analysis of integrated inventory control system in …

ANALYSIS OF INTEGRATED INVENTORY CONTROL

SYSTEM IN SUPPLY CHAIN WITH PARTIAL

BACKLOGGING

C.KARTHEESWARI1 and R. BAKTHAVACHALAM

2

1PG Assistant in Mathematics

Government Hr. Sec. School, Kottairuppu, Thiruppathur, Sivagangai

2Department of Mathematics,

Alagappa University Model Constituent College of Arts and science, Paramakudi.

bakthaa@ yahoo.com

Abstract

In this paper we consider an integrated inventory control system consists

of a Warehouse, Two Distributions Centre’s (DC) each associated with n identical

retailers. A (s, Q) type inventory system with Poisson demand and exponential

distributed lead times for items are assumed at DC (middle echelon). And one-for-

one type inventory policy is assumed at retailer node (lower echelon). Demands

occurring during the stock out periods are partially backlogged at DC. The DC

replenishes their stocks with exponential distributed lead times from warehouse

(upper echelon) has abundant supply source. The measures of system performance

in the steady state are obtained. Numerical examples are provided to illustrate the

proposed model.

Keywords: Supply Chain, Markov process, Inventory control,

Optimization.

1. Introduction

Supply chain is a network of facilities and distribution options that

performs the functions of procurement of materials, transformation of these

materials into intermediate and finished products and the distribution of these

finished products to customers. Supply Chain exists in both service and

manufacturing organizations, but the complexity of the chain may vary greatly

from industry to industry.

Inventory decision is an important component of the supply chain

management, because Inventories exist at each and every stage of the supply chain

as raw material or semi-finished or finished goods. They can also be as Work-in-

process between the stages or stations. Since holding of inventories can cost

International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 1 (2017) © Research India Publications http://www.ripublication.com

618

anywhere between 20% to 40% of their value, their efficient management is

critical in Supply Chain operations

The usual objective for a multi-echelon inventory model is to coordinate

the inventories at the various echelons so as to minimize the total cost associated

with the entire multi-echelon inventory system. This is a natural objective for a

fully integrated corporation that operates the entire system. It might also be a

suitable objective when certain echelons are managed by either the suppliers or

the retailers of the company. The reason is that a key concept of supply chain

management is that a company should strive to develop an informal partnership

relation with its suppliers and retailers that enable them jointly to maximize their

total profit.

Information technology has a substantial impact on supply chains.

Scanners collect sales data at the point-of-sale, and Electronic Data Interchange

(EDI) allows these data to be shared immediately with all stages of the supply

chain.

Multi-echelon inventory system has been studied by many researchers and

its applications in supply chain management has proved worthy in recent

literature. As supply chains integrates many operators in the network and optimize

the total cost involved without compromising as customer service efficiency.

The first quantitative analysis in inventory studies started with the work of

Harris (1915) [9].Clark and Scarf (1960) [5] had put forward the multi-echelon

inventory first. They analyzed a N-echelon pipelining system without considering

a lot size, Recent developments in two-echelon models may be found in Q.M. He

and E.M. Jewkes (2000)[12].Sven Axsäter (1990)[1] proposed an approximate

model of inventory structure in SC. One of the oldest papers in the field of

continuous review multi-echelon inventory system is a basic and seminal paper

written by Sherbrooke [14] in 1968. He assumed (S-1, S) polices in the Deport-

Base systems for repairable items in the American Air Force and could

approximate the average inventory and stock out level in bases.


619

Continuous review models of multi-echelon inventory system in 1980’s

concentrated more on repairable items in a Depot-Base system than as consumable

items (see Graves [7,8], Moinzadeh and Lee [11]). All these papers deal with

repairable items with batch ordering. Jokar and Seifbarghy [13] analyzed a two

echelon inventory system with one warehouse and multiple retailers controlled by

continuous review (R, Q) policy. A Complete review was provided by Benita M.

Beamon (1998)[3]. the supply chain concept grow largely out of two-stage multi-

echelon inventory models, and it is important to note that considerable research in

this area is based on the classic work of Clark and Scarf (1960)[4]. A continuous

review perishable inventory system at Service Facilities was studied by Elango

(2001) [6]. A continuous review (s, S) policy with positive lead times in two-

echelon Supply Chain was considered by Krishnan. K and Elango. C. 2005 [10].

A Modified (Q*, r) Policy for Stochastic Inventory Control Systems in Supply

Chain with lost sale model was considered by Bakthavachalam. R,

Navaneethakrishnan. S, Elango.C,(2012)[2]

The rest of the paper is organized as follows. The model formulation is

described in section 2, along with some important notations used in the paper. In

section 3, both transient and steady state analysis are done. Section 4 deals with

the derivation of operating characteristics of the system. In section 5, the cost

analysis for the operation is derived. Numerical examples and sensitivity analysis

are provided in section 6 and the last section 7 concludes the paper.

2. The Model description

A supply chain system consisting of a manufacturer, warehousing facility,

Two Distributions Centre’s (DC) each associated with n identical retailers dealing

with a single finished product. These finished products moves from the

manufacturer through the network consist of WH, DC, Retailer then the final

customer.

A finished product is supplied from MF to WH which adopts (0, M)

replenishment policy then the product is supplied to DC’s who adopts (s, Q)

policy. The demand at retailer node follows a Poisson distribution with rate


620

1,2,...n) j ; 2,1i(ij . Scanners collect sales data at retailer nodes and Electronic

Data Interchange (EDI) allows these data to be shared to the corresponding DC.

With the strong communication network and transport facility a unit of item is

transferred from Dc to the related retailer with negligible lead time. That is all the

inventory transactions are managed by DC’s. Supply to the Manufacturer in

packets of Q items is administrated with exponential lead time having parameter

) 2,1i(i . The replenishment of items in terms of pockets is made from

Manufacturer to WH is instantaneous. Demands occurring during the stock-out

periods are backlogged up to a specified quantity ‘bi’ at DC, where b is the

backorder level such that Q > b + s, i.e. Q-s > b>0. The maximum inventory level

at DC node S is fixed and the reorder point is s and the ordering quantity is

Q(=S-s) items. The maximum inventory level at Manufacturer is M (M=nQ). The

optimization criterion is to minimize the total cost incurred at all the locations

subject to the performance level constraints.

According to the assumptions the on hand inventory levels at both nodes follows a

Markov process.

We fix the following notations for the forthcoming analysis part of our paper.

[R]ij : The element /sub matrix at (i,j)th position of R.

0 : Zero matrix.

I : Identity matrix.

e : A column vector of 1’s of appropriate dimension.

Ii(t) : On hand inventory level at time t at DC i(i =1,2) .

Iw(t) : On hand inventory level at time t at WH .

hiD : Holding cost per unit of item per unit time at integrated DC i(i =1,2).

hw : Holding cost per unit of item per unit time at WH.

kiD : Fixed ordering cost, regardless of order size at integrated DC i(i =1,2).

kw : Fixed ordering cost for WH.


621

giD : cost per unit shortage cost at DC i(i =1,2).

k

i 1 2 k

i 1

a a a ... a .

nQ

*

i 0

i 0 Q 2Q ... nQ.

3. Analysis

Let ID(t) denote the on hand inventory at warehouse and I1(t) , I2(t) denote

the on hand inventory at DCi (i=1,2 ) respectively at time t . From the assumptions

on the input and output processes, we define I(t) = {(I1(t), I2(t), ID(t)) : t 0} and

we get {I(t) : t 0} = {(I1(t), I2(t), ID(t)) : t 0} as a Markov process with state

space E = {(i, k, m) / i = S,S-1,…, s,s-1,…,2,1,0,-1,-2,…-b; k = S,S-1,… s,s-1, …

,2,1,0,-1,-2,…,-b; m = nQ, (n-1)Q, ... 2Q,Q}.

Since E is finite and all its states are aperiodic, recurrent non-null and also

irreducible. That is all the states are ergodic. Hence the limiting distribution exists

and is independent of the initial state.

The infinitesimal generator of this process A= (a(i,k,m : j,l,n))(i,k,m),(j,l,n)E

can be obtained from the following arguments.

The arrival of a demand for an item at DC1 makes a state transition in the

Markov process from (i, k, m) to (i-1,k,m) with intensity of transition

1j(j=1,2,...,n).

The arrival of a demand for an item at DC2 makes a state transition in the

Markov process from (i, k, m) to (i,k-1,m) with intensity of transition

2j(j=1,2,...,n).

Replenishment of inventory at DC1 makes a state transition from ((i,k,m)

to (i+Q, k, m-Q) with rate of transition µ1(> 0).

Replenishment of inventory at DC2 makes a state transition from (i, k, m)

to (i, k+Q, m-Q) with rate of transition µ2(> 0).

The infinitesimal generator R is given by


622

A000B

BA000

00A00

00BA0

000BA

R

Hence the entities of R are given by

otherwise 0

nQ p ; Q p B

Q ..., 1)Q,-(n , nQ p ; Qq p B

Q ..., 1)Q,-(n , nQ p ; qp A

R pq

The sub matrices of R are given by

otherwise 0

0p; q p D

1, ... ,1s,s p ; q p C

1 , ... 1,-SS, p ; 1q p B

1s , ... 1,-SS, p ; qp A

]A[

1

1

1

1

pq and

otherwise 0

1,0 , ... 1,-ss, p ; Qq p 2M

1,0 , ... 1,-SS, p ; qp M

]B[

1

pq

The sub matrix of A and B are

1 2

2

1 pq 1 2 2

1

( ) p q ; p S,S-1, ... , s 1

p q 1 ; p S,S-1, ... , 1,0,-1,...,-(b-1)

[A ] ( ) p q ; p s,s 1, ... ,1,0,-1,...,-(b-1)

(

2 ) p q ;p 0

0 otherwise

11 pq

p q ; p S,S-1, ... , 1,0,-1,...,-(b-1)[B ]

0 otherwise


623

1 2 2

2

1 pq 1 2 1 2

( ) p q ; p s,s-1, ... ,1,0,-1,...,-(b-1)

p q 1 ; p S,S-1, ... , 1,0,-1,...,-(b-1)

[C ] ( ) p q ; p

1 1 2

s,s 1, ... ,1,0,-1,...,-(b-1)

( ) p q ;p 0

0 otherwise

2 1

2

1 pq 2 1 2

1 2

( ) p q ; p 0

p q 1 ; p S,S-1, ... , 1,0,-1,...,-(b-1)

[D ] ( ) p q ; p s,s 1, ... ,1,0,-1,...,-(b-1)

( )

p q ;p 0

0 otherwise

otherwise 0

1 , ... 1,-SS, p ; qp ]M[

1pq1 and

2

pq

p q Q ; p s,s-1, ... , 1,0[M2]

0 otherwise

3.1. Transient analysis

Define the transient probability function pi,k,m(j, l, n : t) = pr {(I1(t), I2(t),

ID(t)) = (j, l, n) | (I1(0), I2(0), ID(0)) = (i,k,m)}.The transient matrix for t 0 is of

the form P(t) = (pi,k,m(j, l, n : t))(i,k,m)(j,l,n)E satisfies the Kolmogorov- forward

equation R)t(P)t(P , where R is the infinitesimal generator of the process

}0t),t(I{ . The above equation, together with initial condition I)0(P , the

solution can be expressed in the formRtRt ee)0(P)t(P , where the matrix

expansion in power series form is

1n

nnRt

n!

tR+ I = e .

3.2. Steady state analysis:

The structure of the infinitesimal matrix R, reveals that the state space E of the

Markov process {I(t); t 0} is finite and irreducible. Let the limiting probability

distribution of the inventory level process be


624

m,k,i)t(I),t(I),t(I Prlimv D21t

ki

m

where ki

m v is the steady state

probability that the system be in state (i,k,m), (Cinlar [4]).

Let ) .v,v,v,v,v,v(v 0QQ2Q)2n(Q)1n(nQ denote the steady state probability

distribution Where jQ k k k k k k k

S S 1 1 0 1 2 b v v , v , ... , v , v , v , v ..., v for j = 1, 2 ... n and

k = 1,2,…, S for the system under consideration. For each (i,k,m), ki

m v can be

obtained by solving the matrix equation vR = 0

together with normalizing condition m k

j

(i,k,m) E

1

.

Assuming Q = a, we obtain the steady state probabilities

kkiQ )BA( a (-1) , i = 1,2, … n ; k = n-i+1.where a = .)BA( (-1) e

11-n

0i

i1i1

4. Operating characteristics

In this section, we derive some important system performance measures.

4. 1. Mean Reorder Rates:

The event 1 , 2 and w are the Mean reorder rate at DC1, DC2 and WH

are given by

q k

1 1 s 1

q,k

v …4. 1

q s 1

2 2 k

q,k

v

…4.2

and 1

S s S snQ k nQ k

w i 2 i

i 0 k 0, k 0 i 0

v v

…4.3

4. 2. Mean Inventory Levels:

Let iI (i= w,1,2) denote the mean inventory level in the steady state at

WH, DC1 and DC2 i. Thus the mean inventory levels are given by


625

nQ

q k

w i

q Q i,k

I q v

…4.4

S

0i k,q

ki

q1 vi I

…4.5

and

S

0k i,q

ki

q2 vkI … 4.6

4. 3. Mean Shortage Rate:

The demand at DC’s upto ‘b’ items are satisfied by local purchase during

the stock out periods; hence the shortage rate at DC1 and DC2 are given by

q k

1 1 b

k,q

v …4.7

and q b

2 2 i

i,q

v

.

…4.8

5. Cost analysis

In this section we impose a cost structure for the proposed model and

analyze it by the criteria of minimization of long run total expected cost per unit

time.The long run expected cost rate C(s, Q) is given by

w w 1D 1 2D 2 w w 1 1 2 2 1 1D 2 2DC(s,Q) h I h I h I k k k g g …(5.1)

Although we have not proved analytically the convexity of the cost function

C(s,Q), our experience with considerable number of numerical examples indicates

that C(s,Q) for fixed Q appears to be convex in s. In some cases it turned out to be

an increasing function of s. For large number cases of C(s, Q) revealed a locally

convex structure. Hence we adopted the numerical search procedure to determine

the optimal values s*.

6. Numerical Example and Sensitivity Analysis.

In this section we discuss the problem of minimizing the steady state

expected cost rate under the following cost structure. The results we obtained in

the steady state case may be illustrated through the following numerical example.


626

For the following example, we assume that, S = 6, M = 12, 1 = 0.4, 2 = 0.6,

1 = 0.4, 2 = 0.5, hw = 1.25, h1D = 2.65, h2D = 0.195, kw = 0.65, k1D = 0.75,

k2D = 0.7, g1 = 1.25, and g2 = 2.35.

The cost for different reorder levels are given by

s Q C(s,Q)

0 6 74.9010

1 5 74.8781

2 4 74.7423

3* 3* 74.4806*

4 2 74.9460

5 1 74.9819

Table 1 Total expected cost rate as a function s and Q.

For the inventory capacity S, the optimal reorder level‘s*’ and optimal cost C(s,Q)

are indicated by the symbol ‘*’. The graphical representation of the long run

expected cost rate C(s*,Q*) is given below


627

Figure 1 Graphical representation of C(s,Q)

(i) Sensitivity Analysis

The table 2 presents a numerical study to exhibit the sensitivity of the

system on the effect of varying demand rates 1 and 2 with fixed reorder at s=3.

Table 2. The total expected cost vs. demand rates (1 and 2)

1

2 2 3 4 5 6

5 74.4876 77.0681 79.7106 82.3680 85.0314

6 87.9305 90.4541 93.0556 95.6770 98.3055

7 101.3564 103.8341 106.4066 109.0041 111.6096

8 114.7633 117.2020 119.7526 122.3321 124.9206

9 128.1543 130.5580 133.0906 135.6562 138.2317

It is observed that the total expected cost C(s,Q) is increasing with the different

demand rates. Hence the demand rate is a very important parameter of this system.

7. Concluding remarks

In this paper we analyzed a continuous review integrated inventory control

system in a supply chain for single product. We are also proceeding in this multi-

echelon stochastic inventory system with perishable products. This model deals

74.4000

74.5000

74.6000

74.7000

74.8000

74.9000

75.0000

75.1000

0 1 2 3 4 5 6 7 8

C(s,Q)


628

with only tandem network, this structure can be extended to tree structure and to be

more general.

References

[1] Axsäter, S. 1990. Simple solution procedures for a class of two-echelon inventory

problems. Operations research, 38(1), 64-69.

[2] Bakthavachalam.R, Navaneethakrishnan. S, Elango.C,(2012) A Modified (Q, r) Policy for Stochastic Inventory Control Systems in Supply Chain. Proceeding of

International Conference on Advances in Computing," Advances in Intelligent and

Soft Computing",Springer, 247-255.

[3] Benita M. Beamon. 1998. Supply Chain Design and Analysis: Models and

Methods. International Journal of Production Economics.Vol.55, No.3, pp.281

294.

[4] Cinlar, E. Introduction to Stochastic Processes. Prentice Hall, Englewood Cliffs,

NJ, 1975.

[5] Clark, A. J. and H. Scarf, 1960. Optimal Policies for a Multi-Echelon Inventory Problem. Management Science, 6(4): 475-490.

[6] Elango, C., 2001, A continuous review perishable inventory system at service

facilities, unpublished Ph. D., Thesis, Madurai Kamaraj University, Madurai

[7] Graves S. C. (1982), The application of queueing theory to continuous perishable

inventory systems, Management Science, 28,400-406.

[8] Graves, S. C. (1985). A multi-echelon inventory model for a repairable item with one-for-one replenishment. Management Science, 31(10), 1247-1256.

[9] Harris, F., 1915, Operations and costs, Factory management series, A.W. Shah Co., Chicago,48 - 52.

[10] Krishnan.K, 2007, Stochastic Modeling In Supply Chain Management System,

unpublished Ph. D., Thesis, Madurai Kamaraj University, Madurai [11] Moinzadeh, K., & Lee, H. L. (1986). Batch size and stocking levels in multi-

echelon repairable systems. Management Science, 32(12), 1567-1581.

[12] Qi. Ming He and E. M. Jewkes. 2000. Performance measures of a make-to- order inventory- production system. IIE Transactions, 32, 409-419.

[13] Seifbarghy. M, and Jokar. M.R.A, Cost evaluation of a two-echelon inventory

system with lost sales and approximately normal demand. International Journal of Production Economics, 102:244–254, 2006.

[14] Sherbrooke, C., 1968. METRIC: A multi-echelon technique for recoverable item

control. Operations Research. 16(1), 122 - 141.

[15] Svoronos, Antony and Paul Zipkin, 1991. Evaluation of One-for-One

Replenishment Policies for Multiechelon Inventory Systems, Management

Science, 37(1): 68-83.


629

analysis of integrated inventory control system in …

Documents