/j!
r
r
r
retailing price and wholesale price increases as the unit recall cost increases. Furthermore, according to Theorem 1, the optimal profit in
∂ E c∗ (π )
∂ p∗
the centralised supply chain is given by: E c∗ (π ) = 2(a − pc∗ )2 , thus ∂C
= −4(a − pc∗ ) ∂Cc < 0. Similarly, according to Theorem 2,
∂ E ∗ (π )
∂w∗
r
r
∗ ) m < 0. This indicates that the optimal supply chain profit decreases
d
= −2(a − wm
in the decentralised supply chain, we have ∂C
∂Cr
r
as the unit recall cost increases.
∂ p∗
∂ p∗
∂w∗
1 > 0. If
Considering the unit tracking cost, according to Theorems 1 and 2, if CCtr > δ(t, n), we have: ∂C c = ∂C d = ∂C m = 2n
r
tr
tr
tr
∗
∗
∂ pd∗
∂ pd∗
∂ pc∗
∂wm
∂ pc∗
∂wm
Ctr
Ctr
Ctr
1
Cr ≤ ϕ(t, n), we have: ∂Ctr = ∂Ctr = ∂Ctr = 2 > 0. If ϕ(t, n) < Cr ≤ δ(t, n), we have: ∂Ctr = ∂Ctr = ∂Ctr = (− 2m ∗2 +
k
t
∗
λk /k!
kn
kn
tr − C
λ /k! ) ∂m + 1 ∗ . Recall the first order condition C∗2
Cr tk=1
r
k=1 [(k−1)m ∗ +n]2 t λ j /j! = 0, and thus
2m
2[(k−1)m ∗ +n]2 tj=0 λ j /j! ∂Ctr
m
j=0
∗
∂ pd∗
∂ pc∗
∂wm
∂Ctr = ∂Ctr = ∂Ctr > 0. This indicates that the optimal retailing price and wholesale price increases as the unit tracking cost increases.
∂ E ∗ (π )
Furthermore, according to Theorem 1, the optimal profit in the centralised supply chain is given by: E c∗ (π ) = 2(a − pc∗ )2 , thus ∂Cc
=
tr
∗
∂ E d∗ (π )
∂ pc∗
∂wm
∗
∗
= −2(a − wm ) ∂C < 0.
−4(a − pc ) ∂C < 0. Similarly, according to Theorem 2, in the decentralised supply chain, we have ∂C
tr
tr
tr
This indicates that the optimal supply chain profit decreases as the unit tracking cost increases.
Considering the quality inspection threshold, according to Theorems 1 and 2, if CCtr > δ(t, n), we have: pc∗ (t + 1) − pc∗ (t) =
r
k
k
t+1
λk /k!
Ctr
∗ (t + 1) − w ∗ (t) = Cr t+1 λ /k!
− C2r tk=1 tλ /k!j
= C2r λt+1/(t+1)!
(1 − t+1
wm
m
k=1 t+1 j
k=1 t+1 j ) > 0. If C ≤ ϕ(t, n), we
2
j
r
j=0 λ /j!
j=0 λ /j!
j=0 λ /j!
j=0 λ /j!
λk /k!
λk /k!
Cr t+1
Cr t
Cr λt+1 /(t+1)! t+1
k
k
∗
∗
∗
∗
have: pc (t + 1) − pc (t) = wm (t + 1) − wm (t) = 2
k=1 k−1+n t λ j /j! = 2 t+1 λ j /j! ( t+n −
k=1 k−1+n t+1 λ j /j! − 2
j=0
t+1
λk /k!
Ctr
k
k=1 k−1+n t+1 λ j /j! ) > 0. If ϕ(t, n) < Cr ≤ δ(t, n), we have:
j=0
j=0
j=0
f (m ∗ , n, λ, t + 1) − f (m ∗ , n, λ, t)
=
t+1
λk /k!
λk /k!
km ∗
km ∗
.
−
t
t+1 j
∗
∗
(k − 1)m + n
(k − 1)m + n j=0 λ j /j!
k=1
k=1
j=0 λ /j!
t
t
λk /k!
km ∗
λt+1 /(t + 1)! (t + 1)m ∗ −
].
[
= t+1
j
tm ∗ + n
(k − 1)m ∗ + n tj=0 λ j /j!
k=1
j=0 λ /j!
∗
∗
(A16)
∗
∗
(t+1)m
(t+1)m
xm
km
Recall that function (x−1)m
> (k−1)m
>
∗ +n is a strict increasing function of x, thus
∗ +n , therefore
tm ∗ +n
tm ∗ +n
∗
∗
k /k!
∗ ,n,λ,t)
t
∗
∂
p
∂
p
∂
f
(m
λ
km
∗
∗
> 0. Therefore ∂tc = ∂td =
k=1 (k−1)m ∗ +n t λ j /j! , and therefore f (m , n, λ, t + 1) − f (m , n, λ, t) > 0, namely
∂t
j=0
∗
k
∗
∗
∂wm
kn
tr
λ /k! ) ∂m + ∂ f (m ,n,λ,t) . Recall the first order condition C∗2
= (− Ctr∗2 + Cr tk=1
−
∂t
∂t
2[(k−1)m ∗ +n]2 tj=0 λ j /j! ∂Ct
2m
m
∗
∗
∗
k
∂p
∂p
∂w
kn
t λ /k!j
= 0, and thus ∂tc = ∂td = ∂tm > 0. This indicates that the optimal retailing price and wholesale
Cr tk=1
∗
2
[(k−1)m +n]
j=0 λ
/j!
price increases as the quality inspection threshold increases. Furthermore, according to Theorem 1, the optimal profit in the centralised
∂ E c∗ (π )
∂ p∗
supply chain is given by: E c∗ (π ) = 2(a − pc∗ )2 , thus
= −4(a − pc∗ ) ∂tc < 0. Similarly, according to Theorem 2, in the
∂t
∂ E ∗ (π )
d
decentralised supply chain, we have ∂t
quality inspection threshold increases.
∂w∗
∗ ) m < 0. This indicates that the optimal supply chain profit decreases as the
= −2(a − wm
∂t