Journal of Applied Science and Engineering, Vol. 18, No. 3, pp. 213-222 (2015)
DOI: 10.6180/jase.2015.18.3.01
Integrated Supply Chain Model with Price-dependent
Demand and Product Recovery
Yu-Jen Lin1 and Hsien-Jen Lin2*
1
Department of Industrial Engineering and Management, St. John’s University,
Tamsui, Taiwan 251, R.O.C.
2
Department of Applied Mathematics, Department of Finance and Actuarial Science, Aletheia University,
Tamsui, Taiwan 251, R.O.C.
Abstract
This article is concerned with the problem of single-vendor single-buyer integrated supply
chain inventory with price-dependent demand and product recovery. In our model, we assume that the
vendor inspects each lot product in one production run in advance before delivering to the buyer, in
which if defective items are unrecoverable, then he/she disposes of them; otherwise, which are
immediately recovered for reuse. In addition, we consider that the demand rate is a function of unit
price charged by the vendor to the buyer to widen applications to the model. The objective of this
article is to minimize the joint total cost per unit time by simultaneously optimizing the buyer’s order
quantity, the size of each shipment of the vendor, and the number of shipments from the vendor to the
buyer in one production run. Furthermore, an algorithmic procedure of finding the optimal solution is
developed, and finally, some numerical examples are given to illustrate the results.
Key Words: Inventory, Lot Sizing, Price-dependent Demand, Defective Items
1. Introduction
As a consequence of environmental necessities, recovery and recycling play an increasingly important role
in today’s inventory management. In general, the vendor wishes manufactured products delivered to the buyer
are non-defective items. Thus, the vendor inspects each
lot product in every production run in advance before
delivering to the buyer. When the vendor discovers the
defective items, he/she adopts the recovery strategy; the
idea is not only one of the most efficient ways to protect
his/her reputation but also to reduce unnecessary waste.
Schrady [1] was the first to introduce the concept of reusable resource in constant demand and return rates
inventory model. Nahmias and Rivera [2] extended
Schrady’s [1] model by considering an EOQ inventory
model with finite recovery rate. Also, Mabini et al. [3] ex*Corresponding author. E-mail: [email protected]
tended Schrady’s [1] model that presented an EOQ model
with stock-out service level constraints and a multi-item
system. Inderfurth et al. [4] proposed a periodic review
inventory model by considering the product recovery in
stochastic remanufacturing system with multiple reuse
options. Teunter [5] studied an economic ordering quantity inventory model with recoverable items. Koh et al.
[6] modified Nahmias and Rivera’s [2] model. They proposed an inventory model with recovery policy for reusable items in which they assumed that backorders are
not permitted and the defective items adopted a recovery
process. Teunter and Vlachos [7] presented a disposal
option for recovery items that can be remanufactured.
Teunter [8] considered a lot-sizing for inventory model
with the production/procurement of new and recovered
items. Bayindir et al. [9] presented a probabilistic recovery inventory control model. Mitra [10] proposed a remanufactured products pricing model, and maximized
the expected revenue. Mitra [11] constructed a two-eche-
214
Yu-Jen Lin and Hsien-Jen Lin
lon inventory model with returns under generalized conditions, and conducted a deterministic model without
shortage and a stochastic model, in which any shortfall in
end-item demand at stage 1 is backordered. Teng et al.
[12] extended Mitra’s [11] deterministic model to optimize the partial backordering inventory model with product returns. Lin [13] extended Teng et al.’s [12] model
to stochastic demand patterns to accommodate more practical features of the real inventory system. Reviews of
literature on inventory systems with returns are available
in Fleischmann et al. [14] and Guide et al. [15].
In practice, the issue of the integrated inventory system with defective items is paid much attention and
widely discussed. The defective items of systems may
be produced due to the imperfect production process
and/or damage in transit. In general, the defective items
would damage to the company’s reputation. Research
on the defective items in the inventory management has
been mounting steadily for a number of decades. Porteus
[16] is among the first ones who analyzed a significant
relationship between quality imperfection and lot size.
Paknejad et al. [17] proposed a modified EOQ model with
stochastic demand and considered the number of defective items in a lot as a random variable; the defective
items in each lot are discovered and returned to the vendor at the time of delivery of the next lot. Ouyang and
Chang [18] investigated an investing capital in quality
improvement model involving the imperfect production
process with a variable lead time and partial backorders.
Tripathy et al. [19] presented an inventory model with an
imperfect production process, in which the unit production cost is directly related to process reliability and inventory is related to the demand rate. There are other related studies of defective items, such as [20-25], and so
on.
In recent years, many studies focus on the issue of integrated vendor and buyer’s inventory management system. Entrepreneur management of nowadays, one of the
most effective ways to achieve overall maximal profit
seems to be the supply chain management. Goyal [26] is
the first researcher to propose an integrated inventory
model for a vendor-buyer inventory problem such that
the integrated inventory system has minimum total cost.
Banerjee [27] generalized Goyal’s [26] model by considering a joint economic-lot-size model in which a ven-
dor produces on an order of a buyer on a lot-for-lot basis.
Goyal [28] modified Banerjee’s [27] model by relaxing
the assumption of a lot-for-lot vendor policy and suggested that the vendor’s economic production quantity is
a positive integer multiple of the buyer’s purchase quantity. Next, Goyal and Nebebe [29] developed an integrated
inventory model in which the first shipment is smaller
and is followed by shipments of the equal size. Kelle et
al. [30] proposed an integrated inventory model in which
the retailer orders nq quantity per order and the vendor’s
production lot size is mq and delivers in n shipments to
the retailer, where m can differ from n. Further, Ouyang
et al. [31] developed a single-vendor single-buyer integrated inventory model involving defective items. They
applied various methods to manage the defective rate
that involves fuzzy defective rate, fuzzy annual total expected cost, and finally adopted the signed distance procedure to estimate the joint total expected cost. Other related studies of the integrated inventory system include
[32-36], and so on.
In view of the aforementioned papers, we attempt to
establish an integrated vendor-buyer supply chain inventory system with price-sensitive demand and recovery
policy; the vendor and the buyer are devoted to forming
a long-term strategic alliance and trading parties and
then achieving a win-win situation. The objective of this
article is to minimize the joint expected total cost per unit
time of the system, and simultaneously to find the optimal buyer’s order quantity, each shipment size of the
vendor, and number of shipments from the vendor to the
buyer in one production run. Finally, an algorithm is developed to find the optimal solution, and numerical examples are given to demonstrate the proposed model.
This article is organized as follows. In the next section, the notation and assumptions are presented. Section
3 establishes the integrated inventory’s basic model involving three mathematical models: (1) the buyer’s total
cost per unit time; (2) the vendor’s total cost per unit time
and (3) the joint total cost per unit time. And then we derive the optimal buyer’s order quantity, each shipment
size of the vendor and number of shipments from the
vendor to the buyer in one production run. Numerical examples are provided to illustrate the proposed model in
section 4, and section 5 is a summary of the work done in
this paper.
Integrated Supply Chain Model with Price-dependent Demand and Product Recovery
215
2. The Basic Model
In this section we discuss the defective items with a
simple recovery process for an integrated supply chain inventory system. When the buyer orders quantity Q (nondefective items) each time, the vendor delivers the lot of
size Q in m shipments. The first shipment size is q, and
then followed by (m - 1) shipments; each is equal to aq
in (m - 1) shipments, where a = (1 - g)P/D. For more details regarding this production/shipment policy, we refer
the readers to Goyal and Nebebe [29]. Moreover, we
consider that the vendor inspects each batch item in advance before delivering to the buyer. Once the defective
items are discovered, the vendor deals with these defective items as follows: (1) if the defective items can be repaired for reuse, then they are sold to the third-party buyers for unit price, w3, less than p (due to possibly with a
loss of quality) on the different market for profit, and (2)
otherwise, the defective items are disposed. In the following subsections, we build up three types of inventory
mathematical model-determination of the buyer’s total
cost per unit time, determination of the vendor’s total
cost per unit time, as well as determination of the joint total cost per unit time.
2.1 Determination of Buyer’s Total Cost Per Unit
Time
In this subsection, we consider the buyer’s inventory
model. As stated in assumptions 3 and 5, an arriving order items have been inspected by the vendor, and therefore in each shipment items from the vendor to the buyer
are all non-defective items Q = q + (m - 1)aq units.
Additionally, we assume that the buyer receives the
first shipment of quantity q (non-defective items) and then
its length of the shipping cycle is q/D. On the other hand,
the followed by (m - 1) shipments, the number of nondefective items is (m - 1)aq and its length of the shipping
cycle is (m - 1)aq/D. Thus, the buyer’s average inventory of non-defective items per unit time is
(1)
Hence, the expression for the buyer’s total cost per
unit time can be written as follows:
(1a)
2.2 Determination of Vendor’s Total Cost Per Unit
Time
In the supply chain system, the length of cycle for the
vendor is [Q/(1 - g)]/D, the vendor’s set-up cost per setup is S, the transportation cost per delivery is K and the
number of defective items is g[Q/(1 - g)], so that the
vendor’s inspecting cost is w0[Q/(1 - g)], the disposed
treatment cost is lg[Q/(1 - g)]w1, the repaired treatment cost is (1 - l)g[Q/(1 - g)]w2 and the extra profit
for which the vendor sold the defective but recoverable
items to the third-party buyers is (1 - l)g[Q/(1 - g)]w3. On
the other hand, the vendor’s initial stock is [q/(1 - g)]D/P
and the number of stock required by the buyer during the
first shipment is q. Once the production run is started, the
total stock increases at a rate of (P - D) until the complete batch quantity of [q/(1 - g)][1+(m - 1)a] has been
manufactured. Hence, the maximum inventory level in the
supply chain system is [q/(1 - g)]D/P + [q/(1 - g)] [1 + (m
- 1)a][(P - D)/P]. Therefore, the vendor’s average inventory level can be determined by subtracting the buyer’s
average inventory level from the average inventory of
the system. So, the vendor’s average inventory level can
be calculated as follows:
(2)
Hence, the vendor’s average holding cost per unit
time is
(2a)
The resulting vendor’s average total cost per unit time
is therefore.
TCs (m) = production cost + setup cost + transportation
cost + holding cost + inspecting cost + disposed treatment cost + repaired treatment cost - extra profit from
the third-party buyers
216
Yu-Jen Lin and Hsien-Jen Lin
Firstly, for a fixed positive integer m, taking the first
and second order derivatives of JTC (q, m) with respect
to q, we get
(4)
and
(2b)
2.3 Determination of the Joint Total Cost Per Unit
Time
In this subsection, we consider the integrated vendorbuyer cooperative models to achieve overall minimal total cost. On production and inventory strategies, once the
vendor and buyer establish a long-term strategic partnership, they can coordinate their production and inventory
strategies and jointly determine the optimal policy for the
integrated supply chain system.
The joint total cost per unit time for the vendor and
buyer is as follows:
(5)
Consequently, the joint total cost per unit time JTC
(q, m) is a convex function for q when m is fixed. Thus,
for given m, the minimum joint total cost per unit time
JTC (q, m) will occur at the point q which satisfies
¶JTC ( q, m)
= 0. Hence, setting equation (4) equal to
¶q
zero, we obtain
(6)
Substituting equation (6) into equation (3), we obtain
(7)
(3)
where
(3a)
Obviously, we observe that the equation (7) is dependent of m (where m is a positive integer) and a convex
function. The optimal value of m (denoted by m*) can be
obtained when
(8)
(3b)
(3c)
Thus, we establish the following Algorithm to obtain
the optimal solution of (q*, m*). Once q* and m* are
found, the joint total cost per unit time JTC (q*, m*) has a
minimum value.
Integrated Supply Chain Model with Price-dependent Demand and Product Recovery
Algorithm
Step 1. Set m = 1.
Step 2. Utilize equation (7) to compute the corresponding JTC (m).
Step 3. Set m = m + 1, and repeat step 2 to get JTC (m).
Step 4. If JTC (m) £ JTC (m - 1), then go to step 3, otherwise go to step 5.
Step 5. Set JTC (m*) = JTC (m - 1), then m* is the optimal solution.
Step 6. Substituting m* into equation (6) to get q*. The
value (q*, m*) is the optimal solution and hence,
JTC (q*, m*) is the minimum joint total cost per
unit time.
3. Numerical Examples
Example 1
To validate the solution procedure, we consider the
following data: a = 100,000, b = 2.5, p = $4/unit, A =
$100/order, a = 1.5, v = $3/unit, S = $350/setup, K = $25/
shipment, hs = $1/unit/year, hr = $2/unit/year, w0 = $0.5/
unit, w1 = $1/unit, w2 = $1.5/unit, w3 = $3.5/unit, g = 0.005
and l = 0.003. Applying these data to the algorithm procedure, the optimal integrated inventory policy can be
easily found and the results are tabulated in Table 1.
The results of Table 1 show that when the buyer or-
217
ders quantity Q* = 2,863.20 each time, the vendor delivers the lot of the first shipment size is q* = 197.46, followed by nine shipments, each equal to aq* = 269.19,
the optimal number of shipments from the vendor to the
buyer per production run is m* = 10, and the optimal
joint annual total cost is JTC (q, m) = $24,927.84. The
results reveal that the optimal joint annual total cost of
adopting repair and disposal strategy is less than that of
merely adopting disposal strategy.
Example 2
Using the same parameter values as in example 1, we
try to change the values of rate of defective items, g, on
the entire integrated inventory system. Let us consider g
= 0.005(0.005)0.04. We apply the algorithm to obtain the
optimal solutions and the results are given in Table 2.
From Table 2, we observe that when the rate of disposal items is fixed (l = 0.003) and the rate of defective
items increases (therefore, the repaired items increases),
the optimal joint annual total cost and the vendor’s total
cost decrease. Therefore, the vendor should endeavor to
enhance the repair ability so as to reduce his/her cost
and then improve the joint annual total cost of the entire
supply chain system. Besides, it is interesting to see that
when the rate of defective items increases, the optimal
order quantity, Q*, and the total cost of the buyer de-
Table 1. Summary of the optimal solutions for Example 1
aq*
q*
Q*
Total cost ($/year)
m*
Buyer
Vendor
Channel
Defective items adopting repair and disposal strategy
197.46
269.19
2,863.20
10
12,898.52
12,029.32
24,927.84
Defective items adopting disposal strategy
197.46
269.19
2,863.20
12,898.52
12,076.05
24,974.57
10
Table 2. Summary of the optimal solutions for Example 2
g
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
q*
aq*
Q*
m*
TCr (q*)
TCs (m*)
JTC (q*, m*)
197.46
196.52
195.59
194.66
193.74
192.81
191.90
190.98
296.19
294.78
293.39
291.99
290.61
289.22
287.85
286.48
2,863.20
2,849.62
2,836.10
2,822.64
2,809.23
2,795.87
2,782.57
2,769.31
10
10
10
10
10
10
10
10
12,898.52
12,897.67
12,896.83
12,895.99
12,895.16
12,894.35
12,893.54
12,892.73
12,029.32
12,006.31
11,983.33
11,960.38
11,937.47
11,914.59
11,891.74
11,868.92
24,927.84
24,903.98
24,880.16
24,856.38
24,832.64
24,808.94
24,785.28
24,761.66
218
Yu-Jen Lin and Hsien-Jen Lin
crease. In other words, due to the increased rate of defective items, it would induce the buyer to order a smaller
quantity and even turn to other company for ordering.
Thus, buyer’s ordering cost is increasing.
Example 3
Using the same parameter values as in example 1, we
try to change the values of rate of disposal items, l, on
the entire integrated inventory system. Let us consider l
= 0.003(0.08)0.723. We apply the algorithm to obtain the
optimal solutions and the results are given in Table 3.
From Table 3, we observe that when the rate of defective items is fixed (g = 0.005) and the rate of disposal
items increases, in this situation, more the defective items
are disposal, that is, the repaired items decrease, the optimal joint annual total cost and the optimal total cost of
the vendor increase, which means the total cost of the
vendor and the rate of disposal items go in the identical
direction. Therefore, the vendor should endeavor to reduce the rate of disposal items so as to decrease his/her
cost and then improve the joint annual total cost of the
entire supply chain system.
Example 4
In this example, we want to examine the effect of various values of the buyer’s and vendor’s average inventory holding cost per unit time on the joint annual total
cost. Except the different values of the buyer’s and vendor’s average inventory holding costs, we adopt the same
parameter values as in example 1. The computational results are tabulated in Table 4.
From the results in Table 4, it reveals that an increase
in the value of the buyer’s and/or vendor’s average in-
ventory holding cost, results in an increase in the joint
annual total cost. On the other hand, it is interesting to
observe that an increase in the value of the buyer’s average inventory holding cost, hr, results in an increase in
the optimal number of shipments from the vendor to the
buyer per production run, m*, but an increase in the value
of the vendor’s average inventory holding cost, hs, results in a decrease in the optimal number of shipments
from the vendor to the buyer per production run, m*,
which means increasing vendor’s average inventory holding cost, hs, then the vendor tends to ship the larger quantity for each time in one production run.
Example 5
Using the same data proposed in example 1, we perform a sensitivity analysis by changing the unit price
charged by the vendor to the buyer, p, by +50%, +25%,
-25% and -50%, and keeping the remaining parameters
unchanged. The results are shown in Table 5.
On the basis of the results of Table 5, we observed
that D, q*, Q* and JTC (q*, m*) decrease with an increase in the value of the model parameter p, while m* is
Table 4. Computational results for various values of hs
and hr
hs
hr
Q*
m*
JTC (q*, m*)
1
1
2
3
2,871.68
2,863.20
2,925.06
7
10
13
24,760.94
24,927.84
25,055.12
2
2
3
2,030.58
2,055.47
7
9
25,322.01
25,450.09
3
3
1,657.96
7
25,752.54
Table 3. Summary of the optimal solutions for Example 3
l
0.003
0.083
0.163
0.243
0.323
0.403
0.483
0.563
0.643
0.723
q*
aq*
Q*
m*
TCr (q*)
TCs (m*)
JTC (q*, m*)
1,423.92
1,423.92
1,423.92
1,423.92
1,423.92
1,423.92
1,423.92
1,423.92
1,423.92
1,423.92
2,135.88
2,135.88
2,135.88
2,135.88
2,135.88
2,135.88
2,135.88
2,135.88
2,135.88
2,135.88
20,646.88
20,646.88
20,646.88
20,646.88
20,646.88
20,646.88
20,646.88
20,646.88
20,646.88
20,646.88
10
10
10
10
10
10
10
10
10
10
12,555.26
12,555.26
12,555.26
12,555.26
12,555.26
12,555.26
12,555.26
12,555.26
12,555.26
12,555.26
11,062.11
11,065.86
11,069.61
11,073.36
11,077.11
11,080.86
11,084.61
11,088.36
11,092.11
11,095.86
23,617.37
23,621.12
23,624.87
23,628.62
23,632.37
23,636.12
23,639.87
23,643.62
23,647.37
23,651.12
Integrated Supply Chain Model with Price-dependent Demand and Product Recovery
219
Table 5. Effect of change in the parameter of the unit price of Example 5
Parameters
p
% change
+50
+25
-25
-50
% change in
D
q*
Q*
m*
JTC (q*, m*)
--63.71
--42.78
+105.28+465.69-
--39.76
--24.34
+43.27
+137.840
--39.76
--24.34
+43.27
+137.840
0
0
0
0
--53.15
--34.45
+75.76
+303.840
not influenced as the value of the model parameter p
changes. Moreover, D, q*, Q* and JTC (q*, m*) are very
highly sensitive to the changes in p.
Notation and Assumptions
The mathematical models in this paper are developed
on the basis of the following notation and assumptions.
4. Concluding Remarks
Achieving effective cooperation between the vendor and the buyer is a current managerial concern as well
as an important issue. In this paper, we present an integrated vendor-buyer supply chain inventory system with
a combination of price-dependent demand and product
recovery. The vendor inspects each lot product in a production run, if he/she discovers that the defective items
are unrecoverable and then he/she disposes of them,
otherwise, which are immediately recovered for reuse.
Moreover, the supply chain inventory system was considered to protect our living environment because we
adopt the recovery policy to reduce unnecessary material
waste, and so that the supply chain system achieves great
benefits. Besides, in order to widen applications for our
model, we consider that the demand rate is a function of
unit price charged by the vendor to the buyer. Finally, by
analyzing the joint total cost per unit time of supply
chain system, we develop an algorithmic procedure for
finding the optimal solution.
In future research on this problem, it would be interesting to consider the periodic review integrated inventory system. Another possible extension of this work may
be conducted by considering that the vendor provides a
permissible delay payment to the buyer in this integrated
inventory model.
Acknowledgements
The authors greatly appreciate the anonymous referees for their very valuable and helpful suggestions on
an earlier version of the paper.
Notation
P
vendor’s production rate
S
vendor’s set-up cost per set-up
K
transportation cost per delivery from the vendor to
the buyer
v
vendor’s unit production cost
hs
vendor’s average inventory holding cost per unit
per unit time
m
number of shipments from the vendor to the buyer
per production run, a positive integer
g
rate of defective items that the vendor’s production
process can go out of control, 0 £ g < 1
l
rate of disposal items in each inspection lot, 0 £ l
<1
w0 vendor’s unit inspecting cost
w1 vendor’s unit disposed treatment cost
w2 vendor’s unit repaired treatment cost, w1 < w2 < v
w3 vendor’s unit profit for which the defective but recoverable units are sold to the third-party, w3 > w2
a
the ratio between the production rate of non- defective items and the demand rate; i.e., a = (1 - g)P/D
p
unit price charged by the vendor to the buyer, p > v
D(p) buyer’s demand rate; as a function of unit price
charged by the vendor to the buyer. For notational
simplicity, D(p) and D will be used interchangeably in this article
A
buyer’s ordering cost per order
Q
buyer’s order quantity (non-defective items)
q
size of the first shipment from the vendor to the
buyer in a batch
hr
buyer’s average inventory holding cost per unit per
unit time
220
Yu-Jen Lin and Hsien-Jen Lin
Assumptions
1. There is single-vendor and single-buyer for a single
product in this model.
2. Shortages are not allowed.
3. The buyer orders quantity Q each time, and the vendor delivers the lot of size Q in m shipments. The first
shipment size is q, and then this first shipment is followed by (m - 1) shipments; each is equal to aq in (m
- 1) shipments.
4. The vendor’s production process is not perfect. When
the buyer orders quantity Q, the vendor manufactures the lot Q/(1 - g) consisting of defective items of
gQ/(1 - g) and non-defective items of (1 - g)Q/(1 - g),
where g (0 £ g £ 1) represents the defective rate in an
order lot; vendor’s production rate is finite and greater than the buyer’s demand rate, i.e., (1 - g)P > D.
5. Before the buyer receives an arriving lot Q items, the
vendor has quickly inspected a batch of Q/(1 - g)
items. And then the vendor delivers the non-defective
items of Q to the buyer, where inspection is nondestructive and error free.
6. The vendor inspects each lot items Q/(1 - g) and any
defective item is repaired or disposed of immediately.
If the items are defective and unrecoverable then the
vendor disposes these items at a disposed treatment
cost w1 per unit. The others will be repaired at a cost
w2 per unit and then sold to third-party buyers at a unit
price, w3, less than p (possibly with a loss of quality)
on the different market for profit.
7. The demand rate of buyer is a downward sloping function of unit price charged by the vendor to the buyer,
and is given by D(p) = ap-b, where a > 0 is a scaling
factor, and b > 1 is a price-elasticity coefficient.
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1413
Manuscript Received: Oct. 9, 2014
Accepted: Aug. 6, 2015