**Option
****c****ontracts
in ****f****resh
****p****roduce
****s****upply
****c****hain
with ****c****irculation
****l****oss**

Chong Wang^{1,
2},
Xu Chen^{1}

^{1}*University
of Electronic
Science & Technology of China, ^{2}Sichuan Agricultural
University (C*

*hina*

*)*

*Received
September 201**2*

*Accepted** **February
2013*

Wang,
C., & Chen, X. (2013). Option contracts in fresh
produce supply chain with circulation loss. *Journal of Industrial
Engineering and Management, *6(1), 104-112. http://dx.doi.org/10.3926/jiem.667

---------------------

*Abstract:
*

** Purpose:**
The
purpose of this paper is to investigate management decisions via option
contracts in a two-stage supply chain in which a fresh produce supplier
sells
to a retailer, considering the circulation loss of the fresh produce.

** Design/methodology/approach:**
Authors propose a Stackelberg model to analyze the supply chain
members’
decisions in the decentralized supply chain compared with the
integrated one under
the newsvendor framework.

** Findings:**
The
results illustrate that there exists a unique optimal option order
quantity for
the retailer and a unique optimal option order price for the supplier
giving certain
conditions; furthermore, option contracts cannot coordinate the fresh
produce
supply chain when the retailer only orders options.

** Originality/value:**
Agricultural
products especially fresh produce’s characteristics such as circulation
loss and
high risk are considered. Option contracts and game theory are combined
to
manage the fresh produce supply chain’s risk. The proposed tool and
models are
hoped to shed light to the future works in the field of supply chain
risk management.

** Keywords:**
fresh
produce, supply chain, option contracts, management decisions

---------------------

**1. Introduction**

As the competition is intensified in today’s market, many companies realize that the performance of their businesses highly depend on the collaboration and coordination across the supply chain. Unfortunately, the chain members are primarily concerned about their individual interests and that self serving focus often results in poor supply chain performance. Fresh produce’s short life cycles result in little or no salvage value at the end of the selling season as well as amazing circulation loss both in quality and quantity, which makes supply chain coordination more critical. Fresh produce as a necessity of daily life plays an important role in the market. Therefore, it is urgent and important to apply supply chain’s management theories and methods managing fresh produce’s high risk.

Agricultural products supply chain management has generated significant interest lately. As a kind of perishable goods, fresh produce has received a great deal of attention especially. Lowe and Preckel (2004) points out that the supply chain in the food and agribusiness sector is characterized by long supply lead times combined with significant supply and demand uncertainties, and relatively thin margins. They review some of the literature on applications of decision technology tools for a selected set of agribusiness problems and conclude by outlining what they see as some of the significant new problems facing the industry. Ahumada and Villalobos (2009, 2011) review the main contributions in the field of production and distribution planning for agri-foods based on agricultural crops, present an operational model that generates short term planning decisions for the fresh produce industry. Banker and Mitra (2011) investigate the effects of digital trading platforms on commodity prices in agricultural supply chains by comparing transaction data on trading various grades of coffee from a recently implemented digital platform in India with similar transactions from a physical commodity auction held weekly and firm-gate prices in the coffee producing regions of India. Shen, Lai, Leung and Liang (2011) study the inventory replenishment model for perishable agricultural products in a simple two-level supply chain and demonstrate that the supply chain cost decreases with supplier and retailer's collaborative forecasting. However, the above literature do not combine the theory of risk management in the research of fresh produce supply chain.

The research of supply chain risk management involves the application of contracts and options. Lariviere and Porteus (2001) consider a simple supply-chain contract in which a manufacturer sells to a retailer facing a newsvendor problem and the lone contract parameter is a wholesale price. Cachon and Lariviere (2005) discuss revenues-sharing contracts. Barnes-Schuster, Bassok and Anupindi (2002) analyze the role of options in a two-period correlated demand model. They show that channel coordination can be achieved only if the exercise price is piecewise linear and they develop sufficient conditions on the cost parameters so that linear prices coordinate the channel. Li, Ritchken and Wang (2009) investigate the role of forward commitments and option contracts in the presence of asymmetric information. They contrast the role of forward and option contracts and identify cases where combinations of the two are dominant. Zhao, Wang, Cheng, Yang and Huang (2010) take a cooperative game approach to consider the coordination issue using option contracts. They demonstrate that option contracts can coordinate the supply chain and achieve Pareto-improvement. Fu, Lee and Teo (2010) investigate the value of portfolio procurement in a supply chain, where a buyer can either procure parts for future demand from sellers using fixed price contracts or, option contracts or tap into the market for spot purchases. They derive the optimal portfolio procurement strategy for the buyer when both the demand and the spot price are random (either correlated or independent). Chen and Shen (2011) study a one-period two-party supply chain with a service requirement. They derive the retailer’s optimal ordering policy and the supplier’s optimal production policy in the presence of option contracts and a service requirement. However, the above studies do not consider agricultural products especially fresh produce and its characteristics. Considering the characteristics of fresh produce and using game theory, we combine option contracts with supply chain risk management to investigate the fresh produce supply chain management decisions.

The remainder of this paper is organized as follows. Problem description and assumptions are presented in section 2. In section 3, we take the integrated supply chain as the base model. In sections 4, we focus on the retailer’s and supplier’s optimal ordering and pricing policies respectively. Fresh produce supply chain coordination is considered in section 5. We conclude our findings in section 6 and highlight possible future work.

**2. Problem description and
assumption**

We consider a two-stage supply chain in which the
supplier
is a Stackelberg leader selling a fresh produce to the retailer who is
the
follower. There is no opportunity for the retailer to replenish
inventory once
the selling season has begun. Thus, both the supplier and the retailer
make
decisions prior to the season. Before the beginning of the selling
season, the
retailer purchases options from the supplier at per unit price of *w*.
Each option gives the retailer the right (not the obligation) to buy
one unit of
product at the exercise price of *a**w*
after demand has been observed. *a* is an exogenous parameter
and *a*>0. Then, the
supplier decides the
order and exercise prices according to the retailer’s order quantity *Q*
and the stochastic market demand. The supplier delivers the products at
per
unit cost of *c *and the retailer sells the product at per unit
price of *p
*which is known*.*

Considering the features of fresh produce, we
invite *b* (0<*
b**<1) *to imply the circulation loss in quantity
including
the loss caused by nature or artificial factors such as load and
unload, simple
packing, extruding, and transportation et al. range from the initial
ordering
moment to the commodity arrive to the retailer.

Let random variable *D *be the demand which
is non
negative with a mean of *u. *Let *Q _{I}* be the
supply
quantity in an integrated supply chain. The probability density
function is

*f(·)*, cumulative distribution function is

*F(·)*. Let

*F*be differentiable, increasing and

*f(0)=0*. Let [

*x]*.

^{+}=max(0,x),We assume that the unsold fresh produce has no
salvage value
at the end of the selling season, and each of the shortages that do not
meet
demand incurs a unit shortage cost of* g*. To avoid trivialities,
suppose *p>w+**a**w>c.
*The condition ensures
profit for both parties. We also assume the chain members are risk
neutral and information
is symmetric. We start with the discussion of the base model i.e., the
integrated supply chain.

**3. The integrated supply
chain**

To begin, consider the integrated supply chain as
the base
model. This model is a useful benchmark. In the integrated supply
chain, there
is no intermediate links such as wholesale. The integrated supply
chain’s profit
function, denoted *p*_{I}*(Q _{I}),*
is

The first term is the total sells revenue which is
restricted to the total demand and the supply considering the
circulation loss,
and the second term is the supply cost. The last term is the shortage
cost.
Then the integrated supply chain’s expected profit, denoted *E*[*p*_{I}*(Q _{I})],*
is

**Proposition 1.** In the integrated supply
chain, there
exists a unique optimal supply quantity *Q _{I}^{* }*given
by

**Proof. **Because

So, *E*[*p*_{I}*(Q _{I})]*
is concave in

*Q*, i.e., in the integrated supply chain, there exists a unique optimal

_{I}*Q*by solving Eq. (4) =0. The proof is complete.

_{I}^{*}

**4. The decentralized supply
chain**

**4.1. The retailer’s optimal
ordering policy
with option contracts**

Now we consider the retailer’s optimal ordering
policy with option
contracts. The retailer now orders only options. The retailer’s profit
function,
denoted *p*_{R}*(Q)*,
is

The first term is the total sells revenue which is
restricted to the total demand and the ordered options considering the
circulation loss, the second term is the options’ exercise cost, and
the third
term is the shortage cost. The last term is the options’ ordering cost.
Then the
retailer’s expected profit, denoted *E[**p*_{R}*(Q)]*,
is

**Proposition ****2.** With option
contracts in a fresh
produce supply chain, there exists a unique optimal option order
quantity *Q ^{*}*
for the retailer given by

So, *E[**p*_{R}*(Q)]*,
is concave in* Q*, i.e., there exists a unique optimal option
order
quantity *Q*^{* }for the retailer by solving Eq. (8) =0.
The proof
is complete.

**4.2. The supplier’s optimal
pricing policy
with option contracts**

After the fresh produce retailer announces *Q*^{*},
the supplier can decide the option order price *w *and the
option exercise
price *aw*. The
supplier’s profit
function, denoted *p _{s}(w),*
is

The first term is the options’ sells revenue, and
the second
term is the products’ supply cost. The last term is the revenue when
the
options are exercised. Then the supplier’s expected profit, denoted *E[**p _{s}(w]),* is

**Proposition ****3.** With option
contracts in a fresh
produce supply chain, when *w >c*, there exists a unique
optimal option
order price

*w*

^{*}for the supplier given by

And the optimal option exercise price is *a**w*^{*},
meanwhile 0<*a<p/w ^{*}-1*.

**Proof. **Because

Following Kouvelis and Zhao (2011), we define *r(·)=f(·)/(1-F(·))*
as the hazard rate. Many distributions have a non-decreasing hazard
rate. For
example, uniform, normal, logistic, chi-squared, and exponential
distributions
belong to this class of distributions (Bagnoli & Bergstrom, 2005).^{
} From
Eq. (7), ,
and take it into Eq. (14), we have

Taking Eq. (15) and Eq. (16) into Eq. (13), we have

From Eq. (17), when *w >c*, we can
safely get .
So, when

*w*,

__>__c*E[*

*p*

_{s}*(w)]*, is concave in

*w*, i.e., there exists a unique optimal option order price

*w*by taking Eq. (15) into Eq. (12) and solving Eq. (12) =0. Correspondingly, the optimal option exercise price is

^{*}

*a*

*w*

^{*}. As

*c<w*

^{*}+*a*

*w*

^{*}<

*p*and

*a*

*>0, so 0<*

*a*

*<p/*

*w*

^{*}-1. The proof is complete.

Proposition 3 implies that, in order to avoid the market risk caused by demand uncertainty, the premise of the fresh producer willing to provide option contracts is that the option’s order price be no low than the product’s supply cost when the retailer only orders options.

**5. Fresh produce supply
chain coordination
with option contracts**

In** **the previous discussion, we give the
base model and
the chain parties’ decisions with option contracts. We now are
interested on whether
option contracts can coordinate the fresh produce supply chain when the
retailer only orders options.

**Proposition 4.** Option contracts cannot
coordinate the
fresh produce supply chain when the retailer only orders options.

**Proof. **As supply chain coordination
requires that the
decision of a decentralized chain is consist to an integrated chain.
With
proposition 1 and proposition 2, let *Q ^{*}=Q^{*}_{I}*,
we obtain

Thus, when the contracts parameters satisfy *w=c-** ac**w/(p+g)<c*, option
contracts can
coordinate the fresh produce supply chain. But this condition is
contrary to the
condition which the supplier is willing to provide option contracts
i.e.,* w >c
*in proposition 3. So option contracts cannot coordinate the fresh
produce
supply chain when the retailer only orders options. The proof is
complete.

**6. Conclusion and
suggestions for further
research**

In this paper, consider the characteristics of fresh produce and use the Stackelberg model, we investigate the role of option contracts and management decisions for the fresh produce supply chain. Meaningful conclusions are given as follows. 1) With option contracts in a fresh produce supply chain, there exists a unique optimal option order quantity for the retailer and a unique optimal option order price under certain conditions for the supplier. 2) Option contracts cannot coordinate the fresh produce supply chain when the retailer only orders options. This research can be extended to a case that the retailer orders both products and options to see whether option contracts can coordinate the fresh produce supply chain, furthermore extend it to multi-period settings.

**Acknowledgment**

The authors thank the editor and the referees for careful reading the paper. This research is partially supported by the National Natural Sciences Foundation of China (No. 71272128), Youth Foundation for Humanities and Social Sciences of Ministry of Education of China (No. 11YJC630022), the Fundamental Research Funds for the Central Universities (No. ZYGX2009X020）and Sichuan Province Key Technology R&D Program (No. 2012FZ0003).

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Journal of Industrial Engineering and Management, 2008-2021

Online ISSN: 2013-0953; Print ISSN: 2013-8423; Online DL: B-28744-2008

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