INTERNATIONAL JOURNAL OF PRODUCTION ......International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383 (Print), ISSN 0976 – 6391 (Online) Volume 4, Issue

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

(Print), ISSN 0976 – 6391 (Online) Volume 4, Issue 2, May - December (2013), © IAEME

14

OPTIMAL SHIPMENTS, ORDERING AND PAYMENT POLICIES FOR

INTEGRATED SUPPLIER-BUYER DETERIORATING INVENTORY

SYSTEM WITH PRICE-SENSITIVE TRAPEZOIDAL DEMAND AND

NET CREDIT

1Nita H. Shah, 2Digeshkumar B. Shah & 3Dushyantkumar G. Patel

1Department of Mathematics, Gujarat University, Ahmedabad-380009, Gujarat, India

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

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 1M , then the buyer receives a cash discount; otherwise the full payment

must be paid by the time 2M ; where 𝑀2 > 𝑀1 ≥ 0. The joint profit per unit time of supplier-

buyer 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.

INTERNATIONAL JOURNAL OF PRODUCTION TECHNOLOGY AND

MANAGEMENT (IJPTM)

ISSN 0976- 6383 (Print)

ISSN 0976 - 6391 (Online)

Volume 4, Issue 2, May- December (2013), pp. 14-31

© IAEME: www.iaeme.com/ijptm.asp

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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 price-sensitive. 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 single-vendor 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 demand

in which the demand pattern is linearly increasing with time upto some point of time, becomes



16

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 0 1( ) in the purchase price if the buyer pays by

time 1;M otherwise full account is to be settled within allowable credit period 2M ,

where 2 1 0M M . 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 1 2M ,M , a cash flexibility rate scf 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

(0 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.

bA Buyer’s ordering cost per order ($/order)

sA Supplier’s set-up cost ($/setup)

To increase cash inflow and reduce the risk of cash flow shortage, the supplier offers



17

a discount 0 1( ) off the purchase price, if buyer settles the account within time

1M , otherwise, full account is to be settled within an allowable credit period 2M ; where

2 1 0M M .

sC Supplier’s unit manufacturing cost ($/unit)

v Supplier’s unit sale price ($/unit)

P Buyer’s unit sale price ($/unit) (a decision variable)

Note: (1 ) sP v v C −

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

sI Supplier’s carrying charge fraction per unit per year excluding interest charges

spI Supplier’s capital opportunity cost rate per unit /year

scf Supplier’s cash flexibility rate per unit/year

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

bcI Buyer’s interest paid per unit per year

R ( R( P,t ))= Market demand rate (Appendix A), where 0a is scale demand,

1 20 1b ,b are the rates of change of demand, 1 is price-elasticity mark-up and 1u

and 2u 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. 1 is deterministic and constant.

Constant deterioration rate (0 1) of units

T Buyer’s cycle time (a decision variable)

n Number of transfers from a supplier to buyer, n is a positive integer (a decision

variable)

Q Buyer’s procurement quantity during each transfer(a decision variable)

TBP Buyer’s total profit per unit time

TSP 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 atT 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.



18

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 sA . The supplier’s set-up cost per unit time is sA / ( nT ).

FollowingJoglekar (1988), the supplier’s inventory holding cost per unit time is

0

1( ) ( 1)(1 ) ( ) .

T

s s spC I I n I t dtT

+ − − + (See Appendix C for computation of0

T

I( t )dt ).

The purchase cost of an item for the buyer is ( )1 ,jK v− whenaccount is settled at time ;jM

where 1 21,2; 1, 0.j K K= = = Hence, for the permissible delay period, the opportunity cost

per unit time is 1

(1 ) ;j sp jK vI M QT

− where 1 21,2; 1, 0.j K K= = = When the buyer pays at

time 1,M the supplier can use the revenue ( )1 v− to shrinka cash flow crisis during time

2 1M M .− This timely payment acquires gain at the cash flexibility rate per unit time and is

given by 2 11

1 sc( )vf ( M M )Q.T

− − Hence, the supplier’s total profit per unit time is, 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

0

2 1

1 11 1

1 1

Tj s sj s s sp

j sp j sc

( K )vQ C Q ATSP ( n ) C ( I I ) ( n )( ) I( t )dt

T T nT T

( K )vI M Q ( )vf ( M M )Q

T T

−= − − − + − − +

− − −− +

1 21,2; 1, 0 (1)j K K= = =

3.2Buyer’s total profit per unit time

The ordering cost per unit time isbA

T for each transfer of Q units. The buyer’s

purchase cost per unit time is1 j( K )vQ

T

−and inventory holding cost per unit time is

0

(1 ) ( )

;

T

j bK vI I t dt

T

−

where 1 21,2; 1, 0.j K K= = =

On the basis of choice of payment time of the buyer two cases may arise.

1. jT M

2. ; 1,2jT M j = .



19

Case 1: jT M (Fig.2)

Fig. 2: Interest earned when ;jT M 1,2j=

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

unit time is given by0

( , ) ( )

; 1,2.

T

be jPI t R P t dt Q M T

jT

+ −

= (See Appendix D for

0

T

t R( P,t )dt ). Hence, buyer’s total profit per unit timeis

01

0

11

T

j bj b

j

T

be j

( K )vI I( t )dt( K )vQ APQ

TBP ( P,T )T T T T

PI t R( P,t )dt Q( M T )

T

− −

= − − −

+ −

+

1 21,2; 1, 0 (2)j K K= = =

Case 2: ; 1,2jT M j = (Fig.3)



20

Fig. 3 Interestearned and charged when ; 1,2jT M j =

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 beI during

0, ; 1,2jM j = is

1 10

1

1 2 1 20 0 1

1 2

1 2 3 20 1 2

1 PI ( , ) ; 0

1 1( , ) PI ( , ) ( , ) ;

1PI ( , ) ( , ) ( , ) ;

M j

be j

M Muj j

be be ju

Mu u j

be ju u

t R P t dt M uT

PI t R P t dt t R P t dt t R P t dt u M uT T

t R P t dt t R P t dt t R P t dt u M TT

= +

+ +

;where 1 2j ,=

and interest paid per unit time at the rate bcI during , ; 1,2jM T j = is



21

1 2

1 2 3 1

1 2

2

2 3 2

2

3 2

1(1 ) ( ) ( ) ( ) ;

1 1(1 ) ( ) (1 ) ( ) ( ) ;

1(1 ) ( ) ;

u u T

j bc jM u uj

uT T

j bc j bc jM M uj j

T

j bc jM j

K vI I t dt I t dt I t dt M u TT

K vI I t dt K vI I t dt I t dt M u TT T

K vI I t dt u M TT

− + +

− = − +

−

1 21,2; 1, 0j K K= = =

Therefore, total profit of buyer per unit time is

( )

( )( )

( )( )

( )( )

2 1

2 2 1 2

2 2

, : 0

, , :

, : ; 1,2 (3)

j j

j j j

j j

TBP P T M u

TBP P T TBP P T u M u

TBP P T u M T j

=

=

(See Appendix Efor ( )( )2 1, : 0j jTBP P T M u , ( )( )2 1 2, :j jTBP P T u M u ,

( )( )2 2, : ; 1,2j jTBP P T u M T j = ).

The buyer’s total profit per unit time is

( )( )

( )

1

2

, ;,

, ; (4)

j jj

j j

TBP P T T MTBP P T

TBP P T T M

=

3.3 Joint total profit per unit time

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

( )( )

( )

1

2

, , ;, ,

, , ; ; 1,2 (5)

j jj

j j

n P T T Mn P T

n P T T M j

=

=

;where1 1

2 2

( , , ) ( ) ( , )

( , , ) ( ) ( , ); 1,2

j j j

j j j

n P T TSP n TBP P T

n P T TSP n TBP P T j

= +

= + =

The objective is to decide optimal values of discrete variable n and continuous variables

P andT , which maximize ( )j n,P,T , 1 2j ,= .We use following steps to maximize the joint

profit of the supply chain.

4. COMPUTATIONAL PROCEDURE



22

To maximize joint profit, execute following steps:

Step 1: Assign parametric values in proper units to all model parameters.

Step 2: Set 1n = .

Step 3: Solve 0j

P

=

and 0

j

T

=

, 1 2j ,= simultaneously for P andT .

Step 4: Increment n by1.

Step 5: Continue steps 3 and 4 until

( ) ( )( ) ( ) ( ) ( )( )1, 1 , 1 , , 1, 1 , 1 ; 1,2j j jn P n T n n P T n P n T n j − − − + + + =

is satisfied.

Step 6: Stop.

The optimal value of ( ), ,n P T determines the optimal purchase quantityQ (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, 1b = 7%, 2b = 5%, = 1.5, 1u = 15 days, 2u = 45 days, = 0.9, sC = $ 2/unit, v

= $ 4.5/unit, sA = $ 1000/set-up, bA = $ 300/order, sI = 5% /unit/year, bI = 8% /unit/year, spI

= 9% /$/year, bcI = 16%/$/year, beI = 12% /$/year and scf = 17% /$/year and

= 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; 1M and 2M 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.



23

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



24

The last column in table 1 represents percentage of profit gain which is calculated by the

formula Pr ofit with tradecredit

1 100%.Pr ofit without tradecredit

−

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:

Buyer’s profit = ( )( )

( ) ( )

TBP P,Tn,P,T

TBP P,T TSP n

+

=2531917534

17534 4348( )

+= 20288

Supplier’s profit = ( )( )

( ) ( )

TSP nn,P,T

TBP P,T TSP n

+

=253194348

17534 4348( )+= 5031

The reallocated profits for buyer and supplier are exhibited in the last row of Table 2.



25

Table 2: Optimal solutions for different strategies

Strategy Credit

Term

Optimal

Paymen

t

Time

(days)

n P

($)

T

(days

)

R(P,T

)

(units)

Q

(units

)

Profit ($)

Buye

r

Supplie

r Joint

Independen

t

Cash on

delivery 0

1

3

14.1

5 168 198 886 16991 4388

2137

9

Trade

Credit

3/10 net

30

10 1

3

13.7

2 167 206 925 17534 4348

2188

2

Joint Cash on

delivery 0

1

1 6.53 112 283 1878 10336 14463

2480

0

Trade

Credit

3/10 net

30

10 1

0 6.59 122 330 2018 11812 13507

2531

9

Adjuste

d 20288 5031

2531

9

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.



26

Fig. 7 Sensitivity Analysis for Model Parameters on Joint Profit

6. CONCLUSIONS

A co-ordinated supplier-buyer inventory policy is addressed when demand is price-

sensitive 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.

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25150

25200

25250

25300

25350

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25450

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t

Percentage Changes in Affecting Parameters

η

γ

Cs

v

As

Ab

Is

Ib

Isp

Ibc

Ibe

fsc

b1

b2

a



27

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Appendix A : Trapezoidal demand

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

where 1u is time point when the increasing demand function ( )f t changes to constant demand

and 2u is the time point from where constant demand starts decreasing exponentially. In this

( )

( )

( )

1

0 1 2

2

0f t P ; t u

R P,t R P ;u t u

g t P ;u t T

−

−

−

=



30

study, we take ( )f t to be liner in ,t ( ) ( )0 1 2R f u g u= = and ( )g t to be exponentially

decreasing in t . So the demand function is

( )

( )

( )

( )

1 1

2 1 2

3 2

, ; 0

, , ;

, ;

R P t t u

R P t R P t u t u

R P t u t T

=

; where

( ) ( )

( ) ( )

( ) ( ) ( )

1 1

2 1 1

2 23 1 1

1

1

1b t u

R P,t a b t P

R P,t a b u P

R P,t a b u e P

−

−

− − −

= +

= +

= +

Fig. 1 Price sensitive time dependent trapezoidal demand

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

( )( ), ( );0

d I tR P t I t t T

dt= − −

with the initial condition ( ) 0I T = .



31

The solution of the differential equation is

here

1 1

12

2 2

2 2 2 2

1 1 1 1 1

2 2

1 11

1 1

2

1 1

(1 )( )

(1 )

u t u t

u tu t

b ub T T t b u u t

b t b b u baP e e

a b uI t P e e

a b u eP e e

b

− −

−

−

−−

− + − − + −−

+ + − + + − +

+ = − + +

− −

2

2 2

2 2 2 2

1 1

2

1 1

2

(1 )1

( )(1 )

u t

b ub T T t b u u t

aP b ue

I ta b u e

P e eb

−−

− + − − + −−

+ − + +

=

+ − −

2 2

2 21 13

2

(1 )( )

b ub T T t b ta b u e

I t P e eb

− + − −− + = − −

Using ( )0I Q,= we get

1 1

12

2 2

2 2 2 2

1 1 1 1

2 2

1 1

1 1

2

11

(1 )

(1 )

u u

uu

b ub T T b u u

b b u baP e e

a b uQ P e e

a b u eP e e

b

−

−

− + − +−

+ − + + − +

+ = − + +

− −

Appendix C: Computation of total inventory during 0,T

Total inventory during 0,T is given by

( ) ( ) ( ) ( )1 2

1 2 30 0 1 2

u uT T

u u

I t dt I t dt I t dt I t dt= + +

( )

( )

( )

( )

1 1

2 1 2

3 2

; 0

;

;

I t t u

I t I t u t u

I t u t T

=



32

Appendix D: Computation of total demand

( ) ( ) ( ) ( )1 2

0 0 1 2

u uT T

u u

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 ; 1,2jT M j =

( )( ) ( )

1 2

1 2

2 1

0

1 2 3

1

0

1

(1 ) 1, : 0 1 ( )

1(1 ) ( ) ( ) ( )

1 PI ( , )

; 0

j

j

Tj b

j j j b

u u T

j bc

M u u

M

be

j

K vQ APQTBP P T M u K vI I t dt

T T T T

K vI I t dt I t dt I t dtT

t R P t dtT

M u

− = − − − −

− − + +

+

( )( ) ( )2 1 20

2

2 3

2

1

1 20 1

1 2

(1 ) 1, : 1 ( )

1(1 ) ( ) ( )

1PI ( , ) ( , )

;

Tj b

j j j b

u T

j bcM u

j

Mu j

beu

j

K vQ APQTBP P T u M u K vI I t dt

T T T T

K vI I t dt I t dtT

t R P t dt t R P t dtT

u M u

− = − − − −

− − +

+ +

( )( ) ( )2 20

3

1 2

1 2 30 1 2

2

(1 ) 1, : 1 ( )

1(1 ) ( )

( , ) ( , ) ( , )1PI

;

Tj b

j j j b

T

j bcM j

Mu u j

be u u

j

K vQ APQTBP P T u M T K vI I t dt

T T T T

K vI I t dtT

t R P t dt t R P t dt t R P t dt

T

u M T

− = − − − −

− −

+ +

+

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