a two-warehouse inventory model for non-instantaneous ...the optimization prob-lems of the inventory...

ORIGINAL ARTICLE

A two-warehouse inventory model for non-instantaneousdeteriorating items with interval-valued inventory costsand stock-dependent demand under inflationary conditions

Ali Akbar Shaikh1 • Leopoldo Eduardo Cardenas-Barron1 • Sunil Tiwari2,3

Received: 4 August 2016 /Accepted: 14 August 2017 / Published online: 22 August 2017

� The Natural Computing Applications Forum 2017

Abstract This research work develops a two-warehouse

inventory model for non-instantaneous deteriorating items

with interval-valued inventory costs and stock-dependent

demand under inflationary conditions. The proposed

inventory model permits shortages, and the backlogging

rate is variable and dependent on the waiting time for the

next order, and inventory parameters are interval-valued.

The main aim of this research is to obtain the retailer’s

optimal replenishment policy that minimizes the present

worth of total cost per unit time. The optimization prob-

lems of the inventory model have been formulated and

solved using two variants of particle swarm optimization

(PSO) and interval order relations. The efficiency and

effectiveness of the inventory model are validated with

numerical examples and a sensitivity analysis. The pro-

posed inventory model can assist a decision maker in

making important replenishment decisions.

Keywords Inventory � Non-instantaneous deterioration �Two warehouses � Partial backlogging � Stock-dependentdemand � Inflation � Interval-valued cost � Particle swarm

optimization

Mathematics Subject Classification 90B05

1 Introduction and literature review

Inventory models considering deterioration have been

broadly studied under various assumptions in the past few

years. Ghare and Schrader [1] included the concept of

deterioration with the supposition of exponentially decay-

ing items. After, Covert and Phillip [2] revisited and

improved Ghare and Schrader’s [1] inventory model con-

sidering Weibull distribution deterioration. Additional

works in this field are summarized in several review arti-

cles such as Nahmias [3], Raafat [4], Goyal and Giri [5],

Bakker et al. [6] and Janssen et al. [7].

The research articles mentioned previously consider the

general assumption that deterioration of items begins

immediately as these are placed in the warehouse. However,

there exist some products which do not deteriorate instan-

taneously. Such observable fact of deterioration is named as

non-instantaneous deterioration, and the products are called

non-instantaneous deteriorating items.Wu et al. [8] included

the concept of non-instantaneous deterioration into an

inventory model. Further, several researchers such as

Ouyang et al. [9, 10], Wu et al. [11], Jaggi and Verma [12],

Chang et al. [13], Geetha and Uthayakumar [14], Soni and

Patel [15], Maihami and Kamalabadi [16, 17], Shah et al.

[18], Dye [19], Jaggi and Tiwari [20] and Jaggi et al. [21–23]

have elaborated some inventory models for non-instanta-

neous deteriorating products under various situations.

& Leopoldo Eduardo Cardenas-Barron

[email protected]

Ali Akbar Shaikh

[email protected]

Sunil Tiwari

[email protected]

1 School of Engineering and Sciences, Tecnologico de

Monterrey, E. Garza Sada 2501 Sur, C.P. 64849 Monterrey,

Nuevo Leon, Mexico

2 Department of Operational Research, Faculty of

Mathematical Sciences, New Academic Block, University of

Delhi, Delhi 110007, India

3 The Logistics Institute - Asia Pacific, National University of

Singapore, 21 Heng Mui Keng Terrace, Singapore 119613,

Singapore

123

Neural Comput & Applic (2019) 31:1931–1948

DOI 10.1007/s00521-017-3168-4

http://orcid.org/0000-0002-6290-8095

http://crossmark.crossref.org/dialog/?doi=10.1007/s00521-017-3168-4&domain=pdf

http://crossmark.crossref.org/dialog/?doi=10.1007/s00521-017-3168-4&domain=pdf

It is important to remark that in previous studies capacity

of the warehouse is supposed to be unlimited. Further, in the

past few years the concept of two warehouses in inventory

modelling has been explored by several academicians and

researchers under different assumptions. The primary study

in two-warehouse system was considered by Hartley [24].

After that, Sarma [25] built an inventorymodel considering a

limited warehouse capacity. In this inventory model, addi-

tional capacity is acquired by a rented warehouse (RW) due

to the fact that the own warehouse (OW) is of limited

capacity. Yang [26] also developed an inventory model

considering limited capacity, and this inventory model

assumes that the demand rate is constant and the shortages

are fully backordered. Also, the product’s lifetime follows an

exponential distribution (constant rate of hand inventory is

deteriorated). In a subsequent paper, Yang [27] built a sim-

ilar inventory model considering partial backlogging. Wee

et al. [28] studied an inventory model with partial backlog-

ging taking into account that the product lifetime follows the

two-parameter Weibull distribution. Pal et al. [29] studied a

deteriorating product in a two-warehouse system, assuming

that the demand is time dependent and the shortages are

partially backordered. Lee [30] introduced a production–

inventorymodel, considering that the inventory level rises by

a finite production rate. The other assumptions in this pro-

duction model are same as in Yang [26]. Chung and Huang

[31] further developed an inventory model with permissible

delay in payment. Hsieh et al. [32] extended the inventory

model developed by Yang [26] and optimized the inventory

model considering net present value approach. Liao and

Huang [33] investigated the research done by Chung and

Huang [31] by adopting a different approach. It is important

to remark that a number of works in two-warehouse

environment have been published by many researchers over

the last decades. The reader can study the recentworks ofDas

et al. [34], Niu and Xie [35], Rong et al. [36], Dey et al. [37],

Maiti [38], Lee and Hsu [39], Bhunia and Shaikh [40], Liang

and Zhou [41], Bhunia et al. [42], Yang and Chang [43],

Jaggi et al. [44], Bhunia et al. [45, 46], Jaggi et al. [47] and

Tiwari et al. [48]. Table 1 presents a comparison of some

recent papers related to inventorymodels considering one- or

two-warehouse system.

Considering the importance of above facts, this paper

investigates a two-warehouse inventory model for non-in-

stantaneous deteriorating items with interval-valued

inventory costs, partial backlogging and stock-dependent

demand under inflationary conditions. The inventory model

permits shortages and considers that the backlogging rate is

variable and dependent on the waiting time for the next

order, and inventory parameters are interval-valued. The

objective of this research work is to determine the retailer’s

optimal replenishment policy that minimizes the present

worth of total cost per unit time. Then, for each case, the

corresponding optimization problems of the inventory

model have been formulated and solved using two variants

of particle swarm optimization (PSO) and interval order

relations. Numerical examples are solved to demonstrate

the applicability of proposed inventory model, followed by

the sensitivity analysis of the optimal solution with respect

to input parameters of the inventory system. The proposed

inventory model is useful because it helps to the decision

makers in taking important replenishment decisions.

The structure of this paper is as follows. Section 2 gives

detailed description assumptions and notation used in the

paper. Section 3 presents the formulation of the inventory

model. Section 4 proposes two algorithms to find the

Table 1 Comparison of inventory models with one warehouse or two warehouses

References Deterioration Backlogging Inventory costs

(crisp/interval/fuzzy)

Two warehouses/

single warehouse

Computational

technique

Liang and Zhou [41] Constant No Crisp Two warehouses Mathematically

Bhunia et al. [42] Constant Partial backlogging Crisp Two warehouses Genetic algorithm

Yang and Chang [43] Constant Partial backlogging Crisp Two warehouses Mathematically

Jaggi et al. [44] Constant Fully backlogging Crisp Two warehouses Mathematically

Bhunia et al. [45] Constant Partial backlogging Crisp Two warehouses Genetic algorithm

Bhunia et al. [46] Constant Partial backlogging Crisp Two warehouses Particle swarm optimization

Jaggi et al. [47] Constant No Crisp Two warehouses Mathematically

Tiwari et al. [48] Non-instantaneous Fully backlogging Crisp Two warehouses Mathematically

Taleizadeh et al. [49] No No Fuzzy Single warehouse Genetic algorithm and TOPSIS

Taleizadeh et al. [50] No Fully backlogging Fuzzy Single warehouse Genetic algorithm and

particle swarm optimization

Taleizadeh et al. [51] No No Fuzzy Single warehouse Meta-heuristic algorithm

Tat et al. [52] Non-instantneous No Crisp Single warehouse Mathematically

This paper Non-instantaneous Partial backlogging Interval Two warehouses Particle swarm optimization

1932 Neural Comput & Applic (2019) 31:1931–1948

123

optimal solution. Section 5 solves some numerical exam-

ples to validate the inventory model, and further the effi-

cacy of the proposed inventory model is illustrated through

a comprehensive sensitivity analysis. Finally, Sect. 6 pro-

vides some conclusions and future research directions.

2 Assumptions and notation

The mathematical models of the two-warehouse inventory

problems are developed considering the following

assumptions.

1. The costs involved in the inventory model are consid-

ered interval value.

2. Replenishment rate is instantaneous, and lead time is

zero.

3. The own warehouse (OW) has a fixed capacity of W

units, and the rented warehouse (RW) is of unlimited

capacity.

4. The inventory costs in the RW are higher than those in

OW.

5. Demand rate DðtÞ is a function of stock level IðtÞ at

time t which is given by

DðtÞ ¼aþ bIðtÞ; IðtÞ[ 0

a; IðtÞ\0

�;

where a; b are positive constants:

6. The planning horizon of the inventory system is

infinite.

7. Shortages are permitted. Shortages are partially back-

logged, and the fraction of shortages backlogged is a

differentiable and decreasing function of time t, given

by gðtÞ, where t is the waiting time up to the next

order. The partial backlogging rate is gðtÞ ¼ e�dt;

where d is a positive constant.

Additionally, the following notation is used during

development of the inventory model.

Parameter Description

A = [AL, AR] Interval-valued replenishment cost per order ($/

order)

a; b Demand parameters, where a; b[ 0

c = [cL, cR] Interval-valued purchasing cost per unit ($/unit)

c1 = [c1L, c1R] Interval-valued opportunity cost due to lost sale, if

the shortage is lost ($/unit/time unit)

H = [HL, HR] Interval-valued holding cost per unit per unit time in

OW ($/unit/time unit)

F = [FL, FR] Interval-valued holding cost per unit per unit time in

RW, where F[H ($/unit/time unit)

s = [sL, sR] Interval-valued backlogging cost per unit per unit

time, if shortage is backlogged ($/unit/time unit)

Parameter Description

TCi = [TCiL,

TCiR]

Interval-valued total relevant cost per unit time for

case i = 1, 2 ($/time unit)

r Inflation rate (%)

W Capacity of the owned warehouse (units)

I0(t) Inventory level in the OW at any time t where

0 B t B T (units)

Ir(t) Inventory level in the RW at any time t where

0 B t B T units)

BðtÞ Backlogged level at any time t where

tw B t B T (units)

LðtÞ Number of lost sales at any time t where

tw B t B T (units)

td Time period during which no deterioration occurs

(time unit)

a Deterioration rate in OW, where 0 B a\ 1

b Deterioration rate in RW, where 0 B b\ 1; b\ a.

d Backlogging rate, where d[ 0

Decision variables

tr Time at which the inventory level reaches zero in

RW (time unit)

T The length of the replenishment cycle (time unit)

Dependent variables

tw Time at which the inventory level reaches zero in

OW (time unit)

Z Maximum inventory level per cycle (units)

Q Order quantity per cycle (units)

The inventory model considers replenishment cost,

purchasing cost, holding cost, backorder cost and oppor-

tunity cost due to lost sales as interval-valued. The details

of the interval arithmetic and interval order relations are

presented in ‘‘Appendix 1’’.

3 Mathematical formulation of the inventorymodel

This section formulates the mathematical model for two-

warehouse inventory system having non-instantaneous

deteriorating item. In the beginning, an order of Q units

comes into the inventory system. After satisfying the

backorders, Z units are put in storage in the inventory

system, out of which W units are stored in OW and the

remaining Z �Wð Þ units are placed in the RW. Due to

the fact that the deterioration of item is non-instantaneous,

so initially, the units do not deteriorate till the time period

tdð Þ, and after that the deterioration of items starts. Here,

there exist two cases: when td (time during which no

deterioration happens) is less than tr (time during which

inventory in RW becomes zero) and when td is greater

than tr.

Neural Comput & Applic (2019) 31:1931–1948 1933

123

3.1 When td\tr

During the time interval 0; td½ �; there is no deterioration,

and the inventory in RW is consumed only due to demand

while in OW inventory level remains unchanged. In the

time interval td; tr½ �, the inventory level in RW is

decreasing to zero due to the joint effect of demand and

deterioration, and in OW inventory is depleted due to

deterioration only. Further, during the time interval tr; tw½ �reduction in inventory happens in OW due to the joint

effect of demand and deterioration and it attains to zero at

time tw: Likewise, for the duration of the interval tw; T½ �,the demand is backlogged. The inventory level during the

complete cycle 0; T½ � is graphically represented in Fig. 1.

Consequently, the differential equations that model the

inventory level in the RW and OW at any time t during the

period 0; Tð Þ are expressed as:

dIrðtÞdt

¼ � a þ bIrðtÞð Þ; 0� t� td ð1Þ

dIrðtÞdt

þ bIrðtÞ ¼ � a þ bIrðtÞð Þ; td\t� tr ð2Þ

dI0ðtÞdt

þ aI0ðtÞ ¼ 0; td\t� tr ð3Þ

dI0ðtÞdt

þ aI0ðtÞ ¼ � a þ bI0ðtÞð Þ; tr\t� tw ð4Þ

dBðtÞdt

¼ ae�dðT�tÞ; tw\t� T ð5Þ

The solutions of the above five differential equa-

tions (1), (2), (3), (4) and (5) with boundary conditions

Irð0Þ ¼ Z �W ; IrðtrÞ ¼ 0; I0ðtdÞ ¼ W ; I0ðtwÞ ¼ 0 and BðtwÞ¼ 0, respectively, are

IrðtÞ ¼ Z �Wð Þe�bt þ a

be�bt � 1� �

; 0� t� td ð6Þ

IrðtÞ ¼a

bþ be bþbð Þ tr�tð Þ � 1

� �; td\t� tr ð7Þ

I0ðtÞ ¼ Wea td�tð Þ; td\t� tr ð8Þ

I0ðtÞ ¼a

aþ be aþbð Þ tw�tð Þ � 1

� �; tr\t� tw ð9Þ

BðtÞ ¼ a

de�dðT�tÞ � e�dðT�twÞ

n o; tw\t� T ð10Þ

The quantity of lost sales at time t is

LðtÞ ¼Z t

tw

a 1� e�dðT�tÞn o

dt; tw\t� T

LðtÞ ¼ a t � twð Þ � 1


n o� �ð11Þ

Taking into account that there exists continuity of IrðtÞat t ¼ td, it follows from Eqs. (6) and (7) that

Z �Wð Þe�btd þ a

be�btd � 1� �

¼ a

bþ be bþbð Þ tr�tdð Þ � 1

� �ð12Þ

This indicates that the maximum inventory level per

cycle is

Z ¼ W þ a


� �ebtd � a

b1� ebtd� �

ð13Þ

Taking into consideration that there exists continuity of

I0ðtÞ at t ¼ tr, it follows from Eqs. (8) and (9) that

Wea td�trð Þ ¼ a

aþ be aþbð Þ tw�trð Þ � 1

� �

T

W

0 td tr tw

Z-W

ZQ

Lost sales

Time

Fig. 1 Two-warehouse

inventory system when td\tr


123

tw ¼ tr þ1

aþ bln

aþ aþ bð ÞWeaðtd�trÞ

a

ð14Þ

Substituting t ¼ T in Eq. (10), the maximum amount of

demand backlogged per cycle is

B Tð Þ ¼ a

d1� e�dðT�twÞ

� �ð15Þ

As a result, the order quantity during the replenishment

cycle is computed as

Q ¼ Z þ BðTÞ Using Eqs: 13ð Þ and 15ð Þ½ �

Q ¼ W þ a


� �ebtd � a

b1� ebtd� �

þ a


� �ð16Þ

The total cost per cycle is comprised of the following

elements:

1. Present worth of the replenishment cost ¼ A

2. Present worth of the inventory holding cost in RW

¼ F

Ztd0

e�rtIrðtÞdt þZtrtd

e�rtIrðtÞdt

0@

1A

¼ F1

r þ bð Þ Z �W þ a

b

� �1� e� rþbð Þtd

n oþ a

bre�rtd � 1f g

�

þ ae�rtr

bþ bð Þ1

bþ bþ rð Þ eðbþbþrÞ tr�tdð Þ � 1n o�

� 1

rer tr�tdð Þ � 1

n o��

3. Present worth of the inventory holding cost in OW

¼ H

Ztd0

We�rtdt þZtrtd

e�rtI0ðtÞdt þZtwtr

e�rtI0ðtÞdt

0@

1A

¼ HW

r1� e�rtdð Þ þWe�rtd

aþ r1� eðaþrÞ td�trð Þ

� ��

þ ae�rtw

aþ b

1

aþ bþ reðaþbþrÞ tw�trð Þ � 1

� ��

� 1

rer tw�trð Þ � 1

� ��

4. Present worth of the backlogging cost ¼ sRTtw

BðtÞe�rtdt

¼ sa

de�dT 1

d� reðd�rÞT � eðd�rÞtw

n oþ edtw

re�rT � e�rtw

� � �

5. Present worth of the opportunity cost due to lost sales

is

¼ c1e�rT

ZT

tw

1� e�dðT�tÞn o

adt

¼ c1ae�rT T � tw � 1


n o� �

6. Present worth of the deterioration cost is

¼ c bRtrtd

IrðtÞe�rtdt þ aRtwtr

e�rtI0ðtÞdt" #

¼ cae�rtr

bþ bð Þ1

bþ bþ rð Þ eðbþbþrÞ tr�tdð Þ � 1� ��

� 1


� ��

þ e�rtw

aþ b

1


� ��

� 1


� ��

Hence, using the above elements, the present worth of

the total relevant cost per unit time is interval-valued and it

is given by TC1 ¼ ½TC1L; TC1R�.

TC1L tr; Tð Þ ¼ 1

TAL þ FL

1


b


n o��

þ a

bre�rtd � 1f g þ ae�rtr

bþ bð Þ1


� 1


n o�

þ HL

W

r1� e�rtdf g þWe�rtd


n o�

þ ae�rtw

aþ b

1


n o�� 1


n o�

þ sLa

de�dT 1


� �þ edtw

re�rT � e�rtw� ��

þ c1Lae�rT T � tw � 1


� ��

þcLae�rtr

bþ bð Þ1


� 1


� ��þ e�rtw

aþ b

1


� ��

� 1


� ��

ð17Þ


123

and

TC1R tr; Tð Þ ¼ 1

TAR þ FR

1


b


n o��

þ a

bre�rtd � 1f g þ ae�rtr

bþ bð Þ1


� 1


n o�

þ HR

W

r1� e�rtdf g þWe�rtd


n o�

þ ae�rtw

aþ b

1


n o� 1


n o� ��

þ sRa

de�dT 1


� �þ edtw

re�rT � e�rtw� ��

þ c1Rae�rT T � tw � 1


� ��

þcRae�rtr

bþ bð Þ1


� 1


� ��þ e�rtw

aþ b

1


� ��

� 1


� ��

ð18Þ

where Z ¼ W þ abþb

e bþbð Þ tr�tdð Þ � 1� �

ebtd � ab1� ebtd� �

; tw

¼ tr þ 1aþb

lnaþ aþbð ÞWeaðtd�trÞ

a

� �

3.2 When td � tr

The details of the derivation of the mathematical model for

this case are presented in ‘‘Appendix 2’’.

The present worth of the total relevant cost per unit time

is interval-valued, and it is expressed by TC2 ¼½TC2L; TC2R�.

TC2L tr; Tð Þ ¼ 1

TAL þ

FLa

b

1

r þ bð Þ ebtr � e�rtr� ��

þ 1

re�rtr � 1ð Þ

�þ HL

W

r1� e�rtrð Þ þ e�rtr

r þ bW þ a

b

� ��

1� e rþbð Þ tr�tdð Þ� �

þ ae�rtr

brer tr�tdð Þ � 1

� �

þ ae�rtw

aþ b

1

aþ bþ re aþbþrð Þ tw�tdð Þ � 1

� ��

þ 1

r1� er tw�tdð Þ

� ��þ sLa

de�dT

eðd�rÞtw

d� reðd�rÞ T�twð Þ � 1

n oþ edtw

re�rT � e�rtw

� � �

þc1Lae�rT T � tw � 1


n o� �þ cL

ae�rtw

aþ b

1


� �þ 1


� ��

ð19Þ

and

TC2R tr;Tð Þ ¼ 1

TAR þ

FRa

b

1

rþ bð Þ ebtr � e�rtr� ��

þ1

re�rtr � 1ð Þ

�þHR

W

r1� e�rtrð Þ

�

þ e�rtr

rþ bW þ a

b

� �1� e rþbð Þ tr�tdð Þ

� �þ ae�rtr


� �

þae�rtw

aþ b

1


� ��

þ1


� ��þ sRa

de�dT

eðd�rÞtw

d� reðd�rÞ T�twð Þ � 1

n oþ edtw

re�rT � e�rtw

� � �

þc1Rae�rT T � tw � 1


n o� �þ cR

ae�rtw

aþ b

1


� �þ 1


� ��

ð20Þ

where Z ¼ W þ abebtr � 1� �

and tw ¼ td þ 1aþb

ln 1þjaþba

W þ ab

� �eb tr�tdð Þ � a

b

� j

Therefore, the present worth of the total relevant cost

per unit time over the cycle 0; Tð Þ is expressed by

TC tr;Tð Þ ¼TC1L tr;Tð Þ;TC1R tr;Tð Þ½ � if td\tr

TC2L tr;Tð Þ;TC2R tr;Tð Þ½ � if td� tr

�ð21Þ

which is a function of two continuous variables tr and T .

4 Optimization algorithm

In this paper, the aim of the proposed inventory model is to

obtain the optimal values of tr and T which minimize the

present worth of the total relevant cost per unit time. As the

objective functions formulated above are highly nonlinear,

thus, the given optimization problem is solved with the

PSO technique. The details of the PSO are presented in

‘‘Appendix 3’’. Basically, two variants of the PSO tech-

nique are used to solve the optimization problem. These

variants are the particle swarm optimization with con-

striction coefficient (PSO-CO) and weighted quantum

particle swarm optimization (WQPSO). The algorithms

PSO-CO and WQPSO are given in ‘‘Appendix 3’’.

5 Numerical examples and sensitivity analysis

5.1 Numerical examples

To validate the proposed inventory model, three numerical

examples are solved. In determining the solution, the soft


123

computing methods PSO-CO and WQPSO are used. Both

PSO-CO and WQPSO were coded in C programming

language. The computational runs were done on a PC with

Intel Core-2-duo 2.5 GHz Processor in LINUX environ-

ment. For each case, 20 independent runs were made for

each variant of PSO.

For the computational runs, the values for the PSO

parameters are: p_size = 100, m_gen = 100, C1 = 2.05,

C2 = 2.05. The initial velocity is given randomly between

�Vmax and Vmax where Vmax is established to be equal to

20% of the range of each variable in the search domain.

Example 1 The values for the parameter for this example

are as follows: AL = 248, AR = 252, HL = 0.3, HR = 0.7,

FL = 0.5, FR = 0.9, c1L = 4, c1R = 6, cL = 18, cR = 22,

a = 0.05, b = 0.03, r = 0.06, d = 0.9, td = 0.1,

W = 200, sL = 5, sR = 15, a = 80, b = 10 in appropriate

units (Tables 2, 3).



FL = 0.6, FR = 1.1, c1L = 8, c1R = 12, cL = 15, cR = 20,

a = 0.05, b = 0.03, r = 0.06, d = 0.9, td = 0.1,

W = 200, sL = 7, sR = 17, a = 100, b = 12 in appropri-

ate units (Tables 4, 5).



FL = 0.5, FR = 0.9, c1L = 12, c1R = 14, cL = 20,

cR = 25, a = 0.05, b = 0.03, r = 0.06, d = 0.9, td = 0.2,

W = 150, sL = 10, sR = 15, a = 100, b = 5 in appropri-

ate units (Tables 6, 7).

According to the statistical analysis of the results (see

Tables 8, 9, 10, 11, 12, 13), it is easy to see that the both

soft computing methods PSO-CO and WQPSO are

steady.

Table 2 Best found solution of the inventory models for Example 1 by PSO-CO

Case tr tw T Z� B� Q� TC�L TC�

R Centre value of total cost

1 0.10 0.4247 2.5403 331.32 75.65 406.97 334.3136 603.8356 469.07

2 0.10 0.4247 2.5703 213.75 76.00 289.75 338.4069 593.2855 465.85

Table 3 Best found solution of the inventory models for Example 1 by WQPSO

Case tr tw T Z� B� Q� TC�L TC�


1 0.10 0.4247 2.5594 331.34 75.87 407.21 331.3416 603.95 469.05

2 0.10 0.4247 2.5483 213.75 75.74 288.74 338.6174 591.1686 465.85


Case t�r t�w T� Z� B� Q� TC�L TC�


1 0.10 0.3675 2.7738 338.32 64.86 403.18 606.6892 955.6686 781.18

2 0.10 0.3675 2.8104 219.33 64.95 284.28 609.5412 942.5094 776.03

Table 5 Best found solution of the inventory models for Example 2 by WQPSO



1 0.10 0.3675 2.8054 388.33 64.94 453.27 606.7538 955.5281 781.14

2 0.10 0.3675 2.8391 219.36 65.03 284.39 609.5358 940.7254 775.13




1 0.2 0.6255 3.4609 887.25 39.96 927.21 817.3038 970.1118 893.71

2 0.2 0.6265 3.6078 184.36 39.97 224.33 809.2853 943.0838 876.18


123

5.2 Sensitivity analysis

In order to demonstrate the robustness of proposed inven-

tory model and the impact of parameters on the optimal

solution, a sensitivity analysis has been performed. The

percentage changes are considered as measures of sensi-

tivity. The sensitivity analyses are done by varying the

parameters by -20 to ?20%. The results are determined

by changing one parameter at a time and leaving the other

parameters with their original values.

Based on the observation in Tables 14 and 15, the fol-

lowing insights have been drawn:

• One can clearly observe from Tables 14 and 15 that

with the increment in the value of ordering cost (A),

holding cost (H) of OW and holding cost (F) of RW,

Table 7 Best found solution of

the inventory models for

Example 3 by WQPSO



1 0.20 0.6267 3.4242 887.29 39.96 927.25 817.14 970.13 893.64

2 0.20 0.6265 3.6214 184.38 39.98 288.74 809.27 941.74 875.51

Table 8 Statistical analysis of

the results for PSO-CO from

Example 1

Cases 1 2

Mean objective value [334.3136,603.8356] [338.4069,593.2855]

Centre of the objective valuea 469.07 465.85

SD of centre objective valuea 1.1648 9 10-13 1.7474 9 10-13

COV of centre objective valuea 2.4835 9 10-16 3.751 9 10-16

SD of lower bound 1.1649 9 10-13 0.00

SD of upper bound 1.165 9 10-05 0.00

COV of lower bound 3.4845 9 10-16 0.00

COV of upper bound 1.9292 9 10-16 0.00

a SD standard deviation, COV coefficient of variation



Example 2

Cases 1 2





SD of lower bound 1.1649 9 10-13 1.16494 9 10-13

SD of upper bound 5.825 9 10-13 2.33 9 10-13

COV of lower bound 1.92017 9 10-16 1.91118 9 10-16

COV of upper bound 6.09491 9 10-16 2.472 9 10-16




Example 3

Cases 1 2





SD of lower bound 1.1649 9 10-13 2.32989 9 10-13

SD of upper bound 3.495 9 10-13 1.165 9 10-13





123

there is a decrement in the values of optimal cycle

length (T), optimal maximum backorder (B) and opti-

mal order quantity (Q), but present worth of total

optimal cost (TC) increases, because the ordering cost

(A), holding cost (H) of OW and holding cost (F) of

RW increase.

• As purchasing cost (c), backordering cost (s) and

opportunity cost (c1) increase, the optimal cycle length

(T), optimal maximum backorder (B) and optimal order

quantity (Q) increase which results in an increment in

the value present worth of total optimal cost (TC).

• Also, with an increment in the net rate of inflation(r),

the optimal cycle length (T), optimal maximum

backorder (B), optimal order quantity (Q) and present

worth of total optimal cost (TC) decrease.

• With the increment in the value of backlogging

parameter (d), optimal cycle length (T), optimal

maximum backorder (B), optimal order quantity

(Q) and present worth of total optimal cost (TC)

decrease.

• With an increment in the value of non-deteriorating

period (td), it can be perceived that cycle length (T) and

order quantity (Q) increase, but present worth of total

optimal cost (TC) decreases. The result is quite

apparent, because as the non-deteriorating period (td)

increases, the number of deteriorating units decreases


the results for WQPSO from

Example 1

Cases 1 2





SD of lower bound 5.8247 9 10-14 5.8247 9 10-14

SD of upper bound 1.165 9 10-13 2.33 9 10-13






Example 2

Cases 1 2





SD of lower bound 2.32989 9 10-14 1.16494 9 10-13

SD of upper bound 2.33 9 10-13 3.495 9 10-13






Example 3

Cases 1 2



SD of centre objective valuea 2.32989 9 10-13 0.00

COV of centre objective valuea 2.60719 9 10-16 0.00

SD of lower bound 2.32989 9 10-13 1.16494 9 10-13

SD of upper bound 2.33 9 10-13 0.00


COV of upper bound 2.40162 9 10-16 0.00



123

Table 14 Sensitivity analysis with different parameters of the inventory system when td\tr

Parameters % change of parameters % change in

t�w T� B� Z� Q� TC�L TC�

R Centre value

of total cost

AL -20 0.00 3.54 1.36 0.00 0.25 -5.86 0.07 -2.04

-10 0.00 1.80 0.70 0.00 0.13 -2.98 0.05 -1.03

10 0.00 -1.85 -0.76 0.00 -0.14 3.08 -0.07 1.05

20 0.00 -3.83 -1.60 0.00 -0.30 6.28 -0.18 2.12

AR -20 0.00 3.60 1.38 0.00 0.26 -0.22 -3.10 -2.08

-10 0.00 1.83 0.72 0.00 0.13 -0.11 -1.57 -1.05

10 0.00 -1.91 -0.78 0.00 -0.15 0.11 1.60 1.07

20 0.00 -3.87 -1.62 0.00 -0.30 0.20 3.24 2.16

c1L -20 0.00 -3.38 -1.41 0.00 -0.26 -7.21 -0.15 -2.66

-10 0.00 -1.67 -0.68 0.00 -0.13 -3.65 -0.06 -1.34

10 0.00 1.63 0.64 0.00 0.12 3.73 0.04 1.36

20 0.00 3.22 1.24 0.00 0.23 7.54 0.07 2.73

c1R -20 0.64 -5.15 -2.18 0.00 -0.41 0.26 -6.32 -3.97

-10 0.64 -2.52 -1.04 0.00 -0.19 0.14 -3.19 -2.00

10 0.00 2.43 0.95 0.00 0.18 -0.15 3.25 2.04

20 0.00 4.78 1.81 0.00 0.34 -0.31 6.56 4.11

cL -20 0.00 0.23 0.09 0.00 0.02 -0.38 0.01 -0.13

-10 0.00 0.12 0.05 0.00 0.01 -0.19 0.00 -0.06

10 0.00 -0.11 -0.04 0.00 -0.01 0.19 0.00 0.07

20 0.00 -0.22 -0.09 0.00 -0.02 0.38 -0.01 0.13

cR -20 0.00 0.29 0.12 0.00 0.02 -0.02 -0.24 -0.16

-10 0.02 0.14 0.06 0.00 0.01 -0.01 -0.12 -0.08

10 -0.02 -0.13 -0.05 0.00 -0.01 0.01 0.12 0.08

20 -0.07 -0.28 -0.13 0.00 -0.02 0.02 0.24 0.16

td -20 -4.71 -1.54 -0.30 -10.68 -8.75 0.61 0.51 0.55

-10 -2.35 -0.76 -0.14 -5.60 -4.59 0.30 0.25 0.27

10 2.35 0.74 0.14 6.19 5.07 -0.28 -0.24 -0.25

20 4.71 1.41 0.25 13.04 10.66 -0.54 -0.46 -0.49

W -20 -5.02 -1.13 -0.12 -12.07 -9.85 0.27 0.31 0.29

-10 -2.38 -0.49 -0.04 -6.04 -4.92 0.11 0.13 0.12

10 2.14 0.41 0.02 6.04 4.92 -0.06 -0.09 -0.08

20 4.12 0.72 0.01 12.07 9.83 -0.08 -0.14 -0.12

d -20 0.00 17.66 23.70 0.00 4.41 2.32 9.49 6.94

-10 0.00 8.13 10.66 0.00 1.98 0.86 4.16 2.99

10 0.00 -7.09 -8.90 0.00 -1.65 -0.43 -3.31 -2.28

20 0.00 -13.41 -16.48 0.00 -3.06 -0.51 -5.99 -4.04

a -20 5.04 -9.33 -23.67 -13.22 -15.16 -13.33 -17.16 -15.80

-10 2.38 -3.75 -11.57 -6.94 -7.80 -6.87 -8.60 -7.98

10 -2.14 2.73 11.31 7.60 8.29 7.09 8.64 8.09

20 -4.10 4.84 22.49 15.87 17.23 14.32 17.33 16.26

b -20 12.76 4.66 0.98 -3.91 -3.36 -1.82 -1.74 -1.77

-10 5.82 2.16 0.47 -1.63 -1.42 -0.85 -0.81 -0.82

10 -4.97 -1.89 -0.43 1.15 1.02 0.76 0.71 0.73

20 -9.28 -3.58 -0.83 1.93 1.72 1.44 1.34 1.38


123

so the cost incurred due to deterioration also decreases

which accounts for less total cost of the system.

• As per the capacity of the owned warehouse (W) in-

creases, the optimal cycle length (T), optimal maximum

backorder (B) and optimal order quantity (Q) increase,

whereas the value present worth of total optimal cost

(TC) decreases.

• When the parameter (a) increases, the optimal cycle

length (T), optimal maximum backorder (B) and

optimal order quantity (Q) and the value present worth

of total optimal cost (TC) increase, whereas the time at

which the inventory level reaches zero in OW (tw)

decreases.

• With the increment in the value of the parameter (b),

the time at which the inventory level reaches zero in

OW (tw), optimal cycle length (T), optimal maximum

backorder (B) decrease while the optimal order quantity

(Q) and the value present worth of total optimal cost

(TC) increase.

6 Conclusion and directions for future research

6.1 Conclusion

In this paper, a two-warehouse inventory model for non-

instantaneous deteriorating items with interval-valued

inventory costs, partial backlogging and stock-dependent

demand under inflationary conditions has been discussed.

In order to make a more realistic scenario, the shortages are

permitted and are partially backlogged. The present two-

warehouse inventory model is very helpful for retail and

Table 14 continued



R Centre value

of total cost

r -20 0.00 5.11 1.93 0.00 0.36 1.68 2.35 2.11

-10 0.00 2.48 0.97 0.00 0.18 0.81 1.15 1.03

10 0.00 -2.35 -0.97 0.00 -0.18 -0.76 -1.11 -0.98

20 0.00 -4.60 -1.94 0.00 -0.36 -1.46 -2.17 -1.92

HL -20 0.00 0.16 0.06 0.00 0.01 -0.27 0.00 -0.09

-10 0.00 0.08 0.03 0.00 0.01 -0.14 0.00 -0.04

10 0.00 -0.08 -0.03 0.00 -0.01 0.14 0.00 0.05

20 0.00 -0.17 -0.07 0.00 -0.01 0.27 -0.01 0.09

HR -20 0.00 0.40 0.16 0.00 0.03 -0.02 -0.32 -0.22

-10 0.00 0.20 0.08 0.00 0.01 -0.01 -0.16 -0.11

10 0.00 -0.20 -0.08 0.00 -0.01 0.01 0.16 0.11

20 0.00 -0.38 -0.15 0.00 -0.03 0.02 0.33 0.22

FL -20 0.00 0.09 0.04 0.00 0.01 -0.01 -0.09 -0.06

-10 0.00 0.06 0.02 0.00 0.00 0.00 -0.05 -0.03

10 0.00 -0.06 -0.02 0.00 0.00 0.00 0.05 0.03

20 0.00 -0.11 -0.04 0.00 -0.01 0.01 0.09 0.06

FR -20 0.00 5.11 1.93 0.00 0.36 1.68 2.35 2.11

-10 0.00 2.48 0.97 0.00 0.18 0.81 1.15 1.03

10 0.00 -2.35 -0.97 0.00 -0.18 -0.76 -1.11 -0.98

20 0.00 -4.60 -1.94 0.00 -0.36 -1.46 -2.17 -1.92

sL -20 0.00 -0.10 -0.04 0.00 -0.01 -5.86 0.00 -2.09

-10 0.00 -0.02 -0.01 0.00 0.00 -2.93 0.00 -1.04

10 0.00 0.04 0.02 0.00 0.00 2.93 0.00 1.05

20 0.00 0.08 0.03 0.00 0.01 5.87 0.00 2.09

sR -20 0.00 -0.32 -0.13 0.00 -0.02 0.02 -9.76 -6.27

-10 0.00 -0.14 -0.06 0.00 -0.01 0.01 -4.88 -3.14

10 0.00 0.12 0.05 0.00 0.01 -0.01 4.88 3.14

20 0.00 0.22 0.09 0.00 0.02 -0.01 9.76 6.28


123

Table 15 Sensitivity analysis with different parameters of the inventory system when td � tr



R Centre value

of total cost

AL -20 0.00 3.45 1.30 0.00 0.34 -5.78 0.01 -2.04

-10 0.00 1.76 0.68 0.00 0.18 -2.93 0.06 -1.03

10 0.00 -1.77 -0.71 0.00 -0.19 3.03 -0.09 1.04

20 0.00 -3.64 -1.49 0.00 -0.39 6.17 -0.21 2.11

AR -20 0.00 3.50 1.32 0.00 0.35 -0.27 -3.09 -2.07

-10 0.00 1.79 0.69 0.00 0.18 -0.14 -1.56 -1.04

10 0.00 -1.80 -0.72 0.00 -0.19 0.13 1.59 1.06

20 0.00 -3.71 -1.49 0.00 -0.39 0.26 3.22 2.14

c1L -20 0.00 -2.84 -1.16 0.00 -0.30 -7.14 -0.18 -2.71

-10 0.00 -1.18 -0.47 0.00 -0.12 -3.61 -0.08 -1.36

10 0.00 2.02 0.78 0.00 0.20 3.69 0.06 1.38

20 0.00 3.57 1.35 0.00 0.35 7.46 0.09 2.77

c1R -20 0.64 -4.55 -1.89 0.00 -0.50 0.34 -6.53 -4.04

-10 0.64 -2.01 -0.81 0.00 -0.21 0.17 -3.30 -2.04

10 0.00 2.79 1.17 0.00 0.31 -0.18 3.35 2.17

20 0.00 5.10 1.90 0.00 0.50 -0.37 6.76 4.17

cL -20 0.00 0.66 0.26 0.00 0.07 -0.37 0.01 -0.13

-10 0.00 0.53 0.21 0.00 0.05 -0.19 0.00 -0.06

10 0.00 0.34 0.13 0.00 0.03 0.19 0.00 0.07

20 0.00 0.22 0.09 0.00 0.02 0.37 -0.01 0.13

cR -20 0.00 0.71 0.28 0.00 0.07 -0.02 -0.23 -0.16

-10 0.02 0.57 0.22 0.00 0.06 -0.01 -0.12 -0.08

10 -0.02 0.30 0.12 0.00 0.03 0.01 0.12 0.08

20 -0.07 0.17 0.06 0.00 0.02 0.02 0.24 0.16

td -20 -4.71 -1.54 -0.30 -10.68 -8.75 0.61 0.51 0.55

-10 -2.35 -0.76 -0.14 -5.60 -4.59 0.30 0.25 0.27

10 2.35 0.74 0.14 6.19 5.07 -0.28 -0.24 -0.25

20 4.71 1.41 0.25 13.04 10.66 -0.54 -0.46 -0.49

W -20 -5.02 -0.83 0.00 -18.71 -13.81 -0.07 0.65 0.39

-10 -2.38 -0.14 0.01 -9.36 -6.88 -0.06 0.30 0.17

10 2.14 0.92 0.22 9.36 6.96 0.11 -0.26 -0.13

20 4.12 1.31 0.25 18.71 13.88 0.25 -0.48 -0.21

d -20 0.00 17.71 23.72 0.00 6.21 2.14 9.91 7.09

-10 0.00 8.36 10.75 0.00 2.82 0.78 4.36 3.06

10 0.00 -6.44 -8.65 0.00 -2.27 -0.35 -3.49 -2.35

20 0.00 -12.53 -16.14 0.00 -4.23 -0.38 -6.33 -4.17

a -20 5.04 -8.50 -23.29 -1.29 -7.05 -12.94 -17.46 -15.82

-10 2.38 -3.20 -11.33 -0.64 -3.44 -6.67 -8.73 -7.98

10 -2.14 3.13 11.46 0.64 3.48 6.89 8.76 8.08

20 -4.10 5.20 22.61 1.29 6.87 13.92 17.54 16.23

b -20 12.76 5.21 1.18 -0.70 -0.21 -1.78 -1.94 -1.88

-10 5.82 2.64 0.65 -0.36 -0.10 -0.84 -0.90 -0.88

10 -4.97 -1.49 -0.27 0.39 0.22 0.74 0.79 0.77

20 -9.28 -3.20 -0.67 0.81 0.42 1.41 1.49 1.46


123

manufacturing industries in developing countries where

inflation plays a significant role in decision-making. The

aim of this article is to find the retailer’s optimal replen-

ishment policy that minimizes the present worth of total

cost per unit time. The corresponding optimization prob-

lems for the two-warehouse inventory model have been

formulated and solved with two variants of PSO and

interval order relations. The efficiency and effectiveness of

the proposed two-warehouse inventory model are validated

with numerical examples and a sensitivity analysis. The

present paper proves to be very helpful for the decision

maker while considering both phenomena, viz. deteriora-

tion and inflation as these play opposite roles in the deci-

sion-making process. Since deterioration suggests one to

order a smaller lot, high inflation recommends ordering

more.

6.2 Directions for future research

The proposed two-warehouse inventory model can be

extended further in several forms. For example, it can be

formulated under single-level and two-level trade credit

policy. To be more to reality, the inventory model can be

extended with different types of variable demand such as

credit-linked demand and advertisement-dependent

demand, among others. One can also incorporate vendor–

buyer problem by considering same assumptions of present

inventory model.

Acknowledgements The authors are thankful to the anonymous

reviewers for their comments and suggestions which have helped to

improve the quality of the paper. The work was done when the third

author was doing his Ph.D. from University of Delhi. The second

author was supported by the Tecnologico de Monterrey Research

Table 15 continued



R Centre value

of total cost

r -20 0.00 5.11 1.93 0.00 0.36 1.68 2.35 2.11

-10 0.00 2.48 0.97 0.00 0.18 0.81 1.15 1.03

10 0.00 -2.35 -0.97 0.00 -0.18 -0.76 -1.11 -0.98

20 0.00 -4.60 -1.94 0.00 -0.36 -1.46 -2.17 -1.92

HL -20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

HR -20 0.00 0.37 0.14 0.00 0.04 -0.62 0.02 -0.22

-10 0.00 0.18 0.07 0.00 0.02 -0.31 0.01 -0.11

10 0.00 -0.18 -0.07 0.00 -0.02 0.31 -0.01 0.11

20 0.00 -0.37 -0.15 0.00 -0.04 0.63 -0.02 0.22

FL -20 0.00 0.01 0.00 0.00 0.00 -0.01 0.00 0.00

-10 0.00 0.02 0.01 0.00 0.00 0.00 0.00 0.00

10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

20 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00

FR -20 0.00 0.02 0.00 0.00 0.00 0.00 -0.01 0.00

-10 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00

10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

20 0.00 -0.01 0.00 0.00 0.00 0.00 0.01 0.00

sL -20 0.00 -0.01 0.00 0.00 0.00 -5.80 0.00 -2.11

-10 0.00 -0.01 0.00 0.00 0.00 -2.90 0.00 -1.05

10 0.00 0.01 0.00 0.00 0.00 2.90 0.00 1.05

20 0.00 0.01 0.00 0.00 0.00 5.80 0.00 2.11

sR -20 0.00 -0.07 -0.03 0.00 -0.01 0.00 -9.93 -6.32

-10 0.00 -0.03 -0.01 0.00 0.00 0.00 -4.96 -3.16

10 0.00 0.03 0.01 0.00 0.00 0.00 4.96 3.16

20 0.00 0.05 0.02 0.00 0.00 0.00 9.93 6.32


123

Group in Industrial Engineering and Numerical Methods

0822B01006. The third author is grateful to his parents, wife, children

Aditi Tiwari and Aditya Tiwari for their valuable support during the

development of this paper.

Compliance with ethical standards

Conflict of interest The authors declare that there are no conflicts of

interest regarding the publication of this paper.

Appendix 1

This appendix presents a brief overview of interval

arithmetic.

The brief summary of interval mathematics is from

Hansen and Walster [53]. The definitions of interval

arithmetic, interval functions, interval order relations and

central tendencies of interval numbers that are indispens-

able in the formulation of the two-warehouse inventory

model are given below.

An interval number A is defined as A ¼ aL; aR½ � ¼x : aL � x� aR; x 2 Rf g of width aR � aLð Þ. Here, R is the

set of real numbers.

Every real number x 2 R is expressed as a degenerate

interval number [x, x] with zero width. Interval number is

also written in the form of centre and radius of the interval

as A ¼ aC; aWh i ¼ x : aC�aW � x� aC þ aW ; x 2fRg; where aC ¼ aL þ aRð Þ=2 ¼ center of the interval

and aW ¼ aR�aLð Þ=2 ¼ radius of the interval:

Definition 1 Let A ¼ ½aL; aR� and B ¼ ½bL; bR� be two

interval numbers. The addition, subtraction, scalar multi-

plication, multiplication and division of interval numbers

are given below:

1. Addition:

Aþ B ¼ aL; aR½ � þ bL; bR½ � ¼ aL þ bL; aR þ bR½ �:2. Subtraction: A � B ¼ aL; aR½ � � bL; bR½ � ¼ aL; aR½ �

þ �bL;�bR½ � ¼ aL � bL; aR � bR½ �:3. Scalar multiplication:kA ¼ k aL; aR½ � ¼

kaL; kaR½ � if k� 0

kaR; kaL½ � if k\0;

�for any real number k:

4. Multiplication: A� B ¼ min aLbL; aLbR; aRbL;ð½aRbRÞ; max aLbL; aLbR; aRbL; aRbRð Þ�:

5. Division: AB¼ A� 1

B

� �¼ aL; aR½ � � 1

bR; 1bL

h iprovided

0 62 [bL, bR].

Interval order relations

Let A ¼ ½aL; aR� and B ¼ ½bL; bR� be two intervals. Then,

these two intervals can be anyone of the following types:

Type 1: Two intervals are disjoint.

Type 2: Two intervals are partially overlapping.

Type 3: One of the intervals contains the other one.

Some researchers gave the definitions of order rela-

tions between two interval numbers. In recent times,

Sahoo et al. [54] provided the same by modifying the

drawbacks of existing definitions. Their definitions are as

follows:

Definition 2 The order relation [ max among the intervals

A ¼ ½aL; aR� ¼ ac; awh i and B ¼ ½bL; bR� ¼ bc; bwh i; then,

for maximization problems

1. A[ maxB , ac [ bc for Type I and Type II intervals,

2. A[ maxB , either ac � bc ^ aw\bw or ac � bc^aR [ bR for Type III intervals,

According to this definition, the interval A is accepted

for maximization case. Obviously, the order relation

A[ maxB is reflexive and transitive, but not symmetric.

Definition 3 The order relation \min among the intervals

A ¼ ½aL; aR� ¼ ac; awh i and B ¼ ½bL; bR� ¼ bc; bwh i; then,

for minimization problems

1. A\minB , ac\bc for Type I and Type II intervals,

2. A\minB , either ac � bc ^ aw\bw or ac � bc^aL\bL for Type III intervals,

According to this definition, the interval A is accepted

for minimization case. Obviously, the order relation

A\minB is reflexive and transitive, but not symmetric.

Appendix 2

This appendix completes the modelling part of case when

td � tr. In this case, time during which no deterioration

happens is greater than the time during which inventory in

RW attains zero and the behaviour of the inventory model

over the entire cycle 0; T½ � is graphically depicted in Fig. 2.The differential equations that model the inventory level

in the RW and OW at any time t within the period 0; Tð Þare expressed as:

dIrðtÞdt

¼ � a þ bIrðtÞð Þ; 0� t� tr ð22Þ

dI0ðtÞdt

¼ � a þ bI0ðtÞð Þ; tr\t� td ð23Þ

dI0ðtÞdt

þ aI0ðtÞ ¼ � a þ bI0ðtÞð Þ; td\t� tw ð24Þ

dBðtÞdt

¼ ae�dðT�tÞ; tw\t� T ð25Þ

The solutions of the above differential equations with

boundary conditions Ir trð Þ ¼ 0; I0 trð Þ ¼ W ; I0 tWð Þ ¼ 0;B

tWð Þ ¼ 0 are

IrðtÞ ¼a

beb tr�tð Þ � 1

� �; 0� t� tr ð26Þ


123

I0ðtÞ ¼ W þ a

b

� �eb tr�tð Þ � a

b; tr\t� td ð27Þ

I0ðtÞ ¼a

aþ be aþbð Þ tw�tð Þ � 1

� �; td\t� tw ð28Þ

BðtÞ ¼ a


n o; tw\t� T ð29Þ

The quantity of lost sales at time t is

LðtÞ ¼Z t

tw

a 1� e�dðT�tÞn o

dt; tw\t� T

LðtÞ ¼ a t � twð Þ � 1


n o� �ð30Þ

In view that there exists continuity of I0ðtÞ at t ¼ td;,

thus

W þ a

b


b¼ a

aþ be aþbð Þ tw�tdð Þ � 1

� �

tw ¼ td þ1

aþ bln 1þ aþ b

aW þ a

b


b

n o��ð31Þ

Now, using Ir 0ð Þ ¼ Z �W , then the maximum inven-

tory is

Z ¼ W þ a

bebtr � 1� �

ð32Þ

By placing t ¼ T into Eq. (29), the maximum amount

of demand backlogged per cycle is

B Tð Þ ¼ a


� �ð33Þ

As a result, order quantity is Q ¼ Z þ BðTÞ

Q ¼ W þ a

bebtr � 1� �

þ a


� �ð34Þ

Again, the total cost per cycle consists of the following

elements:

1. Present worth of the replenishment cost = A

2. Present worth of the inventory holding cost in

RW = FRtr0

e�rtIrðtÞdt

¼ Fa

b

1

r þ bð Þ ebtr � e�rtr� �

þ 1

re�rtr � 1ð Þ

� �

3. Present worth of the inventory holding cost in OW

¼ H

Ztr0

e�rtWdtþZtdtr

e�rtI0ðtÞdtþZtwtd

e�rtI0ðtÞdt

0@

1A

¼ HW

r1� e�rtrð Þ þ e�rtr

rþ bW þ a

b

� �1� e rþbð Þ tr�tdð Þ

� ��

þae�rtr


� �

þae�rtw

aþ b

1


� ��

þ1


� ��

4. Present worth of the backlogging cost = sRTtW

BðtÞe�rtdt

¼ sa

de�dT 1


n oþ edtw

re�rT � e�rtw

� � �

5. Present worth of opportunity cost due to lost sales ¼

c1e�rT

RTtw

1� e�dðT�tÞ� Ddt

tdtr tw

T

W

Z-W

ZQ

0

Fig. 2 Two-warehouse

inventory system when td [ tr


123

¼ c1ae�rT T � tw � 1


n o� �

6. Present worth of the cost for deteriorated items

¼ caRtwtd

e�rtI0ðtÞdt

¼ cae�rtw

aþ b

1


� ��

þ 1


� ��

Appendix 3: Particle swarm optimization (PSO)

This appendix provides a brief overview of PSO. The concept

of particle swarm optimization algorithm (PSO) was intro-

duced by Eberhart and Kennedy [55] and Kennedy and

Eberhart [56]. This algorithm has been used broadly in

obtaining solutions for optimization problems. The basic

theory of PSO is based on the food-searching activities of

birds. After Eberhart and Kennedy [55] and Kennedy and

Eberhart [56], a lot of work has been published in this field by

many researchers and they developed different variants of

PSO. In this paper, two types of PSO algorithms named as

PSO-CO and WQPSO are applied. Clerc [57] and Clerc and

Kennedy [58] proposed an improved version of PSO, and this

version of PSO is known as PSO-CO, i.e. constriction coef-

ficient-basedPSO,whileXi et al. [59] introduced theweighted

quantum particle swarm optimization (WQPSO).

Below, it is presented the notation used in the PSO

algorithm.

Notation Description

N Dimensionality of the search space

p_size Population size

m_gen Maximum number of generations

v Constriction factor

c1 [ 0ð Þ Cognitive learning rate

c2 [ 0ð Þ Social learning rate

r1, r2 Uniformly distributed random numbers lying in the

interval [0, 1].

vðkÞi

Velocity of ith particle at kth generation/iteration

xðkÞi

Position of ith particle of population at kth generation

pðkÞi

Best previous position of ith particle at kth generation

pðkÞgPosition of the best particle among all the particles in the

population

The algorithms of the PSO-CO and WQPSO are given

below.

Algorithm of particle swarm optimization with constriction (PSO-

CO)

Step 1. Set all PSO parameters and bounds of the decision

variables.

Step 2. Set a population size of particles with random positions

and velocities.

Step 3. Determine the fitness value of all particles.

Step 4. Save track of the locations where each individual has its

highest fitness so far.

Step 5. Save track of the position with the global best fitness.

Step 6. Update the velocity of each particle by using the following

equation:

vðkþ1Þi ¼ v v

ðkÞi þ C1r1 p

ðkÞi � x

ðkÞi

� �þ c2r2 pðkÞg � x

ðkÞi

� �h i

where

v ¼ 2

2�ðc1þc2Þ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðc1þc2Þ2�4ðc1þc2Þ

p�� Step 7. Update the position of each particle by using the following

equation:

xðkþ1Þi ¼ x

ðkÞi þ v

ðkþ1Þi

Step 8. If the stop criterion is reached, go to Step 9, else go to Step

3.

Step 9. Report the position and fitness of global best particle.

Step 10. Stop.

Algorithm of weighted quantum particle swarm optimization

(WQPSO)

Step 1. Set all PSO parameters and bounds for the decision

variables.

Step 2. Set a population size of particles with random positions.

Step 3. Determine the fitness value of each particle.

Step 4. Update the mean best position using the following

equation:

mðkÞ ¼ mðkÞ1 ;m

ðkÞ2 ; . . .;mðkÞ

n

� �¼ 1

psize

Ppsizei¼1

ai1 ~pðkÞi1 ; 1

psize

Ppsizei¼1

ai2 ~pðkÞi2 ;

. . .; 1psize

Ppsizei¼1

ain ~pðkÞin Þ

where ai is the weighted coefficient and aid is the dimension

coefficient of every particle.

Step 5. Compare each particle’s fitness with the particle’s pbest.

Save better one as pbest.

Step 6. Compare current gbest position with earlier gbest position.

Step 7. Update the position of each particle using the following

equation:

xðkÞij ¼ ~p

ðkÞij b0 mðkÞ

j � xðkÞij

�� log 1uj

� �

Step 8. If the stop criterion is met, go to Step 9, else go to Step 3.

Step 9. Report the position and fitness of global best particle.

Step 10. Stop.


123

References

1. Ghare PM, Schrader GF (1963) A model for exponentially

decaying inventory. J Ind Eng 14(5):238–243

2. Covert RP, Philip GC (1973) An EOQ model for items with

Weibull distribution deterioration. AIIE Trans 5(4):323–326

3. Nahmias S (1982) Perishable inventory theory: a review. Oper

Res 30(4):680–708

4. Raafat F (1991) Survey of literature on continuously deteriorating

inventory models. J Oper Res Soc 42(1):27–37

5. Goyal SK, Giri BC (2001) Recent trends in modeling of deteri-

orating inventory. Eur J Oper Res 134(1):1–16

6. Bakker M, Riezebos J, Teunter RH (2012) Review of inventory

systems with deterioration since 2001. Eur J Oper Res

221(2):275–284

7. Janssen L, Claus T, Sauer J (2016) Literature review of deterio-

rating inventory models by key topics from 2012 to 2015. Int J

Prod Econ 182:86–112

8. Wu KS, Ouyang LY, Yang CT (2006) An optimal replenishment

policy for non-instantaneous deteriorating items with stock-de-

pendent demand and partial backlogging. Int J Prod Econ

101(2):369–384

9. Ouyang LY, Wu KS, Yang CT (2006) A study on an inventory

model for non-instantaneous deteriorating items with permissible

delay in payments. Comput Ind Eng 51(4):637–651

10. Ouyang LY, Wu KS, Yang CT (2008) Retailer’s ordering policy

for non-instantaneous deteriorating items with quantity discount,

stock dependent demand and stochastic backorder rate. J Chin

Inst Ind Eng 25(1):62–72

11. Wu KS, Ouyang LY, Yang CT (2009) Coordinating replenish-

ment and pricing policies for non-instantaneous deteriorating

items with price-sensitive demand. Int J Syst Sci

40(12):1273–1281

12. Jaggi CK, Verma P (2010) An optimal replenishment policy for

non-instantaneous deteriorating items with two storage facilities.

Int J Serv Oper Inf 5(3):209–230

13. Chang CT, Teng JT, Goyal SK (2010) Optimal replenishment

policies for non-instantaneous deteriorating items with stock-

dependent demand. Int J Prod Econ 123(1):62–68

14. Geetha KV, Uthayakumar R (2010) Economic design of an

inventory policy for non-instantaneous deteriorating items under

permissible delay in payments. J Comput Appl Math

233(10):2492–2505

15. Soni HN, Patel KA (2012) Optimal pricing and inventory policies

for non-instantaneous deteriorating items with permissible delay

in payment: fuzzy expected value model. Int J Ind Eng Comput

3(3):281–300

16. Maihami R, Kamalabadi IN (2012) Joint pricing and inventory

control for non-instantaneous deteriorating items with partial

backlogging and time and price dependent demands. Int J Prod

Econ 136(1):116–122

17. Maihami R, Kamalabadi IN (2012) Joint control of inventory and

its pricing for non-instantaneously deteriorating items under

permissible delay in payments and partial backlogging. Math

Comput Model 55(5–6):1722–1733

18. Shah NH, Soni HN, Patel KA (2013) Optimizing inventory and

marketing policy for non-instantaneous deteriorating items with

generalized type deterioration and holding cost rates. Omega

41(2):421–430

19. Dye CY (2013) The effect of preservation technology investment

on a non-instantaneous deteriorating inventory model. Omega

41(5):872–880

20. Jaggi CK, Tiwari S (2014) Two warehouse inventory model for

non-instantaneous deteriorating items with price dependent

demand and time varying holding cost. In: Parakash O (ed)

Mathematical modelling and applications. LAMBERT Academic

Publisher, Saarbrucken, pp 225–238

21. Jaggi CK, Sharma A, Tiwari S (2015) Credit financing in eco-

nomic ordering policies for non-instantaneous deteriorating items

with price dependent demand under permissible delay in pay-

ments: a new approach. Int J Ind Eng Comput 6(4):481–502

22. Jaggi CK, Cardenas-Barron LE, Tiwari S, Shafi A (2017) Two-

warehouse inventory model with imperfect quality items under

deteriorating conditions and permissible delay in payments. Sci

Iran Trans E Ind Eng 24(1):390–412

23. Jaggi CK, Tiwari S, Goel SK (2017) Credit financing in economic

ordering policies for non-instantaneous deteriorating items with

price dependent demand and two storage facilities. Ann Oper Res

248(1–2):253–280

24. Hartley VR (1976) Operations research—a managerial emphasis.

Good Year, Santa Monica, pp 315–317

25. Sarma KVS (1987) A deterministic order level inventory model

for deteriorating items with two storage facilities. Eur J Oper Res

29(1):70–73

26. Yang HL (2004) Two-warehouse inventory models for deterio-

rating items with shortages under inflation. Eur J Oper Res

157(2):344–356

27. Yang HL (2006) Two-warehouse partial backlogging inventory

models for deteriorating items under inflation. Int J Prod Econ

103(1):362–370

28. Wee HM, Yu JC, Law ST (2005) Two-warehouse inventory

model with partial backordering and Weibull distribution dete-

rioration under inflation. J Chin Inst Ind Eng 22(6):451–462

29. Pal P, Das CB, Panda A, Bhunia AK (2005) An application of

real-coded genetic algorithm (for mixed integer non-linear pro-

gramming in an optimal two-warehouse inventory policy for

deteriorating items with a linear trend in demand and a fixed

planning horizon). Int J Comput Math 82(2):163–175

30. Lee CC (2006) Two-warehouse inventory model with deteriora-

tion under FIFO dispatching policy. Eur J Oper Res

174(2):861–873

31. Chung KJ, Huang TS (2007) The optimal retailer’s ordering

policies for deteriorating items with limited storage capacityunder trade credit financing. Int J Prod Econ 106(1):127–145

32. Hsieh TP, Dye CY, Ouyang LY (2008) Determining optimal lot

size for a two-warehouse system with deterioration and shortages

using net present value. Eur J Oper Res 191(1):182–192

33. Liao JJ, Huang KN (2010) Deterministic inventory model for

deteriorating items with trade credit financing and capacity con-

straints. Comput Ind Eng 59(4):611–618

34. Das B, Maity K, Maiti M (2007) A two warehouse supply-chain

model under possibility/necessity/credibility measures. Math

Comput Model 46(3–4):398–409

35. Niu B, Xie J (2008) A note on ‘‘Two-warehouse inventory model

with deterioration under FIFO dispatch policy’’. Eur J Oper Res

190(2):571–577

36. Rong M, Mahapatra NK, Maiti M (2008) A two warehouse

inventory model for a deteriorating item with partially/fully

backlogged shortage and fuzzy lead time. Eur J Oper Res

189(1):59–75

37. Dey JK, Mondal SK, Maiti M (2008) Two storage inventory

problem with dynamic demand and interval valued lead-time

over finite time horizon under inflation and time-value of money.

Eur J Oper Res 185(1):170–194

38. Maiti MK (2008) Fuzzy inventory model with two warehouses

under possibility measure on fuzzy goal. Eur J Oper Res

188(3):746–774

39. Lee CC, Hsu SL (2009) A two-warehouse production model for

deteriorating inventory items with time-dependent demands. Eur

J Oper Res 194(3):700–710


123

40. Bhunia AK, Shaikh AA (2011) A two warehouse inventory

model for deteriorating items with time dependent partial back-

logging and variable demand dependent on marketing strategy

and time. Int J Inventory Control Manag 1(2):95–110

41. Liang Y, Zhou F (2011) A two-warehouse inventory model for

deteriorating items under conditionally permissible delay in

payment. Appl Math Model 35(5):2221–2231

42. Bhunia AK, Shaikh AA, Maiti AK, Maiti M (2013) A two

warehouse deterministic inventory model for deteriorating items

with a linear trend in time dependent demand over finite time

horizon by elitist real-coded genetic algorithm. Int J Ind Eng

Comput 4(2):241–258

43. Yang HL, Chang CT (2013) A two-warehouse partial backlog-

ging inventory model for deteriorating items with permissible

delay in payment under inflation. Appl Math Model

37(5):2717–2726

44. Jaggi CK, Pareek S, Khanna A, Sharma R (2014) Credit financing

in a two-warehouse environment for deteriorating items with

price-sensitive demand and fully backlogged shortages. Appl

Math Model 38(21–22):5315–5333

45. Bhunia AK, Shaikh AA, Sharma G, Pareek S (2015) A two

storage inventory model for deteriorating items with variable

demand and partial backlogging. J Ind Prod Eng 32(4):263–272

46. Bhunia AK, Shaikh AA, Sahoo S (2016) A two-warehouse

inventory model for deteriorating item under permissible delay in

payment via particle swarm optimization. Int J Logist Syst Manag

24(1):45–69

47. Jaggi CK, Tiwari S, Shafi A (2015) Effect of deterioration on

two-warehouse inventory model with imperfect quality. Comput

Ind Eng 88:378–385

48. Tiwari S, Cardenas-Barron LE, Khanna A, Jaggi CK (2016)

Impact of trade credit and inflation on retailer’s ordering policies

for non-instantaneous deteriorating items in a two-warehouse

environment. Int J Prod Econ 176:154–169

49. Taleizadeh AA, Niaki STA, Aryanezhad MB (2009) A hybrid

method of Pareto, TOPSIS and genetic algorithm to optimize

multi-product multi-constraint inventory control systems with

random fuzzy replenishments. Math Comput Model

49(5–6):1044–1057

50. Taleizadeh AA, Niaki STA, Meibodi RG (2013) Replenish-up-to

multi-chance-constraint inventory control system under fuzzy

random lost-sale and backordered quantities. Knowl Based Syst

53:147–156

51. Taleizadeh AA, Barzinpour F, Wee HM (2011) Meta-heuristic

algorithms for solving a fuzzy single-period problem. Math

Comput Model 54(5–6):1273–1285

52. Tat R, Taleizadeh AA, Esmaeili M (2015) Developing economic

order quantity model for non-instantaneous deteriorating items in

vendor-managed inventory (VMI) system. Int J Syst Sci

46(7):1257–1268

53. Hansen E, Walster GW (2003) Global optimization using interval

analysis. Marcel Dekker, New York

54. Sahoo L, Bhunia AK, Kapur PK (2012) Genetic algorithm based

multi-objective reliability optimization in interval environment.

Comput Ind Eng 62(1):152–160

55. Eberhart R, Kennedy J (1995) A new optimizer using particle

swarm theory. In: Proceedings of the sixth international sympo-

sium on micro machine and human science, pp 39–43

56. Kennedy J, Eberhart R (1995) Particle swarm optimization. In:

Proceeding of the IEEE international conference on neural net-

works, vol IV, pp 1942–1948

57. Clerc M (1999) The swarm and queen: towards a deterministic

and adaptive particle swarm optimization. In: Proceedings of

IEEE congress on evolutionary computation, pp 1951–1957

58. Clerc M, Kennedy J (2002) The particle swarm: explosion, sta-

bility, and convergence in a multi-dimensional complex space.

IEEE Trans Evol Comput 6(1):58–73

59. Xi M, Sun J, Xu W (2008) An improved quantum-behaved par-

ticle swarm optimization algorithm with weighted mean best

position. Appl Math Comput 205(2):751–759


123

a two-warehouse inventory model for non-instantaneous ...the optimization prob-lems of the inventory...

Documents