30220130403002

International Journal of JOURNAL OF PRODUCTION (IJPTM), ISSN 0976 – 6383
INTERNATIONAL Production Technology and ManagementTECHNOLOGY AND
(Print), ISSN 0976 – 6391 (Online) Volume 4, Issue 3, September - December (2013), © IAEME
MANAGEMENT (IJPTM)

ISSN 0976- 6383 (Print)
ISSN 0976 - 6391 (Online)
Volume 4, Issue 3, September - December (2013), pp. 14-31
© IAEME: www.iaeme.com/ijptm.asp
Journal Impact Factor (2013): 4.3285 (Calculated by GISI)
www.jifactor.com

IJPTM
©IAEME

OPTIMAL SHIPMENTS, ORDERING AND PAYMENT POLICIES FOR
INTEGRATED SUPPLIER-BUYER DETERIORATING INVENTORY
SYSTEM WITH PRICE-SENSITIVE TRAPEZOIDAL DEMAND AND
NET CREDIT
1

Nita H. Shah,

2

Digeshkumar B. Shah

&

3

Dushyantkumar G. Patel

1

Department of Mathematics, Gujarat University, Ahmedabad-380009, Gujarat, India
Department of Mathematics, L.D. Engg. College, Ahmedabad- 380015, Gujarat, India
3
Department of Mathematics, Govt. Poly.for Girls, Ahmedabad- 380015, Gujarat, India
2

ABSTRACT
In this research, an integrated supplier-buyer inventory system is studied when market
demand is price-sensitive trapezoidal and units in inventory are subject to deterioration at a
constant rate. The buyer has an option to choose between discount in unit price and delay in
settling the account against the purchases made offered by the supplier. This type of trade
credit is termed as ‘net credit’. In this scenario, if the buyer settles payment within the
stipulated time period M1 , then the buyer receives a cash discount; otherwise the full payment
must be paid by the time M 2 ; where M 2 > M1 ≥ 0. The joint profit per unit time of supplierbuyer is maximized with respect to selling price, purchase quantity, number of transfers from
the supplier to the buyer and payment time. An algorithm is outlined to obtain optimal
solution. The numerical example is given to validate the proposed formulation. The
managerial issues are deduced through sensitivity analysis of inventory parameters.
Key Words: Integrated Inventory Model, Deterioration, Price-Sensitive Trapezoidal
Demand, Net Credit.

14

International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383

1. INTRODUCTION
The classical EOQ model assumes that the buyer uses cash-on-delivery policy which
is no longer a practice followed by the player of the supply chain. Goyal (1985) first
proposed the concept of delay payment policy opted by the supplier for the buyer to derive
economic order quantity. Thereafter, many researchers analyzed inventory model to study
the effect of delay period payment on stimulating the demand. One can refer review by Shah
et al. (2010) on trade credit and inventory policy. The most of the articles in this review
article established that demand stimulates and lowers on-hand stocking for supplier while
buyer can earn interest on the generated revenue. However, the provision for making early
payment was not addressed.
Ho et al. (2008) observed that the offer of trade credit delays cash-flow and increases
the risk of cash-flow shortage for the supplier. To combat this trade-off, the supplier offers a
cash discount in unit purchase price to attract the buyer for early payment. For example, the
supplier offers 3 % discount on buyer’s unit purchase price if payment is made within 10
days; otherwise the account is to be settled within 30 days for the purchases made. In
financial management, this credit term is regarded as ‘3/10 net 30’. Related articles are by
Liberia and Orgler (1975), Hill and Rainier (1979), Kim and Chung (1990), Arcelus and
Srinivasan (1993), Ouyanget al. (2002), Chang (2002), Huang and Chung (2003). In these
articles, the buyer is the sole decision maker.
Goyal (1976) discussed joint ordering policy for a single-supplier single-buyer. Banerjee
(1986) discussed joint policy when supplier considers a lot-for-lot production. Goyal (1988)
advocated that the holding cost reduces significantly; if the supplier’s economic production
quantity is an integral multiple of the buyer’s order. Bhatnagaret al. (1993), Lu (1995),
Goyal (1995), Vishwanathan (1998), Hill (1997, 1999) Kim and Ha (2003), Li and Liu
(2006) discussed variants of the joint inventory system.Routroy and Sanisetty (2007)
formulated multi-echelon supply chain inventory policies to minimize total joint cost with
respect to economic production/ordering quantity and reorder point.Abad and Jaggi (2003)
discussed a supplier – buyer inventory policy when supplier assumes a lot-for-lot shipment
policy and offers a delay period to the buyer for settling the account against the purchases
made. Shah et al.incorporated deterioration of units in above model where demand is pricesensitive. Some related articles are fromOuyang et al. (2009a, 2009b), Shah et al. (2009),
Shah et al. (2009), Shah et al. (2011), Shah et al. (2011) Shah and Patel (2012) etc. and their
cited references.
Deterioration is defined as the decay, spoilage, evaporation which loses the utility of a
production from the original one. Fruits and vegetables, pharmaceutical drugs, electronic
items, blood components, radioactive chemicals are some of the examples of deteriorating
items. Refer to review articles by Nahmias (1982),Raafat (1991), Shah and Shah (2000) and
Goyal and Giri (2001) on deteriorating inventory models. Yang and Wee (2000) discussed a
heuristic method to model a joint vendor-buyer inventory model for deteriorating items.
Yang and Wee (2005) modelled a win – win strategy for an integrated system of singlevendor single-buyer with deterioration. Shah et al. (2008) extended above model by
incorporating salvage value to the deteriorated items.
Shah et al. (2011) analyzed an integrated inventory policy with ‘two-part’ trade credit
when demand is quadratic. This type of demand is observed in the fashion market, seasonal
products, etc. However, the demand of above mentioned items including branded electronic
items decreases drastically after some time. Cheng et al. (2011) discussed trapezoidal
15


demand in which the demand pattern is linearly increasing with time upto some point of time,
becomes constant in some interval of time and thereafter it decreases exponentially. Shah
and Shah (2012) developed joint optimal inventory policies for two players of the supply
chain when demand is trapezoidal. They (2012) studied effect of deterioration in above
problem.
In this paper, the objective is to analyze an integrated inventory system for
deteriorating items for price-sensitive trapezoidal demand. The units in inventory of both the
players are subject to deterioration at a constant rate. The supplier offers a choice of cash
discount in unit purchase price if payment is settled earlier (specified); otherwise, the buyer
has to make the full payment by the allowable credit period. The joint total profit per unit
time is maximized with respect to payment tie, retail price, purchase quantity and number of
shipments from the supplier to the buyer. The algorithm is proposed to find best optimal
solution. A numerical example is given to validate the developed problem.. Sensitivity
analysis is carried out and managerial issues are discussed.
2. ASSUMPTIONS AND NOTATIONS
2.1 Assumptions
The model is developed with following assumptions.
1. The supply chain comprises of single-supplier single-buyer and for single item.
2. Shortages are not allowed. Lead-time is zero.
3. The demand rate is price-sensitive trapezoidal. (Appendix A)
4. The supplier offers a discount β ( 0 < β < 1 ) in the purchase price if the buyer pays by
time M1; otherwise full account is to be settled within allowable credit period M 2 ,
where M 2 > M1 ≥ 0. The offer of discount in unit purchase price from the supplier
will increase cash in-flow, thereby reducing the risk of cash flow shortage.
5. By offering a trade credit to the buyer, the supplier receives cash at a later date and
hence incurs an opportunity cost during the delivery and payment of the product. On
the buyer’s end, the buyer can generate revenue by selling the items and earning
interest by depositing it in an interest bearing account during this permissible delay
period. At the end of this period, the supplier charges to the buyer on the unsold
stock.
6. During the time [ M1 ,M 2 ] , a cash flexibility rate f sc is used to quantize the favor of
early cash income for the supplier.
7. The units in the inventory system of both the player deteriorate at a constant rate
θ (0 < θ < 1). The deteriorated units can neither be repaired nor replaced during the
period under review.
2.2 Notations
The mathematical concept is developed using following notations.
Ab
Buyer’s ordering cost per order ($/order)

As

Supplier’s set-up cost ($/setup)

16


β

To increase cash inflow and reduce the risk of cash flow shortage, the supplier offers
a discount β ( 0 < β < 1 ) off the purchase price, if buyer settles the account within
time M1 , otherwise, full account is to be settled within an allowable credit period M 2 ;

Cs
v
P

where M 2 > M1 ≥ 0 .
Supplier’s unit manufacturing cost ($/unit)
Supplier’s unit sale price ($/unit)
Buyer’s unit sale price ($/unit) (a decision variable)
Note: P > v > (1 − β )v > Cs

Ib

Buyer’s carrying charge fraction per unit per year excluding interest charges

Is
I sp

Supplier’s carrying charge fraction per unit per year excluding interest charges
Supplier’s capital opportunity cost rate per unit /year

f sc

Supplier’s cash flexibility rate per unit/year

Ibe

Interest earned by the buyer during offered credit period M 2 per unit per year

Ibc
R

Buyer’s interest paid per unit per year
( = R( P,t )) Market demand rate (Appendix A), where a > 0 is scale demand,
0 < b1 ,b2 < 1 are the rates of change of demand, η > 1is price-elasticity mark-up and

γ
θ
T
n

Q
TBP
TSP
π

u1 and u2 are time points at which demand pattern changes. (Fig. 1)
The capacity utilization factor which is the ratio of the market demand rate to
production rate. γ < 1 is deterministic and constant.
Constant deterioration rate (0 < θ < 1) of units
Buyer’s cycle time (a decision variable)
Number of transfers from a supplier to buyer, n is a positive integer (a decision
variable)
Buyer’s procurement quantity during each transfer(a decision variable)
Buyer’s total profit per unit time
Supplier’s total profit per unit time
( = TSP + TBP ) Joint total profit of the integrated system per unit time

3. MATHEMATICAL MODEL
The buyer purchases Q units in each transfer. So the supplier produces in the batches
of size nQ and hoards set-up cost. The supplier tranships Q units manufactured initially and
thereafter, Q units are transported at T time units until the supplier’s inventory depletes to
zero.
The supplier offers the buyer a two-part trade credit period to encourage early
payment reducing risk of cash inflows. During the available credit period buyer earns
interest on the generated revenue. The aim is to maximize the joint profit per unit time of the
integrated system with respect to buyer’s selling price, payment time, procurement quantity
and the number of transfers from the supplier to the buyer.

17


3.1 Supplier’s total profit per unit time
The supplier manufactures nQ units in batches whereQ is defined in Appendix B and
incurs a batch set-up cost As . The supplier’s set-up cost per unit time is As / ( nT ).
FollowingJoglekar (1988), the supplier’s inventory holding cost per unit time is
T
T
1
{Cs ( I s + I sp ) [(n − 1)(1 − γ ) + γ ]} ∫ I (t ) dt. (See Appendix C for computation of ∫ I( t )dt ).
T
0
0

(

)

The purchase cost of an item for the buyer is 1 − K j β v, whenaccount is settled at time

M j ; where j = 1, 2; K1 = 1, K 2 = 0. Hence, for the permissible delay period, the opportunity
1
cost per unit time is (1 − K j β )vI sp M j Q; where j = 1, 2; K1 = 1, K 2 = 0. When the buyer
T
pays at time M1 , the supplier can use the revenue (1 − β ) v to shrinka cash flow crisis during
time M 2 − M1 . This timely payment acquires gain at the cash flexibility rate per unit time
and is given by

1
( 1 − β )vf sc ( M 2 − M1 )Q. Hence, the supplier’s total profit per unit time is,
T

sales revenueplus the interest earned on the timely payment, minus total cost which is sum of
the manufacturing cost, set-up cost, inventory holding cost and opportunity cost,is given by
TSPj ( n ) =

( 1 − K j β )vQ

−

T

T
C Q A
1
− s − s − Cs ( I s + I sp )[( n − 1 )( 1 − γ ) + γ ] ∫ I( t )dt
T
nT T
0

( 1 − K j β )vI sp M j Q
T

{

+

}

( 1 − β )vf sc ( M 2 − M1 )Q
T

j = 1, 2; K1 = 1, K 2 = 0

(1)

3.2Buyer’s total profit per unit time

A
The ordering cost per unit time is b for each transfer of Q units. The buyer’s
purchase cost per unit time is

T
( 1 − K j β )vQ
T

and inventory holding cost per unit time is

T

(1 − K j β )vI b ∫ I (t ) dt
0
; where j = 1, 2; K1 = 1, K 2 = 0.
T
On the basis of choice of payment time of the buyer two cases may arise.
1. T < M j

2. T ≥ M j ; j = 1, 2 .
Case 1: T < M j (Fig.2)

18

(IJPTM),

Fig. 2: Interest earned when T < M j ; j =1,2
In this case, the buyer’s stock level depletes to zero before the permissible delay period. So,
the opportunity cost for the buyer is zero. The interest earned on the generated revenue per

T

PIbe  ∫ t ⋅ R( P, t ) dt + Q ( M j − T ) 
0
 ; j = 1, 2. (See Appendix D for
unit time is given by
T
T

∫ t ⋅ R( P,t )dt ).

Hence, buyer’s total profit per unit timeis

0

T

( 1 − K j β )vI b ∫ I( t )dt

PQ ( 1 − K j β )vQ Ab
−
−
−
T
T
T
T

PIbe  ∫ t ⋅ R( P,t )dt + Q ( M j − T )
Q(
0

+
T

TBPj1 ( P,T ) =

j = 1, 2; K1 = 1, K 2 = 0

(2)

Case 2: T ≥ M j ; j = 1,2 (Fig.3)

19

0

T


Fig. 3 Interestearned and charged when T ≥ M j ; j = 1, 2
In this case, the buyer’s permissible payment time offered by the supplierovers on or
before the cycle time. The interest earned per unit time by the buyer at the rate Ibe during

0, M j  ; j = 1,2 is



Mj
1
PIbe ∫ t ⋅ R1 ( P, t ) dt
; 0 ≤ M j ≤ u1
T
0



Mj
Mj


1
1

u1

PIbe ∫ t ⋅ R( P, t ) dt =  PIbe  ∫ t ⋅ R1 ( P, t ) dt + ∫ t ⋅ R2 ( P, t ) dt 
; u1 ≤ M j ≤ u2
T
u1
0
T
0






Mj
u
u


 1 PI  1 t ⋅ R ( P, t ) dt + 2 t ⋅ R ( P, t ) dt +
∫
∫ t ⋅ R3 ( P, t ) dt  ; u2 ≤ M j ≤ T
∫
1
2
 T be
u1
u2



0



;where j = 1, 2
and interest paid per unit time at the rate Ibc during  M j , T  ; j = 1,2 is



20



 u1

u2
T
 1 (1 − K β )vI  ∫ I (t ) dt + ∫ I (t ) dt + ∫ I (t ) dt  ; M ≤ u ≤ T

j
bc 
j
1
2
3
1
T
u1
u2
M j







T
T
1
1

 u2

(1 − K j β )vI bc ∫ I (t ) dt =  (1 − K j β )vI bc  ∫ I 2 (t ) dt + ∫ I3 (t ) dt 
; M j ≤ u2 ≤ T
T
Mj
u2
T
M j




T
1
; u2 ≤ M j ≤ T
 (1 − K j β )vIbc ∫ I 3 (t ) dt
T
Mj



j = 1, 2; K1 = 1, K 2 = 0
Therefore, total profit of buyer per unit time is

(
(
(

)
)
)

TBPj 2 ( P, T ) : 0 ≤ M j ≤ u1


TBPj 2 ( P, T ) = TBPj 2 ( P, T ) : u1 ≤ M j ≤ u2

TBPj 2 ( P, T ) : u2 ≤ M j ≤ T ; j =1,2

(See

Appendix

(

(

Efor TBPj 2 ( P, T ) : 0 ≤ M j ≤ u1

(3)

)

,

(

TBPj 2 ( P,T ): u1 ≤ M j ≤ u2

)

,

)

TBPj 2 ( P, T ): u2 ≤ M j ≤ T ; j = 1,2 ).
The buyer’s total profit per unit time is
TBPj1 ( P, T ) ; T < M j

TBPj ( P, T ) = 
TBPj 2 ( P, T ) ; T ≥ M j


(4)

3.3 Joint total profit per unit time
The joint profit per unit time of integrated system is given by

π j1 ( n, P, T ) ; T < M j

π j ( n, P, T ) = 
π j 2 ( n, P, T ) ; T ≥ M j ; j = 1, 2

;where

(5)

π j1 (n, P, T ) = TSPj (n) + TBPj1 ( P, T )
π j 2 ( n, P, T ) = TSPj ( n) + TBPj 2 ( P, T ); j = 1, 2

The objective is to decide optimal values of discrete variable n and continuous
variables P and T , which maximize π j ( n,P,T ) , j = 1, 2 .We use following steps to maximize
the joint profit of the supply chain.
21


4. COMPUTATIONAL PROCEDURE
To maximize joint profit, execute following steps:
Step 1: Assign parametric values in proper units to all model parameters.
Step 2: Set n = 1 .
Step 3: Solve

∂π j
∂P

= 0 and

∂π j
= 0 , j = 1, 2 simultaneously for P and T .
∂T

Step 4: Increment n by 1.
Step 5: Continue steps 3 and 4 until

π j ( n − 1, P ( n − 1) , T ( n − 1) ) ≤ π j ( n, P, T ) ≥ π j ( n + 1, P ( n + 1) , T ( n + 1) ) ; j = 1, 2
is satisfied.
Step 6: Stop.
The optimal value of ( n, P , T ) determines the optimal purchase quantity Q (Appendix
B) pertransfer for the buyer.
5. NUMERICAL EXAMPLE
Let us illustrate the developed model with the following numerical values to model
parameters.
a = 1,00,000, b1 = 7%, b2 = 5%, η = 1.5, u1 = 15 days, u2 = 45 days, γ = 0.9, C s = $ 2/unit,
v = $ 4.5/unit, As = $ 1000/set-up, Ab = $ 300/order, Is = 5% /unit/year, Ib = 8% /unit/year,
f sc = 17% /$/year and
I sp = 9% /$/year, I bc = 16%/$/year, I be = 12% /$/year and

θ = 0.12. The supplier offers buyerthe credit term ‘3/10 net 30’ means if buyer pays by 10
days then he will be offered 3% discount in the unit purchase price otherwise the buyer has to
settle the account due against purchases in 30 days.
From Table 1, we see that for 10-shipments, the buyer’s selling price is $ 6.59/unit
and cycle time is122 days maximizing joint total profit of $ 25319 of the integrated system.
The corresponding profit of the supplier is $ 13507 and that of buyer is $ 11812.Each transfer
is of 2018 units. Optimal payment time is 10 days in ‘3/10net 30’credit terms. The concavity
of joint total profit with respect to number of transfers, n and retail sale price, P are shown in
figures 3and 4 respectively. 3-D plot given in figure 5 for n = 10establishes the convavity of
the total joint profit. The variations in permissible delay periods; M 1 and M 2 are worked out
to study the changes in decisionvariable and total joint profit in Table 1. The profit gain is
compared with benchmark of no credit period.

22


Fig.4: Concavity of Joint Profit w.r.t. no. of Shipments (n)

Fig. 5: Concavity of Joint Profit w.r.t. Retail Price (P)

Fig. 6 Concavity of joint profit w.r.t. cycle time and retail price
23


The last column in table 1 represents percentage of profit gain which is calculated by the
 Pr ofit with trade credit

formula 
− 1 ×100%.
 Pr ofit without trade credit 
Table 1: Optimal Solution for Various Credit Terms
M1
(days)

M2
(days)

Optimal
Payment
Time
(days)

n

P
($)

T
(days)

Q
(units)

Profit

Profit (%)

Buyer

Supplier

Joint

Buyer

Supplier

Joint

0

0

0

11

6.53

112

1878

10336

14463

24800

-

-

-

0

30

30

11

6.37

111

1925

10130

15149

25279

-02.03

04.53

01.89

10

30

10

10

6.59

122

2018

11812

13507

25319

12.50

-07.08

02.05

20

30

20

10

6.66

123

2005

12021

13176

25197

14.02

-09.77

01.58

0

60

60

11

6.27

113

2005

10152

15638

25790

-01.81

07.51

03.84

10

60

60

11

6.41

116

1997

11597

14266

25863

10.87

-01.38

04.11

20

60

60

11

6.41

116

1995

11595

14140

25735

10.86

-02.28

03.63

The positive profit gain proves that players of the supply chain are advantageous
under two-level trade credit policy. It is observed that buyer entices to pay at early date in
net credit scenario of ‘3/10net 30’ with maximum profit.
In table 2, independent and joint decisions are compared under different credit terms.
It is seen that the offer of trade credit lowers retail price of the buyer and purchase of larger
order is encouraged. The retail price of the buyer is almost double in independent decision
compared to co-ordinated decision, while procurement quantity is halved. It is observed that
the buyer’s profit decreases and that of supplier increases, which forces buyer to be dominant
player in terms of making decision.Goyal (1976) favored the reallocation of profit for
attracting buyer to opt for joint decision in the supply chain. Reallocate profit of buyer and
supplier as follows:
TBP ( P,T )
Buyer’s profit = π ( n,P,T ) ×
TBP ( P,T ) + TSP ( n ) 


17534
=25319 ×
= 20288
( 17534 + 4348 )
TSP ( n )
Supplier’s profit = π ( n,P,T ) ×
TBP ( P,T ) + TSP ( n ) 


4348
=25319 ×
= 5031
( 17534 + 4348 )
The reallocated profits for buyer and supplier are exhibited in the last row of Table 2.

24


Table 2: Optimal solutions for different strategies
Credit
Term

Strategy
Independent

Joint

Cash on
delivery
Trade
Credit
3/10 net 30
Cash on
delivery
Trade
Credit
3/10 net 30

Optimal
Payment
Time
(days)

n

0

P
($)

T
(days)

R(P,T)
(units)

Q
(units)

13

14.15

168

198

10

13

13.72

167

0

11

6.53

10

10

6.59

Profit ($)
Buyer

Supplier

Joint

886

16991

4388

21379

206

925

17534

4348

21882

112

283

1878

10336

14463

24800

122

330

2018

11812

13507

25319

20288

5031

25319

Adjusted

The sensitivity analysis for model parameters is carried out by changing parameter as
-20%, -10%, 10%, 20%. The figure 6 suggests that joint total profit is very sensitive to
utilization factor and scale demand. This insights that the supplier should maintain production
and demand ratio nearly 1. The joint profit is very sensitive to buyer’s ordering cost. It
directs the buyer to place larger order and do saving in transportation cost. The jointprofit
decreases with increase in mark-up, supplier’s production cost, interest charged to the buyer,
supplier’s opportunity cost and deterioration rate of units in inventory systems of both the
players. The mark-up is controllable because it depends on economy of the business. The
supplier’s opportunity cost depends on when the buyer is willing to pay. However, supplier
can reduce production cost and deterioration rate by using modern machinery and latest
storage facilities. The joint profit increases linearly with time suggesting that supplier and
buyer are benefited when product enters into the system i.e. demand is in increasing phase.

25500

Joint Profit

25450
25400
25350
25300
25250
25200
25150
-20%

-10%

0%

10%

20%

Percentage Changes in Affecting Parameters
Fig. 7 Sensitivity Analysis for Model Parameters on Joint Profit

25

η
γ
Cs
v
As
Ab
Is
Ib
Isp
Ibc
Ibe
fsc
b1
b2
a


6. CONCLUSIONS
A co-ordinated supplier-buyer inventory policy is addressed when demand is pricesensitive trapezoidal and units in inventory deteriorate at a constant rate. The analysis is
focused on two payment scenarios namely ‘net credit’. The total joint profit is maximized
with respect to number of transfers from supplier to the buyer, optimal payment time, the
retail price and cycle time. To attract the buyerfor joint decision, reallocation of the profit
scheme is suggested. This result helps the buyer to make a decision between two
promotional incentives, viz. price discount and permissible delay payment. In future, one can
analyze integrated inventory system for different deterioration rates of units in buyer and
supplier’s warehouses. It is worth incorporating imperfect production processes and
optimizing production.

REFERENCES
1.

2.
3.
4.
5.

6.

7.
8.
9.
10.
11.
12.

13.

14.

Abad, P. L. and Jaggi, C. K. (2003).A joint approach for setting unit price and the
length of the credit period for a seller when the demand is price-sensitive.
International Journal of Production Economics 83(2): 115-122.
Arcelus, F. J. and Srinivasan, G. (1993).Integrating working capital decisions.The
Engineering Economist 39(1): 1-15.
Banerjee, A. (1986). A joint economic-lot-size model for purchaser and
vendor.Decision Sciences 17(3): 292-311.
Bhatnagar, R., Chandra, P. and Goyal, S. K. (1993).Models for multiplant
coordination.European Journal of Operational Research 67(2): 141-160.
Chang, C. T. (2002). Extended economic order quantity model under cash discount
and payment delay. International Journal of Information and Management Sciences
13(3): 57-69.
Cheng, M., Zhang, B. and Wang G. (2011).Optimal policy for deteriorating items with
trapezoidal type demand and partial backlogging. Applied Mathematical Modelling
35: 3552-3560.
Goyal, S. K. (1976). An integrated inventory model for a single supplier single
customer problem.International Journal of Production Research 15 (1): 107-111.
Goyal, S. K. (1985).Economic order quantity under conditions of permissible delay in
payments.Journal of the Operational Research Society 36(4): 335-338.
Goyal, S. K. (1988). A joint economic-lot-size model for purchaser and vendor:
A comment. Decision Sciences 19(1): 236-241.
Goyal, S. K. (1995). A one-vendor multi-buyer integrated inventory model:
A comment. European Journal of Operational Research 82 (2): 209–210.
Goyal S.K. and Giri B.C. (2001).Recent trend in modeling of deteriorating
inventory.European Journal of the Operational Research Society. 134: 1-16.
Hill, R. M. (1997). The single-vendor single-buyer integrated production inventory
model with a generalized policy. European Journal of Operational Research 97 (3):
493-499.
Hill, R. M. (1999). The optimal production and shipment policy for the single-vendor
single-buyer integrated production-inventory problem. International Journal of
Production Research 37 (11): 2463-2475.
Hill, N. C. and Riener, K. D. (1979). Determining the cash discount in the firm's credit
policy.Financial Management 8 (1): 68--73.
26


15. Ho, C. H., Ouyang L. Y., and Su, C. H. (2008). Optimal pricing, shipment and
payment policy for an integrated supplier-buyer inventory model with two-part trade
credit.European Journal of Operational Research 187 (2): 496-510.
16. Huang, Y. F. and Chung, K. J. (2003). Optimal replenishment and payment policies in
the EOQ model under cash discount and trade credit. Asia-Pacific Journal of
Operation Research 20 (2): 177-190.
17. Joglekar, P. N. (1988). Comments on "A quantity discount pricing model to increase
vendor profits".Management Science 34 (11): 1391- 1398.
18. Kim, Y. H. and Chung, K. H. (1990). An integrated evaluation of investment in
inventory and credit: a cash flow approach. Journal of Business Finance and
Accounting 17(3): 381-390.
19. Kim, S. L. and Ha, D. (2003). A JIT lot-splitting model for supply chain management:
enhancing buyer-supplier linkage. International Journal of Production Economics
86(1): 1-10.
20. Li, J. and Liu, L. (2006). Supply chain coordination with quantity discount policy.
International Journal of Production Economics 101(1): 89-98.
21. Lieber, Z. and Orgler, Y. E. (1975).An integrated model for accounts receivable
management.Management Science 22(2): 212-219.
22. Lu, L. (1995). A one-vendor multi-buyer integrated inventory model. European
Journal of Operational Research 81(3): 312-323.
23. Nahmias, S. (1982). Perishable inventory theory: A review. Operations Research 30:
80-708.
24. Ouyang, L. Y., Chen, M. S. and Chuang, K. W. (2002).Economic order quantity model
under cash discount and payment delay.International Journal of Information and
Management Sciences 13(1): 1-10.
25. Ouyang, L. Y., Teng, J. T., Goyal, S. K. And Yang, C. T. (2009a). An economic order
quantity model for deteriorating items with partially permissible delay in payments
linked to order quantity. European Journal of Operational Research 194(2): 418-431.
26. Ouyang, L. Y., Ho, C. H. and Su, C. H. (2009b).An optimization approach for joint
pricing and ordering problem in an integrated inventory system with order-size
dependent trade credit.Computers and Industrial Engineering 57(3): 920-930.
27. Raafat F. (1991). Survey of literature on continuously deteriorating inventory models.
Journal of the Operational Research Society 42(1): 27-37.
28. Routroy, S. and Sanisetty, P., (2007).Inventory planning for a multi-echelon supply
chain. International Journal of Operational Research 2(3): 269-283.
29. Shah, Nita H., Gor, A. S. and Wee , H. M. (2008). Optimal joint vendor-buyer
inventory strategy for deteriorating items with salvage value.Proceedings of 14th
computing:The Australian theory of symposium (CATS) 77: 63-68.
30. Shah, Nita H. and Shah, Y. K. (2000). Literature survey on inventory models for
deteriorating items. Economic Annals 44: 221-237.
31. Shah, N. H., Gor, A. S. and Jhaveri, C. A. (2011).Determination of optimal pricing,
shipment and payment policies for an integrated supplier-buyer deteriorating inventory
model in buoyant market with two-level trade credit. International Journal of
Operational Research 11(2): 119-135.
32. Shah, N. H., Shukla, K. T. and Shah, B. J. (2009). Deteriorating inventory model for
two-level credit linked demand under permissible delay in payments. Australian
Society of Operations Research Bulletin 28(4): 27-36.
27


33. Shah, N. H., Gor, A. S. and Jhaveri, C. A. (2009). Optimal pricing, shipment and
payment policies for an integrated supplier-buyer inventory model in buoyant market
with net credit. International Journal of Business Insights and Transformation
2(2): 3-12.
34. Shah, N. H. ,Gor, A.S. and Wee, H. M. (2010). An integrated approach for optimal
unit price and credit period for deteriorating inventory system when the buyer’s
demand is price-sensitive. American Journal of Mathematical and Management
Sciences 30(3-4): 317-330.
35. Shah, N. H., Shukla, K. T. and Gupta, O. P. (2011). Supply chain inventory model for
deteriorating items under two-level credit policy for declining market. International
Journal of Applied Management Science 3(2): 143-173.
36. Shah, N. H., Gor A. S. and Jhaveri, C. A. (2011). Determination of optimal ordering
and transfer policy for deteriorating inventory system when demand is quadratic.
International Journal of Management Science and Engineering Management
6(4): 278-283.
37. Shah, N. H. ,Soni, H. N. and Jaggi, C. K. (2010). Inventory model and trade credit :
Review. Control and Cybernetics 39(3): 867-884.
38. Shah, N. H. and Patel, A. (2012). Optimal ordering and transfer policy for deteriorating
inventory with stock-dependent demand. International Journal of operations and
Quantitative Management 18(2): 101-115.
39. Shah, N. H. and Shah, D. B. (2012). Vendor-buyer ordering policy when demand is
trapezoidal. International Journal of Industrial Engineering Computation 3(5):
721-730.
40. Shah, N. H. and Shah, D. B. (2013). Joint optimal policy for variable deteriorating
inventory system of vendor-buyer ordering with trapezoidal demand. British Journal of
Applied Science and Technology 3(1): 160-173.
41. Viswanathan, S., (1998). Optimal strategy for the integrated vendor-buyer inventory
model.European Journal of Operational Research 105(1): 38-42.
42. Yang P. C. and Wee H. M. (2005).A win-win strategy for an integrated vendor-buyer
deteriorating inventory system.Mathematical Modelling and Analysis.(Proceedings of
the 10th International Conference MMA2005&CMAM2, Trakai) 541-46.
43. Yang, P. C. and Wee, H. M., 2000. Economic ordering policy of deteriorated item for
vendor and buyer: An integrated approach. Production Planning & Control 11 (5):
474–480.
44. Dowlath Fathima and P.S Sheik Uduman, “Single Period Inventory Model with
Stochastic Demand and Partial Backlogging”, International Journal of Management
(IJM), Volume 4, Issue 1, 2013, pp. 95 - 111, ISSN Print: 0976-6502, ISSN Online:
0976-6510.
45. N.Balaji and Y.Lokeswara Choudary, “An Application of Fuzzy Cognitive Mapping in
Optimization of Inventory Function among Auto Component Manufacturing units in
Sme Sector”, International Journal of Management (IJM), Volume 3, Issue 2, 2013,
pp. 13 - 24, ISSN Print: 0976-6502, ISSN Online: 0976-6510.
46. Deepa H Kandpal and Khimya S Tinani, “Inflationary Inventory Model Under Trade
Credit Subject to Supply Uncertainty”, International Journal of Management (IJM),
Volume 4, Issue 4, 2013, pp. 111 - 118, ISSN Print: 0976-6502, ISSN Online:
0976-6510.

28


Appendix A : Trapezoidal demand
The demand R( P,t ) is considered to be a trapezoidal type whose functional form is

 f ( t ) P −η ; 0 ≤ t ≤ u
1


R ( P,t ) =  R0 P −η ;u1 ≤ t ≤ u2

 g ( t ) P −η ;u2 ≤ t ≤ T

where u1 is time point when the increasing demand function f ( t ) changes to constant demand
and u2 is the time point from where constant demand starts decreasing exponentially. In this
study, we take f (t ) to be liner in t, R0 = f ( u1 ) = g ( u2 ) and g ( t ) to be exponentially
decreasing in t . So the demand function is

 R1 ( P, t ) ; 0 ≤ t ≤ u1

R ( P, t ) =  R2 ( P, t ) ; u1 ≤ t ≤ u2

 R3 ( P, t ) ; u2 ≤ t ≤ T
R1 ( P,t ) = a (1 + b1t ) P −η
−η
; where R2 ( P,t ) = a (1 + b1u1 ) P
−b t − u
R3 ( P,t ) = a (1 + b1u1 ) e 2 ( 2 ) P −η

Fig. 1 Price sensitive time dependent trapezoidal demand
29


Appendix B: Computation of inventory at any instant of time t and purchase quantity

Q
The inventory level in warehouse changes due to price-sensitive trapezoidal demand
and deterioration rate of units in the warehouse. The rate of change of inventory at any
instant of time t is governed by the differential equation
d I (t )
= − R ( P, t ) − θ I (t );0 ≤ t ≤ T
dt
with the initial condition I (T ) = 0 .
The solution of the differential equation is

 I1 ( t ) ; 0 ≤ t ≤ u1

I ( t ) =  I 2 ( t ) ; u1 ≤ t ≤ u2 here

 I 3 ( t ) ; u2 ≤ t ≤ T

 −η  1 + b1t b1 1 + b1u1 θ u1−θ t b1 θ u1−θ t 
+ 2+
− 2 e +
e
aP  −
θ
θ
θ
θ




θ u1−θ t
a(1 + b1u1 )  θ u2 −θ t

I1 (t ) =  P −η
e
−e +


θ

 −η  a(1 + b1u1 )eb2u2  −b T +θ T −θ t − b u +θ u −θ t
2
−e 2 2 2 
P 
 e


θ − b2





 aP −η (1 + b1u1 )
 −1 + eθ u2 −θ t  +



θ

I 2 (t ) = 
b2u2
 P −η  a (1 + b1u1 )e  e − b2T +θ T −θ t − e −b2u2 +θ u2 −θ t 




θ − b2




 a (1 + b1u1 )eb2u2  − b2T +θ T −θ t − b2t
I 3 (t ) = P 
−e 
 e


θ − b2


Using I ( 0 ) = Q, we get
−η

 −η  1 b1 1 + b1u1 θu1 b1 θu1 
e − 2 e +
aP − + 2 +
θ
θ
 θ θ



a(1 + b1u1 )  θ u2 θu1 

Q =  P−η
e −e +


θ

 −η  a(1 + b1u1 )eb2u2  −b T +θT −b u +θ u
 2

−e 2 2 2 
P 
 e
θ − b2




30


Appendix C: Computation of total inventory during [ 0,T ]
Total inventory during [ 0 ,T ] is given by

T

u1

u2

T

0

0

u1

u2

∫ I ( t ) dt = ∫ I1 ( t ) dt + ∫ I 2 ( t ) dt + ∫ I3 ( t ) dt

Appendix D: Computation of total demand
T

u1

u2

T

0

0

u1

u2

∫ t ⋅ R ( P,t ) dt = ∫ t ⋅ R ( P,t ) dt + ∫ t ⋅ R ( P,t ) dt + ∫ t ⋅ R ( P,t ) dt

Appendix E: Buyer’s total profit when T ≥ M j ; j = 1, 2
TBPj 2 ( ( P, T ) : 0 ≤ M j ≤ u1

) = PQ −
T

(1 − K j β )vQ
T

T

−

Ab 1
− (1 − K j β ) vI b ∫ I (t )dt
T T
0

u2
T
1
 u1

 (1 − K j β )vI bc  ∫ I1 (t ) dt + ∫ I 2 (t ) dt + ∫ I 3 (t ) dt 
−


T
u1
u2
M j





1
+ PIbe
T

Mj

∫ t ⋅ R ( P, t ) dt
1

0

; 0 ≤ M j ≤ u1
(1 − K j β )vQ

T
A
1
− b − 1 − K j β vIb ∫ I (t )dt
T
T T
0
u

T
1
 2

− (1 − K j β )vIbc  ∫ I 2 (t ) dt + ∫ I3 (t ) dt 
T
u2
M j



Mj
u1

1


+ PIbe  ∫ t ⋅ R1 ( P, t ) dt + ∫ t ⋅ R2 ( P, t ) dt 
T
u1
0



; u1 ≤ M j ≤ u2
T
PQ (1 − K j β )vQ Ab 1
TBPj 2 ( P, T ) : u2 ≤ M j ≤ T =
−
−
− 1 − K j β vIb ∫ I (t )dt
T
T
T T
0

TBPj 2

( ( P, T ): u1 ≤ M j ≤ u2 ) = PQ −
T

(

(

)

(

−

)

)

T
1
(1 − K j β )vIbc ∫ I3 (t ) dt
T
M
j

Mj
u2
u1



1
 ∫ t ⋅ R1 ( P, t ) dt + ∫ t ⋅ R2 ( P, t ) dt + ∫ t ⋅ R3 ( P, t ) dt 
+ PIbe  0

u1
u2
T






; u2 ≤ M j ≤ T

31

30220130403002

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (7)

Similar to 30220130403002

Similar to 30220130403002 (20)

More from IAEME Publication

More from IAEME Publication (20)

Recently uploaded

Recently uploaded (20)

30220130403002