Int. J. Production Economics 143 (2013) 120–131
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Int. J. Production Economics
journal homepage: www.elsevier.com/locate/ijpe
A coordinated production and shipment model in a supply chain
Onur Kaya n, Deniz Kubalı, Lerzan Örmeci
Department of Industrial Engineering, Koc University, Sariyer, Istanbul 34450, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 7 October 2010
Accepted 13 December 2012
Available online 5 January 2013
In this study, we consider the coordination of transportation and production policies between a single
supplier and a single retailer in a deterministic inventory system. In this supply chain, the customers
are willing to wait at the expense of a waiting cost. Accordingly, the retailer does not hold inventory but
accumulates the customer orders and satisfies them at a later time. The supplier produces the items,
holds the inventory and ships the products to the retailer to satisfy the external demand. We
investigate both a coordinated production/transportation model and a decentralized model. In the
decentralized model, the retailer manages his own system and sends orders to the supplier, while the
supplier determines her own production process and the amount to produce in an inventory
replenishment cycle according to the order quantity of the retailer. However, in the coordinated
model, the supplier makes all the decisions, so that she determines the length of the replenishment and
transportation cycles as well as the shipment quantities to the retailer. We determine the structure of
the optimal replenishment and transportation cycles in both coordinated and decentralized models and
the corresponding costs. Our computational results compare the optimal costs under the coordinated
and decentralized models. We also numerically investigate the effects of several parameters on the
optimal solutions.
& 2013 Elsevier B.V. All rights reserved.
Keywords:
Coordinated production and transportation
Inventory
Deterministic model
1. Introduction
In this study, we consider a deterministic supply chain consisting of a single supplier, who manufactures the items at a finite
rate, and a single retailer, who does not hold inventory but
accumulates the orders. Our model includes the transportation
costs due to shipments as well. Hence, the supplier and the
retailer need to determine an efficient integrated inventory and
transportation policy in order to minimize the total costs of the
supply chain and improve the performance of the system. We
consider both a coordinated (VMI) and a decentralized (non-VMI)
model to analyze the benefits of using an integrated policy.
Under a typical vendor-managed inventory (VMI) agreement,
the supplier decides on the order quantities to be sent to the
retailer and manages the inventory levels at both facilities. There
are several studies in the literature, which show that the integrated policy can improve the supply chain’s performance by
reducing inventory holding costs and increasing service levels.
Bhatnagar et al. (1993), Thomas and Griffin (1996), Sharafali and
Co (2000) and Sucky (2004) provide extensive surveys on coordinating the order and production policies in the single-supplier
n
Corresponding author. Tel.: þ90 212 3381583; fax: þ90 212 3381548.
E-mail addresses: [email protected] (O. Kaya), [email protected] (D. Kubalı),
[email protected] (L. Örmeci).
0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.ijpe.2012.12.020
single-retailer supply chains. Kang and Kim (2010), on the other
hand, consider the coordination of inventory and transportation
managements in a two-level supply chain in which a supplier
serves a group of retailers in a given geographic region and
determines a replenishment plan for each retailer.
Some of the studies ignore the transportation costs associated
with the shipments between the retailer and the supplier. However, these costs may affect the system significantly. For example,
Van der Vlist et al. (2007) introduced shipment costs in the
setting where (Yao et al., 2007) compare the performances of
VMI and non-VMI supply chains in a deterministic environment.
They found that inventory level at the retailer increases and
inventory at the supplier decreases with the VMI setting, contrary
to the findings of Yao et al. (2007). Hence, our model accounts for
the transportation costs, and accordingly we focus on the literature that considers these costs. The models in this context have a
number of distinguishing features: (1) the retailer holds inventory
or collects the demand to satisfy later, (2) the supplier manufactures the items or replenishes them from an ample supplier, and
(3) demand and/or supply is/are stochastic or deterministic. Now
we review the literature by classifying them according to these
features.
We first consider the studies on supply chains with a retailer
who does not keep inventory and a supplier who replenishes the
items instantly from an ample supplier. Companies selling
through sales agents or stores making catalog sales do not hold
O. Kaya et al. / Int. J. Production Economics 143 (2013) 120–131
inventory, but accumulate orders to fulfill later on. Products
which are unreasonable to keep in stock such as large items like
photocopy machines, luxury items like expensive watches or
expensive sports cars provide other typical examples, where the
retailer is reluctant to hold inventory. It is obviously disadvantageous for the retailer to keep such products in stock since the
inventory holding cost for the retailer is very high, whereas
customer waiting costs are relatively low. C
- etinkaya and Lee
(2002) consider a supply chain with such a retailer, where the
supplier is taken as a third-party warehouse that replenishes her
inventory instantly from an ample supplier. They prove that the
transportation cycle lengths are not necessarily equal in a
replenishment cycle. Our study is an extension of C
- etinkaya and
Lee (2002), where the third-party warehouse is replaced by a
capacitated producer. Hence, the items are not replenished
instantly, instead they are produced at a constant rate. Moreover,
we consider an upper bound on the waiting times of the
customers. The supply chains with a retailer who does not keep
inventory and a supplier who can instantly replenish are analyzed
when the demand is stochastic in a series of papers, such as
C
- etinkaya et al. (2006), C
- etinkaya and Lee (2000), Axsater (2001),
C
- etinkaya and Bookbinder (2003), C
- etinkaya et al. (2008) and
Kaya et al. (2013).
The deterministic supply chains with a retailer who keeps
inventory and a supplier who can replenish instantly are studied
extensively. A classic paper by Goyal (1976) analyzes a coordinated model in a deterministic supply chain with a single supplier
that replenishes instantly and a single retailer who keeps inventory to satisfy the external demand immediately. The objective is
to minimize the system-wide costs, which consist of transportation and inventory holding costs. Gupta and Goyal (1989) present
an early survey on buyer–vendor coordination for the transportation and the production policies in the single-supplier singlebuyer supply chains. Toptal et al. (2003) extend the earlier work
by modeling more general transportation considerations. More
explicitly, they consider cargo capacity constraints for both
inbound and outbound transport equipment, as well as a general
transportation cost structure that may explicitly represent a fleet
of vehicles, rather than a single truck. Later, Toptal and Cetinkaya
(2008) document the system-wide cost improvement rates
obtained through coordination for Goyal’s (1976) model analytically, and for Toptal et al.’s (2003) model numerically.
The literature on deterministic supply chains with a single
supplier who produces at a finite rate and a single retailer that
keeps inventory starts with Banerjee (1986). Banerjee (1986)
develops a joint economic-lot-size model under the assumption
of lot-for-lot bases. Goyal (1988) relaxes this assumption to
suggest a more general joint economic-lot-size model. However,
he still assumes that an integer number of equal shipments
should take place, and the whole batch must be produced before
any dispatches. Lu (1995) develops another heuristic method by
relaxing the assumption on finishing the whole batch before
dispatching, while still assuming an integer number of equal
shipments. Goyal (1995) extends Lu’s (1995) policy to a new
shipment policy involving unequal shipment quantities. According to this new policy, successive shipment sizes increase by a
factor equal to the ratio of the vendor’s production rate divided by
the demand rate. However, Hill (1997) illustrates that neither of
the policies of Lu (1995) and Goyal (1995) are optimal, and
proposes another solution method. Goyal and Nebebe (2000)
consider an alternative policy to the problem that is stated in
Hill (1997). Finally, Hill (1999) sets out an algorithm to obtain a
globally optimal solution by combining the policy in Goyal (1995)
and the equal shipment size policy of Lu (1995). In all these
studies, the holding cost of the retailer is assumed to be larger
than that of the vendor. Hill and Omar (2006) extend the work of
121
Hill (1999) when the vendor has larger holding costs than the
retailer.
Coordination of production and shipment decisions arises in
the context of just-in-time (JIT) systems as well. Hahm and Yano
(1992) analyze the joint decisions of production and delivery
scheduling for a single component in a system with a supplier and
a customer, both of which hold inventory. The objective is to
minimize the average cost per unit time, which consists of the
production setup costs and the inventory holding costs at both
the supplier and the customer, as well as the transportation costs.
Hahm and Yano (1995a, 1995c) develop heuristic methods to
solve an economic lot and outbound delivery scheduling problem
for a JIT facility, which allows multiple deliveries per cycle
contrary to the system in Hahm and Yano (1992). Hahm and
Yano (1995b) consider a model with multiple components, and
develop a heuristic procedure to find a cyclic production and
delivery schedule. Osman and Demirli (2012) analyze the economic lot and delivery scheduling problem for a multi-stage
supply chain with multiple items and develop an algorithm to
find the optimal solution for a synchronized replenishment
strategy.
We consider the integrated production and transportation
decisions in a deterministic supply chain in this study, where
the retailer does not hold inventory, and the supplier produces at
a finite rate to ship the products to the retailer. We determine the
optimal production and shipment policies that the companies
should apply for both the integrated control and the decentralized
control cases, while the shipment capacities can be infinite or
finite. When the supplier decides on the production as well as the
timing and quantity of the shipments, significant cost savings can
be realized through integrated production and shipment consolidation. Accordingly, the orders are not sent immediately, but
accumulated for a while in order to satisfy the economies of scale.
We also consider an upper bound on the customer waiting times
since the customers will not accept waiting for a very long time.
We determine the optimal production policy in this setting and we
show that the time between shipments can be of three different
lengths, as opposed to only two in C
- etinkaya and Lee (2002).
Moreover, the optimal policy for this model is significantly
different than the optimal policies found by Hill (1999) and Hill
and Omar (2006) in which the retailer holds inventory to satisfy
the external demand immediately. In Hill (1999) and Hill and
Omar (2006), the optimal shipment policy has two different
phases. In the first phase, the shipments increase by a factor of k,
where k is the ratio of production rate to demand rate. The
shipments of the second phase have equal sizes. The difference
between this model and our model stems from the assumption
that the retailer cannot backlog in Hill (1999) and Hill and Omar
(2006), so that whenever the inventory depletes in the retailer, a
shipment has to be dispatched. In our model, on the other hand,
there is not such a constraint. Hence, the supplier chooses the
shipment times only with the objective of minimizing the
total costs.
As a result, both existence of the production process and
characterization of the retailer with respect to accumulating
orders or holding inventory affect the coordinated policies substantially. In short, this study answers the following questions:
(i) When and how much the supplier should produce, (ii) when to
dispatch a vehicle in order to satisfy the customer orders, and
(iii) in what quantity to dispatch so that economies of scale are
satisfied.
We describe the coordinated supply chain with shipment costs
in the next section. Section 3 presents our main results on this
supply chain. We analyze the decentralized model in Section 4,
while Section 5 considers the supply chain when the transportation capacity is finite. We illustrate the benefits of coordination,
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O. Kaya et al. / Int. J. Production Economics 143 (2013) 120–131
as well as the effects of parameters numerically in Section 6.
Finally, we conclude in Section 7.
2. Coordinated production and shipment model
We analyze the integrated production and transportation
policy of a vendor-managed inventory system with a single
supplier and a single retailer for a specific product. The production and the demand rates of the product are assumed to be
constant and known, denoted by m and l, respectively. We
assume that the supplier has enough capacity to satisfy all
demand, so l o m. The supplier produces the product and carries
inventory to satisfy demand orders from the retailer. The supplier’s cost of carrying one unit of product per unit time is denoted
by h. The retailer faces an external demand from customers and
receives the product from the supplier. Upon arrival of a customer, the firm quotes a waiting time for the delivery of the
product. We assume that the customers are willing to wait for
their orders at most R time units, so that the quoted waiting times
are bounded by R. Accordingly, we consider a system in which the
retailer does not hold any inventory, but the shipment cycle
lengths need to be less than or equal to R. Waiting cost per unit
per unit time, denoted by w, is taken as a penalty associated with
delayed shipment. We do not assume any relation between h and w,
so our results are valid for all possible values of h and w.
We let Kp be the setup cost that the supplier incurs every time
a production process starts. In addition, a fixed transportation
cost, denoted as Kt, is incurred every time a shipment is dispatched to the retailer. Initially, we assume that the trucks used in
the transportation have infinite capacity, which will be relaxed in
Section 5. Finally, the transportation time between the supplier
and the retailer is assumed to be negligible.
In the VMI setting, the supplier has full knowledge on demand.
Consequently, she is responsible for managing the inventory and
determining the transportation schedule and amount, in addition
to the production decisions. Fig. 1 shows how the inventory level
at the supplier and the demand level at the retailer change in a
VMI supply chain. A replenishment cycle denotes the time period
between two consecutive epochs when the production process is
off and the supplier has no inventory. At the beginning of a
replenishment cycle, the supplier decides when to start the
production, which continues until a certain inventory level, Q max,
is reached. Then, the production is stopped for the rest of the
replenishment cycle. Hence, a typical replenishment cycle consists of a production phase and an idle phase. The supplier
dispatches orders to the retailer, regardless of being in production
Fig. 1. Time evolution of the inventory level and the number of waiting
customers.
or idle phase. The length of the time between two successive
dispatches is called a transportation cycle, where we define Ti as
the length of the ith transportation cycle. We denote the length of
an inventory replenishment cycle by T, which consists of several
transportation cycles.
3. Analysis and main results
In this section, we first characterize the optimal policy to
determine when to produce and when and how much to ship to
the retailer. Using this result, we identify the decision variables of
the optimal policy, and derive explicit expressions for all but two
of the decision variables. This enables us to write the cost
function with respect to only two integer parameters. Moreover,
we find upper bounds on these parameters so that they can be
optimized by a two-dimensional line search over a bounded
region.
The optimal production and transportation policy has an
intuitive characterization. A replenishment cycle starts when
the supplier has no production and no inventory. The optimal
policy lets the number of orders at the retailer increase to a
certain level before it starts production. Depending on the level of
accumulated orders at the retailer before production, it may take
a number of shipments until the number of outstanding orders is
dropped to 0. During this initial phase, the supplier is not able to
accumulate any inventory. Afterwards, each shipment satisfies all
the accumulated demand at the retailer, so that the supplier starts
building up inventory. When the inventory level reaches a certain
value Q max, which is a function of the decision variables, production is stopped. Theorem 1 identifies the optimal production and
transportation policy formally. We note that the proofs of all
results are placed in Appendix.
Theorem 1. In the coordinated production and transportation
model, an optimal policy has the following properties:
1. Let m þ 1 denote the number of transportation cycles until the
first time that the number of waiting customers at the retailer
drops to zero. Then, the amount produced in the first m þ 1
transportation cycles is equal to the total demand in those m þ 1
cycles.
2. If m Z 1, then the lengths of all transportation cycles Tk are equal
to each other for all k ¼ 2, . . . ,m þ 1, and they are also equal to the
length of the production time in the first transportation cycle (so
T k rT 1 for all k ¼ 2, . . . ,m þ 1).
3. Let n denote the number of transportation cycles after the first
m þ 1 transportation cycles. Then, the lengths of all the transportation cycles Tk are equal to each other for k ¼ m þ2,
. . . ,m þn þ1.
Fig. 2 illustrates an optimal production and transportation
schedule, as characterized by Theorem 1. We call the time length
between the start of a production cycle and the first time that the
number of waiting customers decreases to 0 as the ‘‘initial phase.’’
Observe that there are m equal transportation cycles with lengths
denoted as Ta, in addition to the first transportation cycle during
the initial phase. Due to statement 1 of Theorem 1, the inventory
level at the supplier and the number of waiting customers at the
retailer are both equal to 0 at the end of the initial phase of the
optimal solution. Then, the total number of customers in the
initial phase, given by lT 1 þ lmT a , is equal to the total number of
produced items, ðm þ 1ÞT a m, which gives the value of Ta as
T a ¼ lT 1 =ðmðm þ1ÞmlÞ. After the initial phase, there are n equal
transportation cycles with lengths denoted as Tb until the end of
O. Kaya et al. / Int. J. Production Economics 143 (2013) 120–131
123
Then, using the relationship T a ¼ lT 1 =ðmðm þ1ÞmlÞ, we can find:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2lðK p þ ðn þm þ 1ÞK t Þððw þ hÞm þ hnðmlÞÞ
Ta ¼
:
mðwþ hÞðhl þwm þ wmðmlÞÞðmnðmlÞ þðm þ n þ1ÞmÞ
ð3:3Þ
Fig. 2. Time evolution of the inventory level and the number of waiting customers
under an optimal policy.
the production cycle. The length of a replenishment cycle is then
given by T ¼ T 1 þmT a þnT b .
Now we can compute the average total cost per unit time,
denoted by Gðm,n,T 1 ,T b Þ, as follows:
Gðm,n,T 1 ,T b Þ ¼
cost of an inventory replenishment cycle
:
length of an inventory replenishment cycle
We compute the total cost incurred during a replenishment cycle,
which has four components: (1) the production set-up costs,
(2) the transportation costs, (3) the inventory holding costs, and
(4) the customer waiting costs. The production is turned on
exactly once in a cycle, so we have Kp for the production set-up
costs. The number of shipments in a cycle is n þ m þ1, incurring a
cost of ðn þ m þ1ÞK t . In order to find the total inventory costs, we
calculate the area below the inventory of the supplier (see Fig. 2),
which gives the following expression:
!
2
ðm þ 1ÞmT 2a ðnðn þ 1Þlmn2 l ÞT 2b
þ
H¼
h:
2
2m
The total waiting costs are computed by calculating the area
below the number of waiting orders
W¼
lðnT 2b þ mT 2a Þ þ lT 21 þ mðm1ÞðmlÞT 2a þ 2 mðmlÞT 2a
2
w:
Then, Gðm,n,T 1 ,T b Þ is given by
Gðm,n,T 1 ,T b Þ ¼
K p þ ðn þ mþ 1ÞK t þ H þ W
,
T
Recall that the customers are willing to wait for their orders until
a reasonable time R, thus, we have the constraints T 1 r R, T a r R
and T b rR. Since T a r T 1 , we can neglect the constraint T a r R. The
values T1 and Tb given in (3.1) and (3.2) will be the optimal cycle
lengths for given m and n values if they satisfy these constraints.
If these constraints are violated, we find the optimal solution by
considering the boundary conditions T 1 ¼ R or T b ¼ R, due to joint
convexity of the objective function.
The cost function, G, can be written as a function of only m and
n, where T1, Tb and Ta can be computed as stated above for given
m and n values. The following lemma provides upper bounds on
m and n, so that the optimal values of m and n can be obtained by
a two-dimensional search in a bounded region.
Lemma 2. The optimal values of m and n are bounded by the
following expressions:
9
8
>
>
>
>
>
>
>
>
=
<
K pm
(
)
and
n r max 1,
>
>
>
Kt
R2 >
>
>
>
>
,
;
: hlðmlÞmin
hm þwl 4
9
8
>
>
>
>
>
>
>
>
=
<
K pl
(
)
m rmax 1,
:
>
>
>
Kt
R2 >
>
>
>
>
,
;
: wmðmlÞmin
hm þ wl 4
Hence we can present our problem as follows:
min Gðm,nÞ subject to
9
8
>
>
>
>
>
>
>
>
=
<
Kpm
(
) ,
0 r n rmax 1,
2 >
>
>
Kt
R >
>
>
>
>
,
;
: hlðmlÞmin
hm þwl 4
9
8
>
>
>
>
>
>
>
>
=
<
Kpl
(
) ,
0 r mr max 1,
2 >
>
>
Kt
R >
>
>
>
>
,
;
: wmðmlÞmin
hm þ wl 4
m,n : integers:
where T ¼ T 1 þ mT a þnT b and T a ¼ lT 1 =ðmðm þ 1ÞmlÞ.
The next lemma shows that for given m and n values, the
objective function is jointly convex in T1 and Tb, which facilitates
solving the problem.
Lemma 1. For a given m and n, G is jointly convex in T1 and Tb.
By Lemma 1, we know G can be optimized by using the first
order conditions. Hence, setting the first derivatives of
Gðm,n,T 1 ,T b Þ with respect to T1 and Tb equal to zero, we obtain
the expressions of T1 and Tb as a function of m and n:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðK p þðn þ m þ1ÞK t Þððwþ hÞm þhnðmlÞÞððm þ1ÞmmlÞ2
,
T1 ¼
lmðw þ hÞðhl þ wm þwmðmlÞÞðmnðmlÞ þ ðm þ n þ 1ÞmÞ
We observe that, as the production rate, m, and the upper
bound on the waiting times, R, increase to infinity, this model
coincides with the model in which the retailer’s orders can be
replenished instantly. Hence, it is not surprising to find that the
resulting cost function G goes to the cost function found in
C
- etinkaya and Lee (2002), as m-1 and R-1. Due to the
dependence of G on n and m in a complex manner, it is difficult
to extract intuitive observations on its behavior with respect to
the system parameters. In Section 6, we numerically explore how
the cost function G changes with respect to the parameters of
the model.
4. Decentralized production and shipment model
ð3:1Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2mðhl þ wm þwmðmlÞÞ½K p þðn þ m þ 1ÞK t Tb ¼
:
lðw þ hÞðmnðmlÞ þ ðm þn þ1ÞmÞððw þ hÞm þ hnðmlÞÞ
ð3:2Þ
In this section, we consider a decentralized model such that
the retailer decides on when and how much to order from the
supplier, and the supplier decides on when and how much to
produce. In this model, we assume that the retailer incurs the
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O. Kaya et al. / Int. J. Production Economics 143 (2013) 120–131
fixed transportation cost and the customer waiting costs, while
the supplier incurs the fixed production costs and the inventory
holding cost. When the retailer decides on his own order quantity,
he places orders based on the economic order quantity (EOQ)
model that minimizes the following cost function subject to the
constraint that the cycle length is less than or equal to R
Gret ðQ Þ ¼
2lK t þ Q 2 w
:
2Q
Then, the optimal order quantity and transportation cycle length
will be as follows:
(rffiffiffiffiffiffiffiffiffiffiffi )
2K t l
Q
Q ret ¼ min
,Rl
and T ret ¼ ret :
w
l
When the number of customer orders reach the EOQ point, a
replenishment request is sent to the supplier, and the supplier
ships the order quantity Q ret immediately to the retailer. For a
given order quantity Q ret determined by the retailer, the supplier
needs to determine the length of the production cycle. Note that,
it is optimal for the supplier to produce an amount equal to nd Q ret
in a production cycle, where nd is an integer. Otherwise, there will
be unnecessary excess inventory left over at the end of the
production cycle, which will increase the inventory costs of the
supplier. Consequently, the time spent for production in a
replenishment cycle is lT ret nd =m. Moreover, the supplier will wait
for t0 time units before she starts the production, so that after the
first shipment the inventory level is exactly 0. Otherwise, excess
inventory will again be created unnecessarily. It is easy to
compute t0 as ðmlÞT ret =m. An illustration of the optimal production and transportation policy for the decentralized model is
shown in Fig. 3.
According to Qret and Tret, determined by the retailer, the
supplier decides on the number of transportation cycles, nd, in a
replenishment cycle to minimize her own cost function, which is
given as follows:
2
Gsup ðnd Þ ¼
2
T ret ðl þ nd ðnd þ 1Þlmn2d l Þ
Kp
þ
h,
2mnd
nd T ret
where the first term is the average production set-up and the
second is the average inventory holding cost. It is straightforward
to show that the cost function Gsup is convex in nd, and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
u2mK p þhl2 T 2
ret
ncont ¼ t
,
2
hT ret lðmlÞ
Fig. 3. Time evolution of the inventory level and the number of waiting customers
for the decentralized model.
minimizes Gsup. However, since nd needs to be integer, comparing
the associated costs Gðbncont cÞ and Gðdncont eÞ will reveal the
optimal value of nd. Then, the total cost for the decentralized
supply chain becomes
2
Gdec ðnd Þ ¼
2
K p þnd K t T ret ðl þ nd ðnd þ1Þlmn2d l Þ
lT ret w
:
þ
hþ
2
2mnd
nd T ret
5. Finite shipment capacity
In this section, we consider the case when the shipments have
a capacity constraint. We assume that the trucks used in the
transportation have a certain capacity C. In reality, we observe
that the wages of the drivers, truck leasing costs and fuel costs
form the main transportation cost components and these are
directly related to the number of vehicles dispatched. The other
fixed costs that are independent of the number of vehicles have a
small portion in the total transportation cost value. Hence, we
denote the transportation cost as pKt, where p is the number of
trucks required to make the shipment and Kt is the transportation
cost per truck. We note that this cost structure is the same cost
structure assumed in C
- etinkaya and Lee (2002).
Our next result shows that it is never optimal to use more than
one truck in a shipment when the shipment capacity is finite:
Lemma 3. In the model with finite shipment capacity, exactly one
truck will be used for each shipment.
We can show that the optimal solution in the finite capacity
case has the same characteristics as in the infinite capacity model.
Hence, the optimal policy still has the same structure as given in
Theorem 1, and the average cost per unit time, G is jointly convex
in T1 and Tb. However, now the shipment amounts are limited by
the capacity C. Hence, in the initial phase of the replenishment
cycle, the shipment amount, which is equal to the total production during a dispatch cycle, should not exceed the capacity of the
truck, so T a m rC. Similarly, in the second phase, the shipment
amount is equal to the total demand accumulated in a cycle and it
should be less than or equal to the capacity of the truck, i.e.,
T b l r C. Then, Tb and Ta are given by Eqs. (3.2) and (3.3),
respectively, if the resulting expressions satisfy the capacity and
waiting time constraints. Otherwise, due to the joint convexity of
the objective function, we analyze the boundary conditions on the
cycle lengths. We first observe that T a r lR=ðmðm þ1ÞmlÞ should
hold to satisfy the constraint T 1 r R, since T a ¼ lT 1 =ðmðm þ1ÞmlÞ.
As a result, we consider the equations
C
lR
T a ¼ min
,
m mðm þ 1Þml
or T b ¼ min fC=l,Rg to determine the optimal solution.
These restrictions increase the upper bounds on m and n,
which are derived in Appendices C.1.1 and C.2.1, since the
number of shipments may be larger in this case.
Using the above properties, we can write our cost function
Gðm,nÞ in terms of m and n as in Section 3. It is straightforward to
obtain the optimal solution for this case by employing a twodimensional search on the integers m and n, as in the infinitetruck capacity model.
In the decentralized model with finite capacity, it is never
optimal for the retailer to use more than one truck in a shipment,
similar to the centralized case. Thus, the retailer will order the
amount as in the infinite capacity case if that amount is less than C,
and otherwise he will order an amount that is exactly equal to C.
Using these results, the next section analyzes the effect of the
shipment capacity on the optimal policies, in addition to the effects
O. Kaya et al. / Int. J. Production Economics 143 (2013) 120–131
of other parameters and the potential savings that can be obtained
by coordination, through computational experiments.
6. Numerical analysis
This section, numerically, analyzes the benefits of the integrated system as well as the impacts of parameters on the optimal
policies. Our base case model has parameters as l ¼ 0:3, m ¼ 0:7,
w¼8, h¼2, K p ¼ 200, K t ¼ 100 with infinite transportation capacity and without an upper bound on the waiting times. We carry
on our sensitivity analysis with respect to each of these parameters by letting them take six different values including the base
value (see Table 1). We report all our findings in Table 2, placed in
Appendix D. Table 2 presents the cost values as well as the
transportation cycle lengths and the number of transportation
cycles in a replenishment cycle for both the coordinated and the
decentralized models for varying parameters. It also includes the
Table 1
Values of the parameters in the computational experiments, where the underlined values correspond to the
base case.
Parameters
Values
h
w
1, 2, 4, 6, 7, 10
0.5, 1, 4, 8, 16, 32
0.1, 0.2, 0:3, 0.5, 0.6, 0.69
0.31, 0.4, 0.6, 0:7, 1, 5
1, 50, 100, 200, 400, 1000
1, 50, 100, 200, 400, 1000
0.1, 1, 2, 3, 4, 1
2, 5, 8, 9, 10, 1
l
m
Kp
Kt
C
R
125
cost savings that can be obtained by using a coordinated model
instead of a decentralized model.
We first explore how the cost function G and Gdec behave with
respect to the model parameters. Fig. 4 presents the cost values in
both the coordinated and decentralized models. Both G and Gdec
increase in all cost parameters, as expected. The surprising
observation is that both cost functions are increasing in the
production rate m, contrary to the expectation that the system
would benefit from higher levels of production capacity. As m
decreases to l, the supplier continues the production for a much
longer period without building up inventory since the demand
and supply rates are very close, which leads to savings from both
the production setup and inventory costs. The behavior of cost
functions with respect to the demand rate l is also interesting. For
low values of l, G and Gdec are increasing in l, since l increases
the transportation and waiting costs. However, as l gets closer to
m, the savings by the better balance between demand and supply
outweigh the increase in costs due to the increased demand.
Hence, similarly to the case when m decreases to l, the production
setup and inventory costs decrease significantly. In fact, the ideal
situation is reached when the ratio l=m reaches 1. We observe
that both cost functions decrease as the truck capacity, C,
increases (see Table 2). Similarly, as the upper bound on the
waiting times of the customers, R, decreases, G increases since the
shipment decisions become more restricted. However, Gdec might
decrease in R since R forces the retailer to order more frequently
in the decentralized model causing the shipment frequency
become closer to its value in the coordinated model, which
decreases the total system costs in some cases.
An interesting comparison involves the individual cost values of
the supplier and the retailer in the coordinated and decentralized
models. In the decentralized model, the shipment frequencies are
determined to minimize the retailer’s cost function, while the
Fig. 4. The cost values in the coordinated model ðGÞ and in the decentralized model ðGdec Þ with respect to (a) holding cost h, (b) waiting cost w, (c) demand arrival rate l,
(d) production rate m, (e) set-up cost of production Kp and (f) transportation cost Kt.
126
O. Kaya et al. / Int. J. Production Economics 143 (2013) 120–131
supplier needs to arrange its production according to the retailer’s
shipment frequencies. As a result, the retailer’s cost is lower and
the supplier’s cost is higher in the decentralized model compared
to the coordinated case. In the coordinated model, the savings
obtained by the supplier through coordination outweighs the
increase in the retailer’s cost function and the total cost values
decrease. In order to convince the retailer to take part in coordination, the supplier needs to pass a part of its savings onto the retailer
which can be easily achieved through a contract mechanism, such
as a simple fixed payment contract, in which the fixed payment
amount is high enough for the retailer to take part in coordination.
We note that there can be many different contracts that might be
used for the coordination of this system, however, analysis of such
contracts is out of the scope of this paper.
Now we consider how the savings obtained by using a
coordinated model instead of a decentralized decision process
vary by the model parameters. We observe that the savings can be
substantially high, as it can increase to more than 50 % when w is
very low. The cost improvements present a monotone behavior
with respect to all parameters, as they increase in h, m or Kt, and
decrease in w, l or Kp. Therefore, the coordination becomes more
important for higher values of h, m or Kt, and lower values of w, l
or Kp. The values of h and w are more critical in the savings that
can be obtained, while the values of Kp and Kt have the least
effects on the savings. Thus, coordination of production and
transportation is more important for the industries with valuable
products (since holding cost is a function of the value of the
product), in which the customers are more patient (w is low).
Moreover, industries with slow moving products with respect to
the production rate (l=m is low) have a significant room for
improving the total system costs. Finally, we consider the effect of
transportation capacity and the customers’ patience on the
savings. Table 2 shows that the coordination is more important
when the transportation capacity is high or the customers are
more patient, as they allow more flexibility for the shipment
policies.
The lengths of the transportation cycles in the coordinated
model present certain relationships as well, since we have
T a r T ret rT b rT 1 in general, but not always (e.g., observe that
T ret 4T 1 when Kp ¼1 in Table 2). It is expected that the capacity
and waiting time constraints decrease all the cycle lengths.
However, these constraints first bound T1 which may result in
higher values of Tb (e.g., observe that Tb increases when R
decreases from 10 to 9 or C decreases from 4 to 3). The
transportation cycle lengths, Ta and Tb, decrease as w increases
or Kt decreases, so that more frequent shipments take place for
higher waiting costs and lower transportation costs, as expected.
The other parameters have varying effects on the transportation
cycles. On the other hand, the length of the replenishment cycle,
T ¼ T 1 þmT a þ nT b , increases as Kp increases. In addition, as h
increases, the length of the replenishment cycle shortens in order
to avoid building up high inventory levels. Finally, we observe
that as w, l or Kp increases or h, m or Kt decreases, the number of
transportation cycles in a replenishment cycle increases.
7. Conclusion
In this study, we determine the optimal integrated production
and dispatch policies in supply chains in which the retailer does
not hold any inventory but accumulates the customer orders to
satisfy at a later time. We present and prove the structure of the
optimal production and transportation policies, so that explicit
expressions for the optimal cost functions can be derived for both
infinite and finite transportation capacity models.
We consider both an integrated model and a decentralized
model for this problem. Our numerical results show that significant cost savings can be realized with the coordinated model
compared to the decentralized model, especially when the ratio
h=w increases or l=m decreases. We also observe that there is
more room for potential savings with coordination when the
transportation capacity and the waiting time constraints are not
very tight. Moreover, our numerical analysis presents the effects
of different parameters on the optimal solutions.
Our work can be extended in various ways: Different types of
transportation cost functions can be considered. Supply chains
with multiple products or multiple retailers present other directions of research. Finally, optimal integrated production and
shipment policies in stochastic environments can be analyzed in
a future study. Even though our results can form a basis for
developing production and shipment policies that can be also
applied in stochastic environments, the performances of such
policies need to be investigated and effective policies need to be
developed for stochastic systems.
Appendix A. Proof of Theorem 1
For the brevity of the paper, we prove only parts 1 and 3 of the
theorem. The proof for part 2 will be similar.
To prove part 1, we consider two cases depending on the value
of m. If m¼ 0, then it means that at the end of the first
transportation cycle, all the orders at the retailer are satisfied
and we claim that in the optimal solution, the supplier will start
the production at a time such that at the end of the first
transportation cycle, his inventory level drops to zero. Assume,
by contradiction, that P is an optimal solution in which the
supplier produces more than the demand accumulated at the
retailer within the first transportation cycle, and then continues
the production until the inventory level at the supplier becomes Q.
Now let P 0 be another solution, which is exactly the same with P
except that it starts the production at a later time such that the
supplier produces an amount that is exactly equal to the demand
accumulated in the first cycle and then continues the production
until the inventory level at the supplier becomes Q. We observe
from Fig. 5 that the supplier needs to pay a larger inventory
holding cost with P compared to P 0 because of the excess inventory
that the supplier needs to carry while everything else remains the
same, which leads to a contradiction.
In the second case, if m 4 0, then we claim that the inventory
level of the supplier will be exactly equal to 0 at the end of the
ðm þ 1Þst transportation cycle. Assume, by contradiction, that P is
an optimal solution in which there are x 40 units of inventory left
over at the supplier when the number of orders at the retailer
drops to zero for the first time (at the end of the ðm þ 1Þst
transportation cycle). Also let P 0 denote another solution which
is exactly the same with P except that the length of the ðmþ 1Þst
transportation cycle is increased by E and the length of the
ðm þ 2Þnd transportation cycle is decreased by E. An illustration
of the ðm þ 1Þst and ðm þ 2Þnd transportation cycles for schedules P
(shown with solid lines) and P 0 (shown with dashed lines) are
shown in Fig. 6.
The cost difference between schedules P and P0 can be written
as: DP0 P ¼ ðh þ wÞððQ xÞEðT b EÞlEÞ where Q is the amount of
inventory at the supplier before the ðm þ1Þst shipment and Tb is
the length of the ðm þ 2Þnd shipment.
Now, similar to P 0 , consider another schedule P00 which is
exactly the same with P except that the length of the ðm þ 1Þst
transportation cycle is decreased by E and the length of the
ðm þ 2Þnd transportation cycle is increased by E. Note that, since
x 40, it is possible to decrease the length of the ðm þ 1Þst
O. Kaya et al. / Int. J. Production Economics 143 (2013) 120–131
Fig. 5. Illustration for the proof of part 1 in Theorem 1 when m¼ 0.
be exactly equal to 0 (x¼0) at the end of the ðm þ 1Þst
transportation cycle.
To prove part 3, we show that a solution with T i 4 T i þ 1 for
some i4 m þ 1 cannot be optimal in detail. We can also show that
a solution which has T i oT i þ 1 for some i 4 m þ1 cannot be
optimal similarly, so we omit that part.
Let P be a policy such that T i 4T i þ 1 for some i 4m þ 1. Let P0 be
an identical policy to P with the exception that the lengths of
these two successive transportation cycles are adjusted so that
T 0i ¼ T i E and T 0i þ 1 ¼ T i þ 1 þ E, where E 40 is a small constant. We
consider the inventory and demand versus time graphs, and
calculate the cost differences between the policies P and P 0 . The
difference in the inventory holding cost, which is shown as the
hatched areas in the corresponding figures between the policies P
and P 0 , is denoted as either loss or gain. The additional area caused
by P 0 compared to P is called gain, and the opposite is called loss.
In order to prove that policy P 0 is better than P, we have to show
that the net area, (net area ¼loss gain), which is the difference
between the loss and the gain, must be positive. For this purpose,
we analyze four cases depending on the location of the cycles Ti
and T i þ 1 . In Case 1, both of these cycles are during the period
when the supplier is making the production. In Case 2, we analyze
the case when Ti is during the production phase, and the production stops during T i þ 1 . In Case 3, we consider the case that the
production stops during Ti and no production is done during T i þ 1
and finally in Case 4, we analyze the case when both Ti and T i þ 1
are in the idle phase.
Case 1: In this case, the two successive transportation cycles
take place in the production phase as seen in Fig. 7. We can
calculate the effects of P0 on the costs as follows: The decrease in
cost caused by P 0 : ðT i EÞlEhþ ðT i EÞlEw ¼ ðT i EÞlEðhþ wÞ. The
increase in cost caused by P 0 : lET i þ 1 h þ lET i þ 1 h ¼ lET i þ 1 ðh þwÞ.
In order to prove that the policy of P0 is better than the P, we
have to show that the net cost difference (net area ¼loss gain) is
positive. But the net difference in the cost is given by
lEðh þ wÞðT i ET i þ 1 Þ, which is always positive. Hence, P0 performs
better than P.
Case 2: The transportation cycle Ti is in the production phase,
while the production stops during the next transportation cycle, T i þ 1 ,
Fig. 6. Illustration for the proof of part 1 in Theorem 1 when m4 0.
transportation cycle by E and the number of orders at the retailer
will still drop to zero at the end of the ðm þ 1Þst transportation
cycle. Then the cost difference between schedules P and P 00 can be
written as DP00 P ¼ ðh þwÞððQ xlEÞE þðT b þ EÞlEÞ.
Observe that a small enough E exists such that either P0 or P00
has a smaller cost than P which contradicts with the assumption
that P is an optimal solution. Thus, a schedule P in which there are
x 4 0 units of inventory left over at the supplier when the number
of orders at the retailer drops to zero for the first time (at the end
of the ðm þ1Þst transportation cycle) cannot be optimal and in
the optimal solution, the inventory level of the supplier will
127
Fig. 7. Illustration of Case 1 for the proof of part 3 in Theorem 1.
128
O. Kaya et al. / Int. J. Production Economics 143 (2013) 120–131
Fig. 9. Illustration of Case 3 for the proof in part 3 of Theorem 1.
Fig. 8. Illustration of Case 2 for the proof of part 3 in Theorem 1.
as illustrated in Fig. 8. Then, the effects of P 0 on the costs are
as follows: The decrease in cost by P 0 : ðT i EÞlEh þðT i EÞ
lEw ¼ ðT i EÞlEðhþ wÞ. Excess inventory holding cost by P 0 :
lET i þ 1 h þ lET i þ 1 h ¼ lET i þ 1 ðh þ wÞ.
Consequently, the net cost difference obtained by P0 is
lEðh þ wÞðT i ET i þ 1 Þ 4 0. Hence, P0 performs better than P.
Case 3: In this case, the production stops during the transportation cycle Ti, so that the transportation cycle, T i þ 1 , lays completely in an idle phase, as shown in Fig. 9. The cost decrease
caused by P 0 is ðT i EÞlEhþ ðT i EÞlEw ¼ ðT i EÞlEðh þwÞ, whereas
the corresponding increase is lET i þ 1 h þ lET i þ 1 h ¼ lET i þ 1 ðh þ wÞ.
Hence, the net decrease in costs due to P 0 is lEðh þ wÞðT i ET i þ 1 Þ 40, as in the previous cases.
Case 4: In this case both transportation cycles are in the idle
phase, as illustrated in Fig. 10. Then, policy P0 decreases the costs
by ðT i EÞlEh þðT i EÞlEw ¼ ðT i EÞlEðh þwÞ, while increasing them
by lET i þ 1 h þ lET i þ 1 h ¼ lET i þ 1 ðh þwÞ. As a result, P0 brings a net
benefit of lEðh þwÞðT i ET i þ 1 Þ 40.
In all the cases, the net benefit, or equivalently the net area, is
positive whenever 0 o E oT i T i þ 1 . Hence, we can always
decrease the costs by decreasing Ti and increasing T i þ 1 , which
implies that a solution in which T i 4 T i þ 1 cannot be optimal. As
we mentioned earlier, we can, similarly, show that a solution
which has T i 4 T i þ 1 for some i 4 m þ1 cannot be optimal. Thus, in
the optimal solution, two successive transportation cycles in the
second phase of the replenishment cycle (i.e., for i4 m þ 1) should
be of equal length.
Fig. 10. Illustration of Case 4 for the proof of part 3 in Theorem 1.
Since 9H11 9 Z 0 and 9H9 Z 0, H is positive semidefinite. Thus G is
jointly convex in T1 and Tb.
Appendix C. Proof of Lemma 2
Appendix B. Proof of Lemma 1
C.1. Upper bound on m
Let H be the Hessian matrix of G with two variables of T1 and
Tb. The determinant of the first element of H, H11, and of H itself
will be as follows:
In the first part of the proof, we compare an optimal solution S,
characterized by optimal (m,n) values, to another solution S0 , and
conclude that optimal m has to be smaller than an expression that
9H11 9 ¼
9H9 ¼
ðm þ1Þmððm þ1ÞmmlÞ½2mðm þ 1Þððm þn þ 1ÞK t þ K p Þ þðw þhÞnT 2b lðmnðmlÞ þ ðm þ n þ 1ÞmÞ
ðmððmþ 1ÞT 1 þnT b Þ þ ðmlÞnmT b ÞÞ3
2
2nmlðm þ 1Þððm þ1ÞmmlÞ ðhþ wÞðmnðmlÞ þ ðm þ n þ 1ÞmÞðK p þ ðn þm þ 1ÞK t Þ
ðmððm þ1ÞT 1 þ nT b Þ þ ðmlÞnmT b Þ4
:
,
O. Kaya et al. / Int. J. Production Economics 143 (2013) 120–131
depends on Ta. In the second part of the proof, we find a lower
bound on Ta by comparing the optimal solution S with another
solution S00 . Combining these two bounds results in the upper
bound expression given in Lemma 1.
If m¼ 0 or m¼1, m is already bounded by 1, so an additional
upper bound is not needed. Hence, we concentrate on the cases
with m Z2.
Let S be the optimal solution of the problem with a cycle
length of T ¼ T 1 þ mT a þ nT b (see Fig. 2). We construct another
solution, S0 , which keeps Ta, Tb and n the same as in S, while
decreasing m by 1 and T1 is shortened accordingly. Recall that T1
can be written in terms of Ta, using the equality lðT 1 þmT a Þ ¼
mðm þ1ÞT a . Hence, T1 is reduced by y ¼ ðmlÞT a =l time units
resulting in T 1 0 ¼ T a ðmmlðm1Þ=lÞ. Moreover, the production
cycle of S0 finishes in T 0 time units, as it decreases by x time units
with x ¼ yþ T a ¼ mT a =l. As a result, when m and T1 are decreased,
the whole production cycle in Fig. 2 shifts to the left by x
time units.
We need to compare the total cost of solutions S and S0 during
the time interval ð0,TÞ. Hence, we need to specify how S0 behaves
in the interval ðT 0 ,TÞ as well. We assume that S0 completes another
129
production cycle in this interval, so that the production starts at
time T 0 þ xðmlÞ=m and finishes at T, lasting for xl=m time units.
All the items produced in this time interval, i.e., lx items, are
shipped to the retailer at time T, which satisfies all the customer
orders that accumulate during x time units.
Observe that the total inventory holding costs at the supplier
in both S and S0 will be the same. Solution S0 has an additional cost
of Kp since it includes two production cycles in the period ð0,TÞ,
while S has only production cycle. On the other hand, S0 will save
wmmT a y of the customer waiting costs, which can be computed
using Fig. 2. Since S is optimal, K p wmmT a y Z0. Thus m r
K p l=wmðmlÞT 2a .
This upper bound on m depends on Ta which is also a decision
variable, so we find a lower bound on Ta in order to obtain an
upper bound on m in terms of the problem parameters. If
T a Z R=2, then R=2 is a lower bound for Ta. Now, we assume that
T a r R=2 and consider another solution S00 which combines two
transportation cycles of length Ta of the solution S into one,
everything else remaining the same. Note that this is always
possible since we assume m Z2 and T a r R=2. Solution S00 saves Kt
since it decreases two shipments to one, but it increases the
Table 2
Summary of numerical results.
Parameters
Coordinated model
m
n
T1
Decentralized model
Ta
Tb
G
Gsup
Gret
n
Tret
Savings
Gdec
Gsup
Gret
%
Base case
h¼1
h¼4
h¼6
h¼7
h ¼ 10
0
0
0
0
1
1
3
4
2
2
1
1
12.31
11.78
12.57
12.51
13.69
13.42
5.28
5.05
5.39
5.36
3.73
3.66
8.12
8.8
7.37
6.34
7.14
6.07
32.72
29.8
36.63
39.69
40.7
43.19
10.38
7.7
13.93
16.49
14.35
16.44
22.34
22.1
22.7
23.19
26.37
26.75
4
5
3
2
2
2
9.13
9.13
9.13
9.13
9.13
9.13
36.68
31.69
44.86
52.23
55.46
65.14
14.77
9.78
22.95
30.32
33.55
43.23
21.91
21.91
21.91
21.91
21.91
21.91
10.8
5.98
18.36
24.02
26.6
33.69
w¼ 0.5
w¼ 1
w¼ 4
w¼ 16
w¼ 32
2
1
0
0
0
0
1
2
4
7
51.43
36.2
17.77
8.33
5.60
10.29
9.87
7.62
3.57
2.40
–
13.5
10.42
6.22
4.38
13.89
16.78
25.90
42.14
55.17
5.86
7.09
9.85
10.88
11.18
8.02
9.69
16.05
31.25
44.0
1
2
3
5
8
36.52
25.82
12.91
6.46
4.56
32.86
29.88
31.72
44.82
56.99
27.39
22.13
16.23
13.83
13.18
5.48
7.75
15.49
30.98
43.82
57.74
43.82
18.36
5.98
3.2
l ¼ 0:1
l ¼ 0:2
l ¼ 0:5
l ¼ 0:6
l ¼ 0:69
0
0
1
1
5
2
2
5
7
23
23.24
15.81
8.8
7.28
5.83
3.32
4.52
4.89
5.46
5.37
14.34
10.54
6.24
5.76
5.39
19.26
27.11
40.1
41.47
39.61
6.15
8.73
10.47
9.81
5.96
13.11
18.37
29.64
31.66
33.65
3
3
6
8
25
15.81
11.18
7.07
6.46
6.02
22.59
31.09
43.02
43.57
40.36
9.94
13.2
14.73
12.59
7.13
12.65
17.89
28.28
30.98
33.23
14.74
12.82
6.77
4.82
1.86
m ¼ 0:31
m ¼ 0:4
m ¼ 0:6
m¼1
m¼5
3
1
0
0
0
15
5
3
3
2
9.18
11.11
12.02
12.85
13.72
8.1
6.67
6.01
3.85
0.82
8.18
8.18
8.32
7.78
8.1
26.9
30.67
32.45
33.15
33.43
4.66
7.86
10.21
10.6
10.59
22.24
22.81
22.24
22.55
22.84
17
6
4
3
3
9.13
9.13
9.13
9.13
9.13
27.59
32.75
35.94
37.98
39.73
5.69
10.84
14.03
16.07
17.82
21.91
21.91
21.91
21.91
21.91
2.51
6.34
9.72
12.7
15.86
Kp ¼ 1
Kp ¼ 50
Kp ¼ 100
Kp ¼ 400
Kp ¼ 1000
Kt ¼1
Kt ¼50
Kt ¼200
Kt ¼400
Kt ¼1000
0
0
0
0
1
6
0
0
0
0
0
1
2
5
8
37
5
2
1
0
8.72
10.24
11.1
14.04
17.95
5.19
9.93
15.7
20.48
30.05
3.74
4.39
4.76
6.02
4.89
0.5
4.25
6.73
8.78
12.88
–
8.14
8.01
7.91
8.01
0.81
5.6
11.32
16.28
–
23.17
27.21
29.5
37.31
46.01
12.75
26.38
41.73
54.41
79.86
1.24
5.15
7.31
14.69
21.76
9.5
10.39
10.34
10.31
10.52
21.93
22.05
22.19
22.61
24.25
3.25
15.99
31.39
44.11
69.34
1
2
3
5
8
37
5
3
2
2
9.13
9.13
9.13
9.13
9.13
0.91
6.46
12.91
18.26
28.87
27.5
31.11
33.39
41.47
51.01
14.18
29.32
47.21
62.21
93.16
5.59
9.19
11.48
19.56
29.1
11.99
13.83
16.23
18.39
23.88
21.91
21.91
21.91
21.91
21.91
2.19
15.49
30.98
43.82
69.28
15.74
12.53
11.62
10.04
9.79
10.08
10.04
11.62
12.53
14.28
C ¼0.1
C ¼1
C ¼2
C ¼3
C ¼4
22
2
1
0
0
91
9
4
3
3
4.52
7.14
10.48
10
12.31
0.14
1.43
2.86
4.29
5.28
0.33
3.33
6.67
8.17
8.12
310.92
44.86
34.19
32.92
32.72
9.5
9.71
9.67
10.89
10.38
301.41
35.14
24.52
22.03
22.34
102
10
5
4
4
0.33
3.33
6.67
9.13
9.13
312.21
46.76
36.89
36.68
36.68
11.81
12.76
13.89
14.77
14.77
300.4
34
23
21.91
21.91
0.41
4.06
7.31
10.23
10.8
R ¼2
R ¼5
R ¼8
R ¼9
R ¼10
0
0
0
0
0
16
6
3
3
3
2
5
8
9
10
0.86
2.14
3.43
3.86
4.29
2
5
8
8.23
8.17
64.03
37.5
33.49
33.15
32.92
11.63
11.5
11.39
11.16
10.89
52.4
26
22.1
22.00
22.03
17
7
4
4
4
2
5
8
9
9.13
64.73
39.31
36.49
36.63
36.68
12.33
13.31
14.39
14.72
14.77
52.4
26
22.1
21.91
21.91
1.08
4.6
8.22
9.49
10.23
130
O. Kaya et al. / Int. J. Production Economics 143 (2013) 120–131
inventory and waiting costs by hmT 2a þwlT 2a (see Fig. 2). Since S is
optimal, K t hmT 2a wlT 2a r0. Thus
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Kt
R
:
if
Ta r
Ta Z
2
hm þwl
Thus
T a Z min
(sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )
Kt
R
,
:
hm þwl 2
By combining the two bounds found above, if m Z 2, we have
mr
K pl
(
Kt
R2
wmðmlÞmin
,
hm þ wl 4
)
then R=2 is a lower bound for Tb. Now, assume that T b rR=2 and
consider a solution S00 , which combines two transportation cycles
of length Tb of the solution S into one, everything else remaining
the same. Note that this is always possible, as we assume n Z 2
and T b r R=2. Solution S00 saves Kt by reducing the number of
transportation cycles by 1, while incurring additional inventory
and waiting costs, hmT 2b þwlT 2b (see Fig. 2). Since S is optimal,
K t hmT 2b wlT 2b r0. Thus
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Kt
R
if T b r :
Tb Z
2
hm þ wl
Thus
T b Z min
(sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )
Kt
R
,
:
hm þ wl 2
As a result, if n Z 2, we have
C.1.1. Upper bound on m for finite shipment capacity case
When the shipment capacity is finite, in order to limit the
shipment size by the capacity C we have T a r C=m. The upper
bound for Ta given in Appendix C.1 is still valid if T a rC=2m since
we can still combine two shipments into one in that case. Hence,
the lower bound for Ta becomes
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi)
(
C R
Kt
, ,
T a Z min
:
2m 2
hm þ wl
Accordingly, if mZ 2, the upper bound on m is given by
mr
(
K pl
C R
wmðmlÞ min
, ,
2m 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi)!2 :
Kt
hm þ wl
C.2. Upper bound on n
The proof is similar to the proof of the upper bound on m. In
the first part, we find an upper bound on n by comparing the total
cost of an optimal solution S with that of another solution S0 . As
this upper bound will include the decision variable Tb, we then
find a lower bound on Tb, which establishes the result of Lemma 2.
If n¼ 0 or n ¼1, n is already bounded by 1, so an additional
upper bound is not needed. Hence, we concentrate on the cases
with n Z 2.
Let S be the optimal solution of the problem with a cycle
length of T ¼ T 1 þ mT a þnT b (see Fig. 2). Consider another solution
S0 which is the same as S, except that S0 stops the production
lT b =m time units before the stopping time of the production in S.
As a result n0 ¼ n1 in S0 , while T1, Ta, Tb and m remains the same.
When n is decreased by 1, the whole production cycle ends at
T 0 ¼ TT b .
In order to compare the costs of S and S0 in the time interval
ð0,TÞ, we assume that S0 completes another production cycle in
ðT 0 ,TÞ. In this new cycle, the production starts at time
T 0 þðT b ðmlÞ=mÞ and lasts for T b l=m time units. Hence, all the
orders accumulated in ðT 0 ,TÞ are produced and shipped to the
retailer at time T.
Observe that the total customer waiting costs at the retailer in
both S and S0 will be the same. Solution S0 has an additional cost of
Kp since it includes two production cycles. On the other hand,
there will be a saving of nhlðmlÞT 2b =m in the inventory holding
costs at the supplier (see Fig. 2), since production is stopped at an
earlier time and n0 ¼ n1. As S is optimal, K p ðnhlðmlÞT 2b Þ=m Z 0.
Thus n r K p m=hlðmlÞT 2b .
This upper bound depends on Tb which is also a decision
variable. Thus, we find a lower bound on Tb in order to obtain an
upper bound on n in terms of the problem parameters. If T b Z R=2,
K pm
(
nr
hlðmlÞmin
Kt
R2
,
hm þ wl 4
):
C.2.1. Upper bound on n for finite shipment capacity case
When the shipment capacity is finite, in order to limit the
shipment size by the capacity C we have T b rC=l. The upper
bound for Tb given in Appendix C.2 is still valid if T b r C=2l since
we can still combine two shipments into one in that case. Hence,
the lower bound for Tb becomes
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi)
(
C R
Kt
:
, ,
T b Z min
2l 2
hm þwl
Accordingly, if n Z 2, the upper bound on n is given by
nr
(
Kpm
C R
hlðmlÞ min
, ,
2l 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi)!2 :
Kt
hm þwl
Appendix D. Proof of Lemma 3
Assume with contradiction that p trucks are used in a shipment with p 4 1. Then, instead of waiting for the load of p trucks
to accumulate and sending all of them in a single shipment with p
trucks, dividing this shipment into p by making a shipment
immediately every time a single truck load is accumulated and
making the pth shipment exactly at the same time of the original
shipment, will lead to lower inventory and waiting costs while
the transportation costs remain the same. Thus, using p trucks in
a shipment with p 4 1 cannot be optimal.
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