# Imperfect Reworking Process Consideration in Integrated

Authors: Ming-Cheng Lo; Ming-Feng Yang
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

This study develops an improved inventory model to help the enterprises to advance their profit increasing and cost reduction in a single
vendor single-buyer environment with general demand curve, adjustable production rate, and imperfect reworking process under permissible delay in payments. For advancing practical use in a real world,
we are concerned with the following strategy determining, which includes the buyer_s optimal selling price, order quantity, and the number of shipments per production run from the vendor to the buyer. An algorithm and numerical analysis are used to illustrate the solution procedure.


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## Body

## 1. Introduction

In this highly competitive globalized
environment, enterprises
are forced to pace their supply according to the requirements of customers. Their
initiative to have quick
customer response will help them to occupy the market and become the market
leaders. Many enterprises attempt to manage their supply chain
effectively. One useful technique to achieve this target is to
use just-in-time (JIT) and the key to a successful JIT system is to be
able to benefit both the vendor and buyer. This is done through the mutual
negotiations and agreements on how the savings are divided (Hahn et al. [1]).
JIT systems in today’s supply chain environment require the
creation of a new spirit of cooperation
between the buyer and the vendor to gain and maintain a competitive advantage.
As Ha and Kim [2] have pointed out, the integrated inventory model can
contribute significantly to this vendor-buyer relationship. In
Ohta and Furutani’s [3] model, a supply chain
system, which consists of the supplier, the buyer, and the customer, where the
buyer corresponds to a wholesaler, analyzes the effect of customer order
cancellations on (s,S) inventory policies for the supplier and
the buyer in the supply chain system.The
spirit of cooperation among enterprises is needed to improve the
effectiveness of the supply chain. One of the common
strategies in the
business cooperation is that the buyers are offered a permissible delay period to pay back for the goods
bought without paying any interest. During this period, the buyer does not need
to pay interest on goods kept in stock. However,
higher interest is charged if the payment for the goods is not paid by the end
of this period. For the
vendor, he has the benefit of attracting the buyer to purchase his goods in
large batches. Therefore, the existence of the
permissible delay period will promote
a vendor’s selling and reduce on-hand stock level. Simultaneously, a buyer can
earn the interest of the sales revenue and reduce the holding stock because of
the reduced amount of capital invested in stock for the duration of the
permissible delay period. Goyal and Cárdenas-Barrón [4] first developed an EOQ model with constant
demand rate under conditions of permissible delay in payments. He supposed that
no deterioration occurs and the capacity of the warehouse is unlimited.
Besides, he also disregarded the difference between the selling price and the
purchase cost, and concluded that the economic replenishment time interval and
order quantity usually increase marginally under permissible delay in payments.
Aggarwal and Jaggi [5] extended the Goyal’s model to deteriorating items. Jamal
et al. [6] farther extended the model of Goyal [7] to permit shortage and
deterioration. Yang and Wee [8] developed a
single-vendor, multibuyers inventory policy of a deteriorating item with a
constant demand rate. Recently, Teng [9]
amended the Goyal’s model by considering the difference between the selling
price and the purchase cost, and found an alternative conclusion. Abad and
Jaggi [10] provided an integrated approach to the vendor for determining his
pricing and credit policy when end demand is price sensitive. They considered the vendor-buyer
relationship under a noncooperative as well as a cooperative situation and
supposed that the vendor follows a lot-for-lot shipment policy. Huang and Yao
[11] aimed at optimally coordinating inventory for a
deteriorating item among all the partners in a supply chain system with a
single vendor and multiple buyers so as to minimize the average total costs. Teng et al. [12] then improved Teng [9] by supposing that
demand rate is price sensitive. Ouyang et al. [13] proposed a model with
adjustable production rate under the condition of permissible delay in
payments. Huang et al. [14] want to extend that fully permissible delay
in payments to the supplier would offer the retailer partially permissible
delay in payments. The retailer must make a partial payment to the supplier
when the order is received. Then the retailer must pay off the remaining
balance at the end of the permissible delay period. Their
research showed that the trade credit strategy, as permissible delay in
payments, could be a win-win strategy. Their analysis also identified that the
total channel profit would increase while the vendor and the buyer could
cooperate to share necessary business information with each other and balance
the rate between production and market demand.It is
impractical that the above integrated vendor-buyer inventory models are assumed that the produced or received products are perfect without any imperfect
quality item. In fact, due to
the deteriorating production process of the vendor
and the damage during the
transportation process from the vendor to the buyer,
an arrival order batch for the buyer may contain some
percentage defectives. Therefore,
the conventional integrated
inventory model without quality consideration
is inappropriate for the situation in which an arrival batch contains
some imperfect quality items. Porteus
[15] first incorporated the effect of defective items into the classical EOQ
model and introduced the alternative of investing in process quality improvement through reducing
uncontrollable process quality parameters. Rosenblatt and Lee [16] also
considered the effect of an unreliable production process into the EPQ model. Their
results showed that the
average percentage of defective items would be increased by reducing the lot size. Lee and
Rosenblatt [17] added process inspection consideration into production runs so
that the change, which could move to the process out of control, could be
inspected and restored earlier than classical EOQ models. Schwaller [18]
extended the EOQ model by joining a known defective rate assumption into the
incoming batches and that fixed and variable screening costs are incurred in finding
and expelling. Zhang and Gerchak [19] considered a joint lot sizing and
inspection policy in an EOQ model where a percentage defective is random. Cheng
[20] recommended an EOQ model with demand-dependent unit production cost and
imperfect production processes. He formulated the problem as a geometric
programing and solved it to get closed-form optimal solutions. Recently,
Ben-Daya and Hariga [21] examined the effect of defective items on production
scheduling and established a mathematical model to illustrate the scheduling
questions. Salameh and Jaber [22] considered a joint lot sizing and inspection
policy under an EOQ model for items with imperfect quality. Their results showed that economic lot
size quantity tends to increase as the average percentage of imperfect quality
items increase. This contradicts with the finding of Rosenblatt and Lee [16].
They also considered that poor-quality items should be sold as a single batch at a discounted price prior to receiving the next shipment. Hayek and Salameh [23] studied an inventory operating
policy under the condition that imperfect quality items would be reworked where
shortages are allowed and backordered. Goyal  [7] proposed a
simple approach to determine the economic production quantity for items with
imperfect quality.From
the above-mentioned arguments, for advancing practical use in a real world, this paper develops an
integrated inventory model with process unreliability consideration and
permissible delay in payments. Imperfect quality items are handled in the same
way as proposed in Salameh and Jaber [22]. Yu et al. [24] developed a production-inventory model
considering a deteriorating item with imperfect quality and partial
backordering. This paper further extends the
model of Ouyang et al. [13] to imperfect quality items. The main purpose is to maximize the
joint total profit from the perspective of both the vendor and the buyer with the following strategy determining, which includes the
buyer’s optimal selling price, order quantity, and the number of shipments per
production run.The rest of this paper is
organized as follows. The
following section describes the notations and assumptions made herein. Section3 reports on the proposed mathematical model
and Section 4 establishes
the solution procedure. Section 5 provides numerical examples
to illustrate the analysis of Sections 3 and 4. The final
section draws the
research conclusions.


## 2. Notations and Assumptions

To establish the proposed model, the following notations
are used.Notations
D(p): Average demand per year, as a
function of the selling price
PA: Vendor’s production rate,
A>DQ: Buyer’s order quantity per ordercV: The unit production cost for the
vendorcB: The unit purchasing cost for the
buyerP: The unit selling price for the
buyer, a decision variableSV: Setup cost per production run for
the vendorSB: Ordering cost per order for the
buyerhV: The unit holding cost rate for
the vendor excluding interest chargeshB: The unit holding cost rate for
the buyer excluding interest chargesF: Transportation cost per shipmentn: The total number of shipments per
production run from the vendor to the buyer, a positive integer and a
decision variableL: Buyer’s replenishment time
interval between successive deliveries and a decision variablem: Buyer’s permissible delay period
offered by the vendor per orderIVp: Vendor’s capital opportunity cost
per dollar per yearIBP: Buyer’s capital opportunity cost
per dollar per yearIBe: Buyer’s interest earned per dollar
per yearZ: Percentage of defective items in Q,
a random variablef(z): Probability density function of
zω: The unit inspecting costcR: Repair cost per item of imperfect
quality for the vendorTPV(n,p,Q): The vendor’s total annual profitTPB(p,Q): The buyer’s total annual profitJTP(n,p,Q): The joint total annual profitETPV(n,p,Q): The expected vendor’s total
annual profit with
ZEJTP(n,p,Q):
The expected joint total annual
profit with
Z.The assumptions made in the paper are as follows.Assumptions
(1) There is a single vendor and single buyer for
a single product.
(2) The isoelastic curve the most conventionally
assumed is selected as a price-demand function form throughout this model and
we setD(p)=γp−β, where γ>0 is a scaling factor, and β≥1 is an index of price elasticity.
(3) The production rate,A, is adjustable.
The ratio between the production rate
A
and the demand rate D
was set to be
A/D=λ, where λ>1 is a constant and
A=λD.
(4) The buyer orders a quantity ofQ
for
each order with an ordering cost SB; the vendor manufactures
at rate
A, in batches of size
nQ
with a lot setup cost SV;
each batch is delivered to the buyer in
n
equally sized shipments. For
each shipment, the buyer brings a transportation cost F.
(5) Successive shipments are scheduled so that
the next one arrives at the buyer when his stock from previous shipment has
just been consumed.
(6) Shortages are not allowed.
(7) The relationship between the buyer’s selling
pricep, buyer’s purchasing cost
cB
and vendor’s
production cost cV is p≥cB≥cV.
(8) The vendor offers the buyer a permissible
delay periodm. During this permissible delay period, the buyer sells
the items and uses the sales revenue to earn interest at a rate of
IBe.
At the end of this time period, the buyer pays the purchasing cost to the
vendor and the items still in stock bring a capital opportunity cost at a rate
of IBp.
(9) During the vendor’s production process, the
produced items are continuously reviewed.
(10) Under JIT manufacturing concept, defective
items are not allowed. For maintaining JIT spirit and conforming to the truth,
we assume that the reworking of defective items starts instantly they are fund
in the same batch cycle and these reworked items are of perfect quality.
(11) In a single batch at the
end of the vendor’s 100% inspecting process, if imperfect
quality items are
found and the repair cost must be paid.
(12) The time horizon is infinite.


## 3. Model Formulation

In this section, we formulate the model for the reality
assuming that the vendor offers the permissible delay in payments to the buyer
and imperfect quality items can be produced during a production run. We make
use of the imperfect quality items consideration to extend the integrated inventory
model established by Ouyang et al. [13]. Imperfect quality items are occurred in the vendor’s production process, and these items can be reworked immediately in the same batch cycle. These imperfect quality items
being reworked are of perfect quality. So the vendor delivers an order quantity
of
Q
with perfect quality to the buyer and the buyer accepts it over
n
times.Figure1 depicts the behavior of inventory levels for both
the vendor and the buyer, which is along the notations and the assumptions
shown above. The joint total annual profit for the vendor and the buyer
consists of (3.1) the vendor’s total
annual profit, and (3.2) the buyer’s total
annual profit.Figure 1
An integrated inventory system for the vendor and the
buyer.


### 3.1. The Vendor’s Total Annual Profit

In each production run, the vendor produces the item in the
quantity ofnQ
with the rate of
A
and brings a setup cost
SV
as the buyer places an order of quantity Q over
n
times. But, the vendor’s manufacturing will
produce some imperfect quality items. It is assumed that each batch of size
Q
produced contains percentage defectives
ZQ. From assumption 10 stated above, the quantity of
ZQ
with imperfect quality must be reworked instantly
when they are fund. These reworked items are
excellent in quality. So the vendor’s production quantity can be divided into
two parts: the quantity of n(1−Z)Q with perfect quality and the reworked quantity of
nZQ
with perfect
quality. Therefore, the total production quantity for the vendor is still nQ and the buyer would receive it in
n
batches, which each has a quantity
of
Q
with no defect. Following the above notations and assumptions, the components in the vendor’s total annual profit function are(i)
sales revenue per year =D(cB−cV),(ii)
setup
cost per year =SV/nL=SVD/nQ,(iii)
inventory holding cost including financing cost per year =(cVhV+cVIVp)×(Q/2)[n(1−1/λ)−1+2/λ],(iv)
opportunity
cost per year for offering the permissible delay periodm = cBIVp×Dm,(v)
inspecting
cost per year =cR×nZQ/nL=cRZD.Thus, the vendor’s total annual profit,TPV(n,p,Q), can be shown to be (3.1)TPV(n,p,Q)=sales  revenue-setup  cost-inspecting  cost-holding   cost−opportunity  cost=D(cB−cV)−SVDnQ−Dω−cRZD−cVQ(hV+IVp)2[n(1−1λ)−1+2λ]−cBIVpDm.


### 3.2. The Buyer’s Total Annual Profit

In this model, the buyer’s replenishment time interval
between successive deliveries isL=Q/D.
The buyer brings an ordering cost
SB
and a transportation
cost
F
for each order of quantity Q. The buyer’s total annual
profit consists of the following components:(i)
sales revenue per year =D(p−cB),(ii)
cost of placing orders per year =SB/L=SBD/Q,(iii)
transportation cost per year =F/L=FD/Q,(iv)
inventory holding cost
per year =cBhB×Q/2,(v)
interest earned from the
sales revenue received during the permissible delay periodm,(vi)
capital
opportunity cost payable for the items unsold after the permissible delay
periodm (notice that this cost only exists if L≥m).Considering the components (v)
interest earned, and (vi) capital opportunity
cost, the model has the following two possible cases based on the values ofL
and m. These two
cases are depicted graphically in
Figure 2.Inventory systems for the buyer based on the
relation betweenL and m.
(a)(b)Case 1  (L<m ).
We first consider Case1 in Figure 2, where L<m,
the component (v), the interest earned per year at a rate of IBe in the time span [0,m] is pIBe(Dm−Q/2).
In addition, the component (vi), capital opportunity cost payable per year
during the time span [0,m] does not exist.
From the above discussions, total profit per year for the
buyer,TPB1(p,Q), is given by (3.2)TPB1(p,Q)=sales  revenue-ordering   cost-transportation  cost-holding   cost+interest  earned=D(p−cB)−SBDQ−FDQ−cBhBQ2+pIBe(Dm−Q2).Case 2  (L≥m).
For Case2 in Figure 2, where L≥m,
the component (v), the interest earned per year at a rate of
IBe
during the time span [0,m] is pIBe(Dm)2/2Q. Next, the
component (vi), capital opportunity cost payable per year during the time span
[m,L] is cBIBp(Q−Dm)2/2Q.
As a result, total profit per year for the buyer,TPB2(p,Q), is (3.3)TPB2(p,Q)=sales  revenue-ordering  cost-transportation  cost-holding  cost+interest  earned−opportunity  cost=D(p−cB)−SBDQ−FDQ−cBhBQ2+pIBe(Dm)22Q−cBIBp(Q−Dm)22Q.


### 3.3. The Expected Joint Total Annual Profit

Hence, the joint total annual profit function,JTP(n,p,Q), can be expressed as(3.4)JTP(n,p,Q)={JTP1(n,p,Q)=TPV(n,p,Q)+TPB1(p,Q),ifL<mJTP2(n,p,Q)=TPV(n,p,Q)+TPB2(p,Q),ifL≥m,  where(3.5)JTP1(n,p,Q)=D(p−cV−ω−cRZ)−DQ(SVn+SB+F)−cVQ(hV+IVp)2[n(1−1λ)−1+2λ]−cBIVpDm−cBhBQ2+pIBe(Dm−Q2),JTP2(n,p,Q)=D(p−cV−ω−cRZ)−DQ(SVn+SB+F)−cVQ(hV+IVp)2[n(1−1λ)−1+2λ]−cBIVpDm−cBhBQ2+pIBe(Dm)22Q−cBIBp(Q−Dm)22Q.To reduce the notations used by (3.5), we set  Y≡cV(hV+IVp).
We also replace Q=DL and D=D(p)=γp−β into the joint total annual profit function
JTP(n,p,Q). Given that
Z
is a random variable with a known probability
density function f(z). Then we set the expected value of
Z, μ=E(Z) and the expected value of (3.4), EJTP(n,p,L), is given as(3.6)EJTP(n,p,L)={EJTP1(n,p,L)=ETPV(n,p,L)+TPB1(p,L)ifL<m,EJTP2(n,p,L)=ETPV(n,p,L)+TPB2(p,L)ifL≥m,    where (3.7)EJTP1(n,p,L)=γp−β{p−cV−ω−cRμ+(pIBe−cBIVp)m−L2{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}}−1L(SVn+SB+F),(3.8)EJTP2(n,p,L)=γp−β{p−cV−ω−cRμ+cB(IBp−IVp)m−L2{cB(hB+IBp)+Y[n(1−1λ)−1+2λ]}+(pIBe−cBIBp)m22L}−1L(SVn+SB+F).


## 3.1. The Vendor’s Total Annual Profit

In each production run, the vendor produces the item in the
quantity ofnQ
with the rate of
A
and brings a setup cost
SV
as the buyer places an order of quantity Q over
n
times. But, the vendor’s manufacturing will
produce some imperfect quality items. It is assumed that each batch of size
Q
produced contains percentage defectives
ZQ. From assumption 10 stated above, the quantity of
ZQ
with imperfect quality must be reworked instantly
when they are fund. These reworked items are
excellent in quality. So the vendor’s production quantity can be divided into
two parts: the quantity of n(1−Z)Q with perfect quality and the reworked quantity of
nZQ
with perfect
quality. Therefore, the total production quantity for the vendor is still nQ and the buyer would receive it in
n
batches, which each has a quantity
of
Q
with no defect. Following the above notations and assumptions, the components in the vendor’s total annual profit function are(i)
sales revenue per year =D(cB−cV),(ii)
setup
cost per year =SV/nL=SVD/nQ,(iii)
inventory holding cost including financing cost per year =(cVhV+cVIVp)×(Q/2)[n(1−1/λ)−1+2/λ],(iv)
opportunity
cost per year for offering the permissible delay periodm = cBIVp×Dm,(v)
inspecting
cost per year =cR×nZQ/nL=cRZD.Thus, the vendor’s total annual profit,TPV(n,p,Q), can be shown to be (3.1)TPV(n,p,Q)=sales  revenue-setup  cost-inspecting  cost-holding   cost−opportunity  cost=D(cB−cV)−SVDnQ−Dω−cRZD−cVQ(hV+IVp)2[n(1−1λ)−1+2λ]−cBIVpDm.


## 3.2. The Buyer’s Total Annual Profit

In this model, the buyer’s replenishment time interval
between successive deliveries isL=Q/D.
The buyer brings an ordering cost
SB
and a transportation
cost
F
for each order of quantity Q. The buyer’s total annual
profit consists of the following components:(i)
sales revenue per year =D(p−cB),(ii)
cost of placing orders per year =SB/L=SBD/Q,(iii)
transportation cost per year =F/L=FD/Q,(iv)
inventory holding cost
per year =cBhB×Q/2,(v)
interest earned from the
sales revenue received during the permissible delay periodm,(vi)
capital
opportunity cost payable for the items unsold after the permissible delay
periodm (notice that this cost only exists if L≥m).Considering the components (v)
interest earned, and (vi) capital opportunity
cost, the model has the following two possible cases based on the values ofL
and m. These two
cases are depicted graphically in
Figure 2.Inventory systems for the buyer based on the
relation betweenL and m.
(a)(b)Case 1  (L<m ).
We first consider Case1 in Figure 2, where L<m,
the component (v), the interest earned per year at a rate of IBe in the time span [0,m] is pIBe(Dm−Q/2).
In addition, the component (vi), capital opportunity cost payable per year
during the time span [0,m] does not exist.
From the above discussions, total profit per year for the
buyer,TPB1(p,Q), is given by (3.2)TPB1(p,Q)=sales  revenue-ordering   cost-transportation  cost-holding   cost+interest  earned=D(p−cB)−SBDQ−FDQ−cBhBQ2+pIBe(Dm−Q2).Case 2  (L≥m).
For Case2 in Figure 2, where L≥m,
the component (v), the interest earned per year at a rate of
IBe
during the time span [0,m] is pIBe(Dm)2/2Q. Next, the
component (vi), capital opportunity cost payable per year during the time span
[m,L] is cBIBp(Q−Dm)2/2Q.
As a result, total profit per year for the buyer,TPB2(p,Q), is (3.3)TPB2(p,Q)=sales  revenue-ordering  cost-transportation  cost-holding  cost+interest  earned−opportunity  cost=D(p−cB)−SBDQ−FDQ−cBhBQ2+pIBe(Dm)22Q−cBIBp(Q−Dm)22Q.


## 3.3. The Expected Joint Total Annual Profit

Hence, the joint total annual profit function,JTP(n,p,Q), can be expressed as(3.4)JTP(n,p,Q)={JTP1(n,p,Q)=TPV(n,p,Q)+TPB1(p,Q),ifL<mJTP2(n,p,Q)=TPV(n,p,Q)+TPB2(p,Q),ifL≥m,  where(3.5)JTP1(n,p,Q)=D(p−cV−ω−cRZ)−DQ(SVn+SB+F)−cVQ(hV+IVp)2[n(1−1λ)−1+2λ]−cBIVpDm−cBhBQ2+pIBe(Dm−Q2),JTP2(n,p,Q)=D(p−cV−ω−cRZ)−DQ(SVn+SB+F)−cVQ(hV+IVp)2[n(1−1λ)−1+2λ]−cBIVpDm−cBhBQ2+pIBe(Dm)22Q−cBIBp(Q−Dm)22Q.To reduce the notations used by (3.5), we set  Y≡cV(hV+IVp).
We also replace Q=DL and D=D(p)=γp−β into the joint total annual profit function
JTP(n,p,Q). Given that
Z
is a random variable with a known probability
density function f(z). Then we set the expected value of
Z, μ=E(Z) and the expected value of (3.4), EJTP(n,p,L), is given as(3.6)EJTP(n,p,L)={EJTP1(n,p,L)=ETPV(n,p,L)+TPB1(p,L)ifL<m,EJTP2(n,p,L)=ETPV(n,p,L)+TPB2(p,L)ifL≥m,    where (3.7)EJTP1(n,p,L)=γp−β{p−cV−ω−cRμ+(pIBe−cBIVp)m−L2{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}}−1L(SVn+SB+F),(3.8)EJTP2(n,p,L)=γp−β{p−cV−ω−cRμ+cB(IBp−IVp)m−L2{cB(hB+IBp)+Y[n(1−1λ)−1+2λ]}+(pIBe−cBIBp)m22L}−1L(SVn+SB+F).


## 4. Methodology

The objective of this paper is to find an optimal inventory
policy to maximize the joint total annual profit between a vendor and a buyer.


### 4.1. Determination of the Optimal Number of Shipments N for Any Given P and L

Firstly, taking the first-order and second-order partial
derivatives ofEJTPi(n,p,L), for i=1,2,
with respect to n, we obtain(4.1)∂EJTP(n,p,L)∂n={∂EJTP1(n,p,L)∂n∂EJTP2(n,p,L)∂n=−γp−βLY2(1−1λ)+SVn2L,∂2EJTP(n,p,L)∂n2={∂2EJTP1(n,p,L)∂n2∂2EJTP2(n,p,L)∂n2=−2SVn3L<0.Therefore, for fixedp and L, EJTPi(n,p,L) is strongly concave on n>0 for i=1,2.
Thus, in each production run, determining the optimal number of shipments n∗,
is simplified to obtain a global optimum.


### 4.2. Determination of the Optimal Replenishment Time Interval L for Any Given N and P

By taking the first-order and second-order partial
derivatives ofEJTPi(n,p,L), for i=1,2,
with respect to L, we have(4.2)∂EJTP1(n,p,L)∂L=−γp−β2{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}+1L2(SVn+SB+F),(4.3)∂2EJTP1(n,p,L)∂L2=−2L3(SVn+SB+F)<0,(4.4)∂EJTP2(n,p,L)∂L=−γp−β2{cB(hB+IBp)+Y[n(1−1λ)−1+2λ]+(pIBe−cBIBp)m2L2}+1L2(SVn+SB+F),(4.5)∂2EJTP2(n,p,L)∂L2=−1L3[γp−βm2(cBIBp−pIBe)+2(SVn+SB+F)].Consequently,EJTP1(n,p,L) is
strongly concave on L for fixed
n and p. So there exists a
unique replenishment time interval value of L1, which
maximizes  EJTP1(n,p,L).  The value of L1 can be
found by equating (4.2) to be zero, and we obtain(4.6)L1=(SV/n+SB+F)(γp−β/2){cBhB+pIBe+Y[n(1−1/λ)−1+2/λ]}.To make certainL1<m, we
exchange (4.6) into inequality L1<m, and get that  (4.7)iff(SVn+SB+F)<γp−βm22{cBhB+pIBe+Y[n(1−1λ)−1+2λ]},then⁢L1<m.Then substituting (4.6) into
(3.7) and rearranging the result
leads to(4.8)EJTP1(n,p)≡EJTP1(n,p,L1)=γp−β[p−cV−ω−cRμ+(pIBe−cBIVp)m]−2γp−β(SVn+SB+F){cBhB+pIBe+Y[n(1−1λ)−1+2λ]}.From the inequality (4.7), we know that if L2≥m,
it means that(4.9)(SVn+SB+F)≥γp−βm22{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}.Next, we need to check the second-order partial derivative
ofEJTP2(n,p,L) for concavity with respect to L.
The part Y[n(1−1/λ)−1+2/λ]=Y[((n−1)(λ−1)+1)/λ],
where n≥1 and λ>1, so it is certainly positive. Accordingly, it follows that
(4.10)γp−βm2(cBIBp−pIBe)+2(SVn+SB+F)≥γp−βm2{cB(hB+IBp)+Y[n(1−1λ)−1+2λ]}>0,and the polynomial (4.5) is negative. Therefore, EJTP2(n,p,L) is also strongly concave on L for fixed n and p.
Similarly, we can obtain an optimum of L2
which maximizes EJTP2(n,p,L). Solving for L2 by equating
(4.4) to be zero, we
have(4.11)L2=(SV/n+SB+F)+(γp−βm2/2)(cBIBp−pIBe)(γp−β/2){cB(hB+IBp)+Y[n(1−1/λ)−1+2/λ]}.To make certainL2≥m,
we exchange (4.11) into inequality L2≥m,
and get that(4.12)iff(SVn+SB+F)≥γp−βm22{cBhB+pIBe+Y[n(1−1λ)−1+2λ]},thenL2≥m.Substituting (4.11) into (3.8) and rearranging(4.13)EJTP2(n,p)≡EJTP2(n,p,L2)=γp−β[p−cV−ω−cR+cB(IBp−IVp)m]−2γp−β[(SVn+SB+F)+γp−βm22(cBIBp−pIBe)]{cB(hB+IBp)+Y[n(1−1λ)−1+2λ]}.Hence, the above processes onL lead to the
following theorem.Theorem 4.1.
For any given n and
p, we can get the
following results.
(i)
If(SV/n+SB+F)<(γp−βm2/2){cBhB+pIBe+Y[n(1−1/λ)−1+2/λ]},
then L*=L1.(ii)
If(SV/n+SB+F)≥(γp−βm2/2){cBhB+pIBe+Y[n(1−1/λ)−1+2/λ]},
then L*=L2.(iii)
If(SV/n+SB+F)=(γp−βm2/2){cBhB+pIBe+Y[n(1−1/λ)−1+2/λ]},  then
L*=m.Proof.
The above processes
onL imply that
Theorem 4.1 holds.


### 4.3. Determination of the Optimal Selling Price P

According to Theorem4.1, we set a function of p, ψ(p),
as a distinction function
which is given to be(4.14)ψ(p)=γp−βm22{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}.ψ(p) is a
monotonically decreasing function of p, and p is a monotonic
variable, where given any p+>p−   such that
ψ(p+)<ψ(p−), because(4.15)dψ(p)dp=−γp−β−1m22{β{cBhB+Y[n(1−1λ)−1+2λ]}+(β−1)pIBe}=−γp−β−1m22{β{cBhB+Y[(n−1)(λ−1)+1λ]}+(β−1)pIBe}<0,  where n ≥ 1, λ > 1 and
β > 1.Utilizing the results in Theorem4.1, we set p0 such
that(4.16)(SVn+SB+F)=ψ(p0)=γp0−βm22{cBhB+p0IBe+Y[n(1−1λ)−1+2λ]}.Then for any givenp which is substituting into ψ(p),
we get(4.17)(SVn+SB+F){<ψ(p),ifp<p0≥ψ(p),ifp≥p0.By comparing (4.7),
(4.12), and
(4.17), the following results can be yielded(4.18)iffp<p0,thenL1<m.iffp≥p0,thenL2≥m.Consequently, we know from (3.6),
(4.8), (4.13),
and (4.18)
that  (4.19)EJTP(n,p)={EJTP1(n,p)=EJTP1(n,p,L1)ifp<p0,EJTP2(n,p)=EJTP2(n,p,L2)ifp≥p0,  where n is fixed.Solving for the optimal selling pricep*, by taking
the first-order partial derivative of (4.8) with respect to p and
equating the result to be zero, we obtain(4.20)∂EJTP1(n,p)∂p=γp−β(1−β)(1+IBem)+βγp−β−1(cV+ω+cRμ+cBIVpm)+γp−β(SV/n+SB+F)2×(β/p){cBhB+pIBe+Y[n(1−1/λ)−1+2/λ]}−IBecBhB+pIBe+Y[n(1−1/λ)−1+2/λ]=0.Then we need to verify the second-order partial derivative
condition for concavity, as(4.21)∂2EJTP1(n,p)∂p2=−βγp−β−2[p(1−β)(1+IBem)+(β+1)(cV+ω+cRμ+cBIVpm)]−γp−β−4(SV/n+SB+F)8×{(β+1)pIBe{2β{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}−(β+1)pIBe}+β(β+2){cBhB+Y[n(1−1λ)−1+2λ]}2}×{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}−3/2<0.Likewise, taking the first-order partial derivative of (4.13) with
respect to p and equating the result to be zero, we get(4.22)∂EJTP2(n,p)∂p=βγp−β−1[cV+ω+cRμ+cB(IVp−IBp)m]+γp−β(1−β)+{βp(SVn+SB+F)+γp−β−1m2[βcBIBp+(12−β)pIBe]}×(γp−β/2){cB(hB+IBp)+Y[n(1−1/λ)−1+2/λ]}(SV/n+SB+F)+(γp−βm2/2)(cBIBp−pIBe)=0.
The second-order condition for concavity that we need to
verify is(4.23)∂2EJTP2(n,p)∂p2=−γp−β−2{β{(β+1)[cV+ω+cRμ+cB(IVp−IBp)m]+(1−β)p}+γp−β{cB(hB+IBp)+Y[n(1−1/λ)−1+2/λ]}[2(SV/n+SB+F)+m2(cBIBp−pIBe)]3/2×{β(β+2)γp−β(SVn+SB+F)2+3βm2(SVn+SB+F)[cB(β+1)IBp−βpIBe]+γp−βm4[(β2−14)(pIBe)2+β(2β+1)cBIBppIBe−β(β+1)(cBIBp)2]}}<0.


### 4.4. Optimal Solution Procedure

Thus, we can use the following solution procedure to find
optimal valuesn,
p, and L for this model. The solution
procedure is commonly known as dichotomy, as in Algorithm 1.Algorithm 1
Step 1: setn=1.Step 2: determine thep0 by solving
(4.16).Step 3: if there exists ap1 where p1≤p0,
and satisfies both the first-order condition as in (4.20)
andthe second-order
condition as in (4.21), then we compute L1(p1)
by (4.6) andEJTP1(n,p1,L1(p1))
by (4.8). If not, we set EJTP1(n,p1,L1(p1))=0.Step 4: if there exists ap2 where p2≥p0,
and satisfies both the first-order condition as in (4.22) andthe second-order
condition as in (4.23), then we compute L2(p2)
by (4.11) andEJTP2(n,p2,L2(p2))
by (4.13). If not, we set EJTP2(n,p2,L2(p2))=0.Step 5:  ifEJTP1(n,p1,L1(p1))≥EJTP2(n,p2,L2(p2)). Set EJTP(n,p[n],L[n])=EJTP1(n,p1,L1(p1)),then (p[n],L[n]) is an optimal solution for a
given n. If not, EJTP(n,p[n],L[n])=EJTP2(n,p2,L2(p2)).Step 6: setn=n+1,
repeat steps 2–5 to obtain EJTP(n,p[n],L[n]).Step 7: ifEJTP(n,p[n],L[n])≥EJTP(n−1,p[n−1],L[n−1]), go to step 6. If not, go to step 8and stop.Step 8: setEJTP(n∗,p∗,L∗)=EJTP(n−1,p[n−1],L[n−1]),
so (n∗,p∗,L∗) is an optimal solution.Consequently,
the buyer’s optimal order quantity per order is Q∗=D(p∗)L∗.


## 4.1. Determination of the Optimal Number of Shipments N for Any Given P and L

Firstly, taking the first-order and second-order partial
derivatives ofEJTPi(n,p,L), for i=1,2,
with respect to n, we obtain(4.1)∂EJTP(n,p,L)∂n={∂EJTP1(n,p,L)∂n∂EJTP2(n,p,L)∂n=−γp−βLY2(1−1λ)+SVn2L,∂2EJTP(n,p,L)∂n2={∂2EJTP1(n,p,L)∂n2∂2EJTP2(n,p,L)∂n2=−2SVn3L<0.Therefore, for fixedp and L, EJTPi(n,p,L) is strongly concave on n>0 for i=1,2.
Thus, in each production run, determining the optimal number of shipments n∗,
is simplified to obtain a global optimum.


## 4.2. Determination of the Optimal Replenishment Time Interval L for Any Given N and P

By taking the first-order and second-order partial
derivatives ofEJTPi(n,p,L), for i=1,2,
with respect to L, we have(4.2)∂EJTP1(n,p,L)∂L=−γp−β2{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}+1L2(SVn+SB+F),(4.3)∂2EJTP1(n,p,L)∂L2=−2L3(SVn+SB+F)<0,(4.4)∂EJTP2(n,p,L)∂L=−γp−β2{cB(hB+IBp)+Y[n(1−1λ)−1+2λ]+(pIBe−cBIBp)m2L2}+1L2(SVn+SB+F),(4.5)∂2EJTP2(n,p,L)∂L2=−1L3[γp−βm2(cBIBp−pIBe)+2(SVn+SB+F)].Consequently,EJTP1(n,p,L) is
strongly concave on L for fixed
n and p. So there exists a
unique replenishment time interval value of L1, which
maximizes  EJTP1(n,p,L).  The value of L1 can be
found by equating (4.2) to be zero, and we obtain(4.6)L1=(SV/n+SB+F)(γp−β/2){cBhB+pIBe+Y[n(1−1/λ)−1+2/λ]}.To make certainL1<m, we
exchange (4.6) into inequality L1<m, and get that  (4.7)iff(SVn+SB+F)<γp−βm22{cBhB+pIBe+Y[n(1−1λ)−1+2λ]},then⁢L1<m.Then substituting (4.6) into
(3.7) and rearranging the result
leads to(4.8)EJTP1(n,p)≡EJTP1(n,p,L1)=γp−β[p−cV−ω−cRμ+(pIBe−cBIVp)m]−2γp−β(SVn+SB+F){cBhB+pIBe+Y[n(1−1λ)−1+2λ]}.From the inequality (4.7), we know that if L2≥m,
it means that(4.9)(SVn+SB+F)≥γp−βm22{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}.Next, we need to check the second-order partial derivative
ofEJTP2(n,p,L) for concavity with respect to L.
The part Y[n(1−1/λ)−1+2/λ]=Y[((n−1)(λ−1)+1)/λ],
where n≥1 and λ>1, so it is certainly positive. Accordingly, it follows that
(4.10)γp−βm2(cBIBp−pIBe)+2(SVn+SB+F)≥γp−βm2{cB(hB+IBp)+Y[n(1−1λ)−1+2λ]}>0,and the polynomial (4.5) is negative. Therefore, EJTP2(n,p,L) is also strongly concave on L for fixed n and p.
Similarly, we can obtain an optimum of L2
which maximizes EJTP2(n,p,L). Solving for L2 by equating
(4.4) to be zero, we
have(4.11)L2=(SV/n+SB+F)+(γp−βm2/2)(cBIBp−pIBe)(γp−β/2){cB(hB+IBp)+Y[n(1−1/λ)−1+2/λ]}.To make certainL2≥m,
we exchange (4.11) into inequality L2≥m,
and get that(4.12)iff(SVn+SB+F)≥γp−βm22{cBhB+pIBe+Y[n(1−1λ)−1+2λ]},thenL2≥m.Substituting (4.11) into (3.8) and rearranging(4.13)EJTP2(n,p)≡EJTP2(n,p,L2)=γp−β[p−cV−ω−cR+cB(IBp−IVp)m]−2γp−β[(SVn+SB+F)+γp−βm22(cBIBp−pIBe)]{cB(hB+IBp)+Y[n(1−1λ)−1+2λ]}.Hence, the above processes onL lead to the
following theorem.Theorem 4.1.
For any given n and
p, we can get the
following results.
(i)
If(SV/n+SB+F)<(γp−βm2/2){cBhB+pIBe+Y[n(1−1/λ)−1+2/λ]},
then L*=L1.(ii)
If(SV/n+SB+F)≥(γp−βm2/2){cBhB+pIBe+Y[n(1−1/λ)−1+2/λ]},
then L*=L2.(iii)
If(SV/n+SB+F)=(γp−βm2/2){cBhB+pIBe+Y[n(1−1/λ)−1+2/λ]},  then
L*=m.Proof.
The above processes
onL imply that
Theorem 4.1 holds.


## 4.3. Determination of the Optimal Selling Price P

According to Theorem4.1, we set a function of p, ψ(p),
as a distinction function
which is given to be(4.14)ψ(p)=γp−βm22{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}.ψ(p) is a
monotonically decreasing function of p, and p is a monotonic
variable, where given any p+>p−   such that
ψ(p+)<ψ(p−), because(4.15)dψ(p)dp=−γp−β−1m22{β{cBhB+Y[n(1−1λ)−1+2λ]}+(β−1)pIBe}=−γp−β−1m22{β{cBhB+Y[(n−1)(λ−1)+1λ]}+(β−1)pIBe}<0,  where n ≥ 1, λ > 1 and
β > 1.Utilizing the results in Theorem4.1, we set p0 such
that(4.16)(SVn+SB+F)=ψ(p0)=γp0−βm22{cBhB+p0IBe+Y[n(1−1λ)−1+2λ]}.Then for any givenp which is substituting into ψ(p),
we get(4.17)(SVn+SB+F){<ψ(p),ifp<p0≥ψ(p),ifp≥p0.By comparing (4.7),
(4.12), and
(4.17), the following results can be yielded(4.18)iffp<p0,thenL1<m.iffp≥p0,thenL2≥m.Consequently, we know from (3.6),
(4.8), (4.13),
and (4.18)
that  (4.19)EJTP(n,p)={EJTP1(n,p)=EJTP1(n,p,L1)ifp<p0,EJTP2(n,p)=EJTP2(n,p,L2)ifp≥p0,  where n is fixed.Solving for the optimal selling pricep*, by taking
the first-order partial derivative of (4.8) with respect to p and
equating the result to be zero, we obtain(4.20)∂EJTP1(n,p)∂p=γp−β(1−β)(1+IBem)+βγp−β−1(cV+ω+cRμ+cBIVpm)+γp−β(SV/n+SB+F)2×(β/p){cBhB+pIBe+Y[n(1−1/λ)−1+2/λ]}−IBecBhB+pIBe+Y[n(1−1/λ)−1+2/λ]=0.Then we need to verify the second-order partial derivative
condition for concavity, as(4.21)∂2EJTP1(n,p)∂p2=−βγp−β−2[p(1−β)(1+IBem)+(β+1)(cV+ω+cRμ+cBIVpm)]−γp−β−4(SV/n+SB+F)8×{(β+1)pIBe{2β{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}−(β+1)pIBe}+β(β+2){cBhB+Y[n(1−1λ)−1+2λ]}2}×{cBhB+pIBe+Y[n(1−1λ)−1+2λ]}−3/2<0.Likewise, taking the first-order partial derivative of (4.13) with
respect to p and equating the result to be zero, we get(4.22)∂EJTP2(n,p)∂p=βγp−β−1[cV+ω+cRμ+cB(IVp−IBp)m]+γp−β(1−β)+{βp(SVn+SB+F)+γp−β−1m2[βcBIBp+(12−β)pIBe]}×(γp−β/2){cB(hB+IBp)+Y[n(1−1/λ)−1+2/λ]}(SV/n+SB+F)+(γp−βm2/2)(cBIBp−pIBe)=0.
The second-order condition for concavity that we need to
verify is(4.23)∂2EJTP2(n,p)∂p2=−γp−β−2{β{(β+1)[cV+ω+cRμ+cB(IVp−IBp)m]+(1−β)p}+γp−β{cB(hB+IBp)+Y[n(1−1/λ)−1+2/λ]}[2(SV/n+SB+F)+m2(cBIBp−pIBe)]3/2×{β(β+2)γp−β(SVn+SB+F)2+3βm2(SVn+SB+F)[cB(β+1)IBp−βpIBe]+γp−βm4[(β2−14)(pIBe)2+β(2β+1)cBIBppIBe−β(β+1)(cBIBp)2]}}<0.


## 4.4. Optimal Solution Procedure

Thus, we can use the following solution procedure to find
optimal valuesn,
p, and L for this model. The solution
procedure is commonly known as dichotomy, as in Algorithm 1.Algorithm 1
Step 1: setn=1.Step 2: determine thep0 by solving
(4.16).Step 3: if there exists ap1 where p1≤p0,
and satisfies both the first-order condition as in (4.20)
andthe second-order
condition as in (4.21), then we compute L1(p1)
by (4.6) andEJTP1(n,p1,L1(p1))
by (4.8). If not, we set EJTP1(n,p1,L1(p1))=0.Step 4: if there exists ap2 where p2≥p0,
and satisfies both the first-order condition as in (4.22) andthe second-order
condition as in (4.23), then we compute L2(p2)
by (4.11) andEJTP2(n,p2,L2(p2))
by (4.13). If not, we set EJTP2(n,p2,L2(p2))=0.Step 5:  ifEJTP1(n,p1,L1(p1))≥EJTP2(n,p2,L2(p2)). Set EJTP(n,p[n],L[n])=EJTP1(n,p1,L1(p1)),then (p[n],L[n]) is an optimal solution for a
given n. If not, EJTP(n,p[n],L[n])=EJTP2(n,p2,L2(p2)).Step 6: setn=n+1,
repeat steps 2–5 to obtain EJTP(n,p[n],L[n]).Step 7: ifEJTP(n,p[n],L[n])≥EJTP(n−1,p[n−1],L[n−1]), go to step 6. If not, go to step 8and stop.Step 8: setEJTP(n∗,p∗,L∗)=EJTP(n−1,p[n−1],L[n−1]),
so (n∗,p∗,L∗) is an optimal solution.Consequently,
the buyer’s optimal order quantity per order is Q∗=D(p∗)L∗.


## 5. Numerical Examples

Example 5.1.
Consider an inventory situation with the following
parametric values partially adopted in Ouyang et al. [13] and Salameh and Jaber
[22]:
(i) scaling factorγ=100000,
(ii) index of price elasticityβ=1.5,
(iii) ratio between the production rate and the demand rateλ=1.5,
(iv) purchasing cost for the buyercB  =
$4.5/unit,
(v) ordering cost for the buyerSB  =
$10/order,
(vi) buyer’s unit holding cost ratehB  =
0.111,
(vii) buyer’s capital opportunity costIBp = 0.08/$/yr,
(viii) buyer’s interest earned rateIBe =
0.06/$/yr,
(ix) transportation costF = $50/shipment,
(x) production cost for the vendorcV =
$2.2/unit,
(xi) setup cost per production run for the vendorSV = $350/setup,
(xii) vendor’s unit holding cost ratehV = 0.046,
(xiii) vendor’s capital opportunity costIVp = 0.03/$/yr,
(xiv) inspecting costω = $0.5/unit,
(xv) repair
cost per imperfect quality itemcR = $2/unit.
The percentage defective random variable,Z, is
uniformly distributed with its probability density function (PDF) as
(5.1)f(z)={25,0≤z≤0.040,otherwise.Therefore,μ=E[Z]=∫00.0425zdz=0.02.The above solution algorithm is applied to get the
computational results for various values of permissible delay periodm as shown in
Table 1.Table 1
Optimal solutions for various values of permissible delay periodm(λ=1.50).
mn∗p0p∗L∗ (day)D(p∗)Q∗n∗Q∗Profit ($/yr)(day)VendorBuyerJoint010—p2  =
8.6191L2 =
65.95213951.9107714.07347140.73446542.774315639.383122182.15745100.2313p2 =
8.6097L2 =
65.87803958.3844714.43937144.39306546.517715648.152522194.670210100.5903p2 =
8.6007L2 =
65.76603964.5993714.34437143.44346549.811415658.035422207.846815101.0290p2 =
8.5920L2 =
65.61573970.6225713.79497137.94886552.770615668.916322221.686920101.5361p2 =
8.5837L2 =
65.42783976.3830712.78387127.83816555.274415680.917022236.191425102.1074p2 =
8.5758L2 =
65.20243981.8788711.30947113.09366557.319415694.042922251.362330102.7421p2 =
8.5683L2 =
64.93923987.1080709.36887093.68816558.902015708.301122267.203140104.2066p2 =
8.5545L2 =
64.29883996.7599704.07367040.73646560.663415740.251622300.915041104.3680p2 =
8.5532L2 =
64.22633997.6711703.43987034.39776560.745015743.691622304.436642104.5321p2 =
8.5519L2 =
64.15213998.5827702.78777027.87686560.826615747.159322307.985943104.6991p2 =
8.5506L2 =
64.07633999.4946702.11737021.17326560.908015750.654722311.562744104.8689p2 =
8.5494L2 =
63.99944000.3367701.42217014.22136560.8706(b)15754.296722315.167345105.0417p2 =
8.5482L2 =
63.92084001.1791700.70857007.08546560.833015757.966822318.799850105.9502p2 =
8.5424L2 =
63.50414005.2548696.84976968.49676560.402815776.979422337.382260118.4755p2 =
8.5341L2 =
60.27244011.0993662.35227285.87376556.344815820.448322376.7931701111.0359p1 =
8.5309L1 =
60.23434013.3564662.3060(a)7285.36556545.316915872.911222418.2281801114.0380p1 =
8.5280L1 =
60.22214015.4037662.50927287.60086533.908615925.757522459.6661901117.5552p1 =
8.5250L1 =
60.20944017.5234662.71957289.91476522.606415978.499722501.1061(a)  When 0 < m ≤ 70, the optimal order quantity Q∗ is negatively
correlated to the length of the permissible delay period m.(b) When m <
44, the vendor’s expected total annual profit is positively correlated to the
length of the permissible delay period m but as m ≥ 44, it is reverse.Table1 shows that (1) the expected joint total annual
profit increases when the permissible delay period m increases, (2) the
optimal selling price p∗ and the optimal replenishment time interval L∗ are decreasing with the increasing of permissible delay period
m, (3) as
the annual demand D(p∗) is increasing with the decreasing of p∗,
the buyer’s expected total annual profit increases as well as the expected
joint total annual profit, and (4) the optimal order quantity Q∗ decreases with the increasing of permissible delay period within the range of 0
< m ≤ 70. Generally speaking, a
longer permissible delay period offered may motivate the buyer to carry out
frequent shipments in small batches. It also can shorten the replenishment time
interval to utilize the credit period in the profit increasing and cost
reduction more and more. In addition, it also can be observed from Table 1 that
when m < 44, the vendor’s expected total annual profit follows the
value of permissible delay period m increasing. But when m ≥ 44, the expected total annual profit of the vendor
decreases as the value of permissible delay period m increases. These
results indicate that the buyer can always profit from the permissible delay
period. For the vendor, he can also profit from the permissible delay in
payments strategy while the credit period of time is not longer than 44 days.
But on the contrary, if the permissible delay period m is greater than
44 days, the vendor’s expected total annual profit decreases through his sales
revenue by permitting the buyer a credit period of time cannot disburse his
opportunity cost. Based on the above discussions, it illustrates that applying
the permissible delay in payments strategy in an integrated inventory model
would advance the profit increasing and cost reduction.Proceeding to the next, we compare the proposed model
herein with the model established by Ouyang et al. [13]. The two models are
mainly different from considering imperfect quality items or not. These
comparison results are presented in Table 2.Table 2
Comparison results with Ouyang et al. (2005).
Profit ($/yr)m
(day)JointVendorBuyerThis paperOuyang et al.Improved (%)This paperOuyang et al.Improved (%)This paperOuyang et al.Improved (%)022182.157424691.0000−10.166542.774312100.0000−45.9315639.383112591.000024.211022207.846824725.0000−10.186549.811412127.0000−45.9915658.035412598.000024.293022267.203124799.0000−10.216558.902012157.0000−46.0515708.301112642.000024.256022376.793124921.0000−10.216556.344812138.0000−45.9815820.448312783.000023.76λ = 1.50.Clearly, it is seen that imperfect quality items cause a
significant profit loss. The improvement of the joint profit is
greater than 10%. Besides, we also compare the relevant profit of the vendor
and the buyer in the proposed model with Ouyang et al. [13] further. The
results show that the vendor’s profit is downward obviously due to
the effect of imperfect quality items. His profit improvement is very big and the improved range is
greater than 45%. Table 1 reveals that the optimal selling prices
p∗s in
the situations where the values of permissible delay period m∈{0,10,30,60} are all higher than them as in Ouyang et al.
[13]. Hence, this
causes the annual
demand D(p∗) and the optimal order quantity
Q∗ to be smaller
than them as in Ouyang et al. [13], this result also induces the damage of the
vendor’s sales revenue. Furthermore, the process unreliability consideration
between the vendor and the buyer will incur the vendor to bear the warranty
cost. So the vendor’s profit in the proposed model is smaller than it as in
Ouyang et al. [13]. But, the buyer’s profit is upward, and his profit improvement is greater than 23%. The results reveal that the buyer’s
profit increment is from his profit raise owing to the higher optimal selling
price p∗
which
can pay for the total of his demand decrement owing to the higher optimal
selling price p∗ and the inspecting cost incurred in finding and
expelling imperfect quality items. Therefore, the buyer’s profit in the
proposed model is larger than it as in Ouyang et al. [13]. Then from the above
discussions in Table 2, it demonstrates that the proposed model
produces a significant profit loss when comparing with the joint total annual
profit without considering imperfect quality items. These results have really
met the truth.Example 5.2.
We take the same values for the parameters as in Example5.1. Suppose the value of permissible delay period m = 30, we investigate
the effect of the ratio between the production rate and the demand rate, λ.
Similarly, we also compare the proposed model herein with the model of Ouyang
et al. [13]. Following the above solution procedure, the computational results
for various values of the ratio λ are presented in Table 3.Table 3
Optimal solutions for various values ofλ (m = 30).
λn∗p0p∗L∗
(day)D(p∗)Q∗n∗Q∗Profit ($/yr)VendorBuyerJoint1.01612.8477p2 =
8.4577L2 =
60.94044065.5713678.787641406.04216986.033015571.660222557.69321.10202.8641p2 =
8.5068L2 =
61.93664030.4234683.919613678.39156794.958415632.921622427.88001.50102.7421p2 =
8.5683L2 =
64.93923987.1080709.36887093.68816558.902015708.301122267.20312.0082.6533p2 =
8.5947L2 =
66.82793968.7516726.63975813.11796459.024515739.696722198.72123.0072.6202p2 =
8.6158L2 =
67.75103954.1814733.97195137.80346380.221815764.246422144.4682Table3 reveals that the expected total annual profit of
the vendor and the whole integrated inventory model increase as the value of
the ratio λ is close to 1. On the contrary, the buyer’s profit
decreases. These results are the same as the conclusions in Ouyang et al. [13].
The results imply that if the JIT cooperation between the vendor and the buyer
could be implemented successfully, the vendor’s profit and the joint profit
will increase following that the vendor can get the real time demand rate
through the buyer and adjust his production rate to the demand rate. Comparison
results between the proposed model and the model of Ouyang et al. [13] are
shown in Table 4. The results reveal that the profit improvement of the joint
profit is greater than 10%. Obviously, imperfect quality items can lead to a noticeable profit loss.Table 4
Comparison results with
Ouyang et al. (2005).
λProfit ($/yr)JointVendorBuyerThis paperOuyang et al.Improved (%)This paperOuyang et al.Improved (%)This paperOuyang et al.Improved (%)1.0122557.693225140.0000−10.276986.033012758.0000−45.2415571.660212382.000025.761.1022427.880024988.0000−10.256794.958412489.0000−45.5915632.921612499.000025.071.5022267.203124799.0000−10.216558.902012157.0000−46.0515708.301112642.000024.252.0022198.721224719.0000−10.206459.024512017.0000−46.2515739.696712702.000023.923.0022144.468224655.0000−10.186380.221811906.0000−46.4115764.246412749.000023.65m = 30 days.


## 6. Conclusions

This paper investigates a production/inventory situation
which producing process would go out of control under permissible delay in
payments. In this research, we assume that in the vendor’s production process,
the imperfect quality items are reworked immediately as they are found and meantime the vendor must bear the repair cost. This new proposed model herein
shows a different thought on inventory modeling. The expected joint total
annual profit function has been derived. Then by analyzing this derived
function, we can obtain the unique closed-form optimal solution for the
replenishment time interval and develop a simple solution procedure to determine
the buyer’s optimal selling price, order quantity, and the number of shipments
per production runs from the vendor to the buyer. Finally, the numerical
examples adopted in the Ouyang et al. [13] and Salameh and Jaber [22] explain
the solution algorithm. These results reveal that applying the permissible
delay in payments strategy between the vendor and the buyer can promote the
profit increasing and cost reduction. They also indicate that the successful
implementation of JIT cooperation in an integrated inventory model leads to the
profit rise of the whole inventory model. Besides, the proposed model generates
an impressive profit loss when compared with the joint total annual
profit without incorporating imperfect quality items into consideration.



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*Source: 386252-2008-06-18.xml*