User Phase Information Based Inventory Policy for Supply Chain Systems with Remanufacturing
User Phase Information Based Inventory Policy for Supply Chain
Systems with Remanufacturing
Ming Dong1*, Feng Zhu2, and Dali Zhang2
* Corresponding author: Tel.: +86-21-5230-1193; Mobile: +86-158-2192-9669; E-mail: mdong@sjtu.edu.cn
1Antai College of Economics & Management
Shanghai Jiao Tong University
Shanghai, 200052, P. R. China
2Sino-US Global Logistics Institute
Shanghai Jiao Tong University
Shanghai, 200230, P. R. China
ABSTRACT
Most of the literatures for inventory management policies of supply chain systems with remanufacturing had
focused solely on warehouse operation. The existence of its optimal policy has yet been confirmed. In addition, the
IT platform nowadays has been well developed so that user phase alignment could be executed at relatively low
cost and it is a global phenomenon for companies to pursue cross-tier stretch along its value chain. In this study, a
supply chain model comprised of three major biz entities (warehouse, remanufacturing facility and user phase) is
presented. An optimal (s,S) type policy for this model is first introduced under the non-leadtime assumption and
then extended to the leadtime scenario. The theoretically optimal policy is not applicable as a consequence of
lacking information with regard to the failure product’s return flow. In order to solve this issue, data mining within
the user phase is executed and the specification of user phase information that serves to implementing the optimal
policy is articulated. Additionally, numerical experiments are presented to verify the optimality of the (s,S) type
policy and the contribution of user phase information sharing to inventory cost reduction.
1. INTRODUCTION
The optimality of (s,S) policy for a broad range of stochastic inventory problems is one of the most essential results
in inventory management theories, where s denotes the ordering threshold and S is the order-up-to stock level. Under
the assumption that all cost functions are convex, Scarf [1] proved the optimality of (s,S) policy in inventory problems
with finite planning horizons. Veinott [2] extended Scarf’s work to inventory models with uni-modal holding and
penalty costs. Zipkin and Federgruens [3] verified the optimality of (s,S) policy in infinite planning horizon scenarios.
Based on the contribution of these scholars, (s,S) policy had become a full-fledged inventory management theory.
With the pervasive penetration of sustainability philosophy in modern industries, immense amount of capital is
being invested by various categories of companies to enhance the recyclability within their operation. Increasing
number of firms is chasing an integrated business model aiming at salvaging more residual value from the failure
products. In order to achieve this, the majority of companies are seeking opportunities of achieving more presence on
the user phase through collaborative inventory management tactics or proper recycling incentive schemes.
Remanufacturing is becoming an important source to replenish inventory other than procurement. However, no optimal
policy has currently been confirmed dealing with this hybrid manufacturing/remanufacturing inventory problems. On
the other hand, conventional inventory optimization strategies, including the (s,S) policy, place exclusive emphasis on
the warehouse operation, which sometimes are unsuitable nowadays when put into real practice. Hence, the
conventional (s,S) type inventory policy, from the inventory management perspective of view, needs to be revised in
order to conquer these new business situations.
The research of inventory policies for supply chain systems with remanufacturing dated back to the 1960’s and
there are basically two primary research lines: deterministic models and stochastic models. Schrady [4] established an
EOQ-based remanufacturing inventory model that inspired a series of consequent researches in this area. Nahmias and
Rivera [5] published a remanufacturing inventory management model with remanufacturing capacity constraints.
Dobos and Richter [6] introduced the disposal option into traditional remanufacturing models that extends the research
framework of remanufacturing supply chain systems. Van der Lann and Teunter [7] proposed a heuristic inventory
Flexible Automation and Intelligent Manufacturing, FAIM2014
management strategy for deterministic supply chain systems with remanufacturing. Along the stochastic research line,
Simpson [8] proposed a remanufacturing supply chain management policy within limited planning horizon, and
initiated the application of dynamic programming in solving stochastic inventory problems. Inderfurth [9] provided an
inventory control strategy that includes the disposal option of returned products under the random demand and
recovery assumption and also provided heuristic approaches for acquiring the decision parameters. Kiesmuller [10]
brought up a remanufacturing supply chain model based on newsvendor models. Ahiska and King [11] discussed the
optimal inventory policy in supply chain systems with a special ordering and remanufacturing pattern.
As we could conclude from existing literatures of inventory policy research for supply chain systems with
remanufacturing, only very few papers applied the stochastic periodical review policies. Also, the existence of the
optimal inventory policy has yet been confirmed. For the depiction of the user phase demand and the return products,
current literatures simply uses certain distributions (e.g. Poission distribution) to fit the inflow and outflow of products
from the warehouse managers’ standpoint, however, the deficit of referable decision information hinders the
implementation of the optimal inventory policy. Hence, here are the issues that would be stressed in this paper: 1) The
existence of (s,S) type optimal inventory policy in supply chain systems with remanufacturing. 2) The contribution of
user phase information to the implementation of the optimal inventory policy. 3) Specification of the user phase
information would be incurred in order to execute the optimal inventory policy.
2. INVENTORY POLICY FOR SUPPLY CHAIN SYSTEMS WITH REMANUFACTURING
2.1. FRAMEWORK
The stochastic inventory model focuses on the derivation of replenishment decisions over a finite discrete planning
horizon under stochastic demands. In this paper, we discuss a supply chain system involving two distinct sources of
replenishing the inventory of the serviceable products. Except for traditional procurement, the remanufacturing of
failure products is essentially a complementary source of restocking and the quality of remanufacturing products are
assumed to be the same as procured ones. Refer to the Figure.1 for the framework of supply chain systems with
remanufacturing.
Figure 1. Framework of supply chain systems with remanufacturing.
Following notations are included in the model:
K : Setup cost incurred whenever an order is placed
cp : Unit purchasing cost
cr : Unit remanufacturing cost
h: Unit holding cost of serviceable products
v : Unit backorder cost
ξt : Stochastic demand in period t
User Phase Information Based Inventory Policy for Supply Chain Systems with Remanufacturing
rt : Remanufacturing products in period t
pt : Procured products in period t
λ : Procurement leadtime
ϕ (⋅) : Probability density function
xt : Initial stock level at the beginning of period t
yt : Stock level before demand order is delivered
μ : Remanufacturing leadtime
ρ : Remanufacturing rate of used products
τ : Average lifetime of products serving the user phase
N : Number of decision periods
The demand over the planning horizon is a sequence of random variables that are not necessarily identically nor
independent. We consider an inventory management problem with n planning periods from t =1 to t = n . The
timeline of pertinent events are as follows:
1) At the beginning of the t th period, the inventory manager faces an initial inventory level of xt .
2) After checking out the initial inventory, returned products rt whose remanufacturing process has been completed
would be shipped to the warehouse and the inventory level becomes xt + rt .
3) After receiving the remanufacturing products, the inventory manager places an order with quantity pt which
would be arrived after a fixed procurement leadtime of λ periods. Orders made at the beginning of the t th
period will arrive at the beginning of t + λ th period.
4) After issuing the order, outside procurement pt−λ made λ periods ago is delivered and the inventory level
becomes xt + rt + pt−λ .
5) The demand for the current period is realized at the end of the period and then the inventory level becomes
xt + rt + pt−λ −ξt .
We assume that the replenishment of the inventory would be ceased at the end of the planning horizon, so the last
order is placed in period n −λ . Without loss of generality, we assume that purchasing cost for inventory is charged at
the time of order. The case where purchasing cost is charged at the time of delivery could be transform by a
straightforward shift of cost indices. This action has no influence on the inventory level. Excess inventory is carried to
the next period, incurring a per-unit holding cost. On the other hand, each unit of unsatisfied demand is backlogged to
the next period with a per-unit backlog penalty cost. In the last period, the penalty of lost sales can be accounted through
backlog cost.
According to the given timeline of all the pertinent events included in the model, the system dynamics of the
inventory is:
xt+1 = xt + rt + pt−λ − ξt
(1)
Orders made before period t =1 are pipeline inventories when the planning horizon initiates. Notice that for λ ≥1,
the prior procurement quantities pt−λ (t = 1, 2,...,λ ) are known values.
For returned products shipped to the remanufacturing facility, only certain percentage could be recycled. Assume
arbitrarily that 1− ρ percentage of returned products have to be disposed due to irreversible failure. In other words,
only ρ percentage of the return products ret−μ collected from user phase μ periods ago could be remanufactured into
serviceable products in period t . The remanufacturing process for ret−μ could be completed and the products would
delivered to the warehouse at the beginning of period t (denoted as rt ), such that rt = ρ ⋅ ret−μ . Assuming that there is
no difference between the procurement and remanufacturing products in serviceable inventory.
A linear cost structure is assumed and all cost parameters are non-negative. A fixed setup cost is incurred whenever
an outside procurement is triggered. Purchasing cost in period t involves a setup cost K and a unit purchasing cost pt :
P p (2) t ( ) =
K + cp ⋅ pt pt > 0
0 pt = 0
Flexible Automation and Intelligent Manufacturing, FAIM2014
Holding cost of serviceable products in the warehouse is charged proportional to the time and the quantity.
Unsatisfied demands are backordered with a linear punish cost which allows the current inventory level to be negative.
Holding/backlog cost of serviceable products is charged according to the stock level at the end of the period. This
expenditure is proportional to the on-hand stock or unfulfilled demands. Thus, the loss function L(⋅) is defined
corresponds to two contingencies whether demand exceeded inventory or inventory was sufficient to fulfill the orders
in period t .
(3)
The loss function is convex since both the holding and penalty costs are convex increasing functions. The
remanufacturing cost in period t is defined as:
T rt ( ) = cr ⋅ rt (4)
Then, the total cost over n planning horizon could be formulated as:
(5)
We discretize the planning horizon into n periods and the inventory control policy established in this paper is on
periodic review basis. The objective is to determine the dynamic ordering quantities pt so as to minimize the total
expected cost over the planning horizon in response to the stochastic demands. Though a deterministic leadtime λ
exists between an order is made and delivered, we get started with the non-leadtime case for simplification purposes.
2.2. NON-LEADTIME CASE
In this section, dynamic programming is the methodology we implemented here to derive the optimal policy
tackling supply chain systems with remanufacturing. In non-leadtime cases, we acquired the full awareness of the
quantity of serviceable products rt that would be delivered in period t. The remanufacturing cost T rt ( ) is determined
as long as the remanufacturing rate ρ is fixed. Then the procurement quantity pt is the only decision variable we need
to take into consideration in the objective function. Hence, the recursive structure of the dynamic programming could
be modeled as:
(6)
where yt = xt + rt + pt .
To prove the optimality of (s,S) type ordering policy, we first define that:
(7)
Lemma 1 It is economically efficient to make an order from xt , if and only if there is some yt > xt + pt satisfying that
gt xt + pt ( ) > K + gt yt ( ) . And if such an order is indeed placed, the order-up-to level is the yt that minimizes gt yt ( ).
The logic behind Lemma 1 is straightforward. Denote where the minimum of gt (⋅) is obtained to be St . If the
negative gt (⋅) is uni-modal, finding a unique st st < St ( )satisfying gt st ( ) = gt St ( ) + Kindicates that st, St ( )policy is
indeed optimal in period t . However, further scrutinize of the behavior of gt (⋅) reveals that gt (⋅)actually has a couple
of maxima and minima. Thus in this inventory problem of supply chain systems with remanufacturing, we need to prove
the optimality of (s,S) type policy by showing that the oscillations along with these maxima and minima are not large
enough to cause a deviation from (s,S) type.
Lemma 2 gt (x)(t = 1, 2,..., n)are K-convex functions[1].
The necessary and sufficient condition for the optimality of (s,S) type policy is:
K + g(a + x) ≥ g(x) + ag' (x)(a ≥ 0) (8)
With the K-convexity of gt (⋅) being confirmed, we obtain K + g(a + x) ≥ g(x) + ag' (x)by letting b→0 . By
now, we succeed to demonstrate that (s,S) type ordering policy is indeed optimal in the non-leadtime case.
Denote St as the global minimum of gt (⋅) and st st < St ( )is the point that satisfies K + gt St ( ) = gt st ( ). We have the
following optimal ordering policy in the non-leadtime case.
L yt ( ) = h ⋅ yt ( −ξ )ϕ (ξ )dξ
0
yt∫
+ v ⋅ ξ − yt ( )ϕ (ξ )dξ
yt
∞
∫
f = L yt ( ) + P pt ( ) + T rt ( )
t=1
nΣ
ft xt + rt ( ) = min
pt ≥0
P pt ( ) + L yt ( ) + ft+1 yt − ξt
+ rt+1 ( )ϕ ξt
( )ϕ rt+1 ( )dξt
drt+1
0
∞∫
0
∞
∫
gt yt ( ) = cp ⋅ yt + L yt ( ) + ft+1 yt −ξt + rt+1 ( )ϕ ξt ( )ϕ rt+1 ( )dξt drt+1
0
∞∫
0
∞
∫
User Phase Information Based Inventory Policy for Supply Chain Systems with Remanufacturing
Theorem 1
Ordering policy in non-leadtime case:
If xt + rt ≥ st , make no order.
If xt + rt < st , then order up to St .
In period t , issue an order to replenish the inventory to St if the stock level after the delivery of remanufacturing
product xt + rt is lower than st , make no order otherwise.
2.3. LEADTIME CASE
A crucial reason that complicates the ordering process of inventory management problems is the time lag involved
in effecting procurement policies. There are situations in which this lag is sufficiently small and it could be disregarded.
However, this sort of delay is actually significant in most practical cases and neglecting it would lead to inappropriate
inventory strategies. Hence, it is mandate to extend our results to leadtime scenarios. As we have defined, the time lag
between placing an order and its subsequent delivery to the warehouse is the procurement leadtime λ .
Notice that the initial stock level xt comprises the remanufacturing products of previous periods. Assume that
orders pt−λ (t = 1, 2,...,λ )made before period t =1 are already available for the warehouse manager. The recursive
equation of the objective function in the leadtime case could be modeled as follows:
(9)
Lemma 3 The objective function could be transformed into following form:
(10)
where ht (z) satisfies:
(11)
We start our proof of Lemma 3 by taking an inspection into the properties of objective function ft (⋅), it could be
rewrite in following form:
(12)
Assuming differentiable, its first-order derivative contains a variable set of xt + rt + pt−λ + rt+1, pt−λ +1,..., pt−1 { } .
Hence, when the global minimum is obtained, the optimal ordering quantity pt
* is a function that follows the form
pt
* = pt
* xt + rt + pt−λ + rt+1, pt−λ +1,..., pt−1 ( ). This equation shows the fact that the optimal ordering quantity depends on
pt−λ and rt+1 through xt + rt + pt −λ + rt +1 . Continue our deduction process following this logic we would come to the
conclusion that the optimal ordering quantity pt
* = pt
* xt + pt−λ +1 + pt−λ +2 + ... + p2 + Rλ ,t ( ). Thus, we complete the proof
of lemma 3. Notice that ht (z) has exactly the same structure in comparison with the objective function given in
non-leadtime case except that the loss function is replaced by a multiple integral, which is also convex. The optimality
of (s,S) type policy holds as long as the loss function is convex, hence the optimal policy is characterized by st and St :
Theorem 2
Ordering policy in leadtime case:
ft xt + rt, pt−λ ,..., pt−1 ( ) = min
pt ≥0
P pt ( ) + L xt + rt + pt−λ { ( ) + ft+1 xt + rt + pt−λ − ξt
+ rt+1, pt−λ +1,..., pt−1 ( )ϕ ξt
( )ϕ rt+1 ( )dξt
drt+1}
0
∞∫
0
∞∫
ft xt + rt, pt−λ ,..., pt−1 ( ) = min
pt ≥0
P pt ( ) + L xt + rt + pt−λ ( ) + Ert+1 L xt + rt + pt−λ −ξt + rt+1 ( )
ϕ ξt ( )dξt
0
∞∫
+ ...
+ Ert+1,rt+2 L xt + rt + pt−λ +i −ξt+i + rt+1+i ( )
i=0
1Σ
ϕ ξt ( )ϕ ξt+1 ( )dξt dξt+1
0
∞∫
0
∞∫
+ ...
+ ... Ert+1,...,rt+λ
0
∞∫
0
∞∫
L xt + rt + pt−λ +i −ξt+i + rt+1+i ( )
i=1
λ −1 Σ
ϕ ξt ( )...ϕ ξt+λ ( )dξt...dξt+λ + ht xt + r1 + pt−λ +i
i=0
λ −1 Σ
ht (z) = min
pt ≥0
P pt ( ) + ht+1 z + pt −ξt+λ ( )ϕ ξt+λ ( )dξt+λ
0
∞
∫
+ ... Ert+1,...,rt+λ L z + pt − ξt+i
0
λ −1 Σ + rt+ j
j=1
λ Σ
0
∞
∫
0
∞
∫ ϕ ξt ( )...ϕ ξt+λ −1 ( )dξt ...dξt+λ −1
ft xt + rt, pt−λ ,..., pt−1 ( )
= min
p≥0
P pt ( ) + L xt + rt + pt−λ ( ) + ft+1 xt + rt + pt−λ − ξt
+ rt+1, pt−λ +1,..., pt−1 ( )ϕ ξt
( )ϕ rt+1 ( )dξ drt+1
0
∞∫
0
∞∫
= L xt + rt + pt−λ ( ) + min
p≥0
P pt ( ) +
0
∞∫
ft+1 xt + p1 + r1 − ξt
+ rt+1, pt−λ +1,..., pt−1 ( )ϕ ξt
( )ϕ rt+1 ( )dξ drt+1
0
∞∫
Flexible Automation and Intelligent Manufacturing, FAIM2014
If xt + pt−λ +i + Rλ ,t
i=1
λ −1 Σ ≥ st , make no order.
If
xt + pt −λ +i + Rλ ,t
i=1
λ −1 Σ < st , then order up to St
In period t , issue an order to replenish the inventory to St if the summation of initial inventory xt , pipeline
inventory pt−λ +1 + pt−λ +2 + ... + pt−1and the expected quantity of remanufacturing products within the procurement
leadtime Rλ ,t of is lower than st , make no order otherwise.
From the structure of optimal policy in Theorem 2, we could conclude that there indeed exists an optimal (s,S) type
inventory policy for supply chain systems with remanufacturing. However, the unavailability of expected quantity of
remanufacturing products within the procurement leadtime Rλ ,t makes the theoretical optimal policy inapplicable
based just on the warehouse’s perception. Chapter 3 would illustrate how to implement this theory with the help of user
phase characterization.
3. USER PHASE CHARACTERIZATION
The result in chapter 2 indicates that it is impossible to implement the theoretical optimal inventory policy simply
based on in-house information of the warehouse. The lack of information with regard to remanufacturing within the
procurement leadtime reveals the essence of further data mining in the user phase. Zhang and Liu [12] discuss the
synergy of reliability theory in supply chain system management. User phase characterization is the approach this paper
used to implement the theoretical optimal policy.
The overall off-warehouse timespan for products in the remanufacturing supply chain model as described in figure.1
is comprised of two parts: serving time in the user phase and remanufacturing time in the remanufacturing facility.
Assume that the warehouse manager is of full awareness of the process time of failure products in the remanufacturing
facility, which is usually true since both the warehouse and the remanufacturing facility are under the operation of the
same company in most cases. On this basis, the time that products spent in the user phase is the only issue we need to
account and we could assume that the remanufacturing process could be finished in negligible time μ = 0 . This means
that the value of Rλ ,t is determined only by the product’s failure behavior in the user phase. Denote that the failure time
of a product in the user phase to be τ and the quantity of products serving the user phase by period t to be Nt . Thus,
the quantity of serviceable products which is also the quantity of failed products from the user phase from period t
through the procurement leadtime is:
(13)
As we could conclude from equation (13), when the procurement leadtime is larger than the expected lifetime of a
product serving the user phase, the ordering behavior of the user phase is needed in order to estimate Rλ ,t . The
expected value of Rλ ,t is defined as:
(14)
For the lifetime variable τ , we could refer to the failure log in the user phase for more insight of the average serving
time in real practice. Discretize the failure time on period basis and define p(k) as the probability of a product that has
a lifetime of k periods. Then we could rewrite the expected value of Rλ ,t as:
(15)
R(τ , t;λ ) =
N t+λ −τ − N t−τ λ ≤ τ
N t − N t−τ + ξi
i=t
t+λ −τ Σ λ > τ
Rλ ,t = Eξ Eτ R τ , t;λ ( )
= Eξ R(τ , t;λ )ϕ (τ )dτ
0
∞∫
= Nt+λ −τ − Nt−τ ( )ϕ (τ )dτ
L
∞∫
+ Nt − Nt−τ + ξi
i=t
t+λ −τ
Σ
ϕ ξ ( )ϕ (τ )dξ dτ
0
τ∫
0
∞∫
Rλ ,t = Eξ Ek R k,t;λ ( )
= Nt+λ −k − Nt−k ( ) p(k)
k=λ
∞Σ
+ Nt − Nt−k + ξi
i=t
t+λ −k Σ
k=0
λ Σ p(k)ϕ (ξ )dξ
0
∞
∫
User Phase Information Based Inventory Policy for Supply Chain Systems with Remanufacturing
By now, the missing information in implementing the theoretical optimal policy is fulfilled by exploiting the failure
and ordering behavior in the user phase. This methodology opens up an alternative approach for inventory management
research not only for remanufacturing systems but also for inventory studies focus solely on the warehouse operation.
The alignment of reliability theory in solving inventory management problems leaves space for further exploration.
4. A NUMERICAL EXAMPLE
A numerical example is introduced here to illustrate the application of (s,S) type policy for supply chain systems
with remanufacturing and further verify the essentiality of user phase information. In the numerical example, the
demand is assumed to be Poisson distributed with the mean quantity of 20. The policy parameters st and St for each
period are illustrated in Figure 2. The trigger threshold st is stable in comparison with the severe frustration of the
order-up-to level St . Figure 3 illuminates the contribution of user phase information to the inventory cost reduction
under different procurement leadtimes. In a 100-decision-period model, user phase information sharing could save
almost half of the total inventory cost in a scenario with leatime of 1/4 decision horizon, i.e. 25 periods. On this basis,
the alignment of remanufacturing and user phase characterization is indicated as an economically efficient way for
inventory cost optimization.
Figure 2. Decision parameter of the optimal (s,S) type policy over the planning horizon.
Figure 3. The contribution of user phase information sharing to inventory cost reduction under various leadtimes.
Flexible Automation and Intelligent Manufacturing, FAIM2014
Table 1. Simulation Parameter for the numerical example.
Parameter N K h v cp ρ
Value 100 10 0.1 5 1 0.2
5. CONCLUSIONS
The user phase is innovatively introduced to the conventional supply chain models and the optimality of (s,S) type
policy is proved in a stochastic inventory model with procurement leadtime and fixed order set-up cost. In order to
implement the theoretically optimal but practical inapplicable policy, data mining in the user phase is executed to fulfill
the missing information with regard to the remanufacturing. User phase alignment is of necessity in order to acquire the
information with regards to the failure behavior of servicing products. A numerical example is then used to verify its
implementation and the essentiality of information sharing of the user phase.
ACKNOWLEDGEMENTS
This study is supported by the National Natural Science Foundation of China under Contract No. 71131005.
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