Generalization of cost optimization in (S-1, S) lost sales inventory model

Generalization of cost optimization in (S-1, S) lost sales inventory model

Vinod Kumar Mishra1, Lal Sahab Singh2

1, 2

Department of Mathematics & Statistics,

Dr.Ram Manohar Lohia Avadh University, Faizabad-224001, (Uttar Pradesh) INDIA 1

Department of Computer Science & Engineering,

Kumaon Engineering College, Dwarahat, Almora, – 263653 (Uttarakhand), INDIA [email protected], [email protected]

Abstract

This paper consider the problem of finding optimal critical levels for an inventory system for multiitem with multiple demand classes, Poisson demand process, lost sales and ample supply. The ordering policy is assumed to be the

Ý

S

?

1,

S

Þ

type i.e. replacement item is ordered as soon as a unit of stock is used and represent three accurate and efficient heuristic algorithms at a given base stock level.

Keywords: Inventory System, Multi-item, multiple demand, and optimal critical level. 1. Introduction

In this paper, we study the

Ý

S

?

1,

S

Þ

lost sales inventory model for multiitem that is demanded by different demand classes that have different penalty cost values. A penalty is incurred if a demand is not fulfilled from stock. Within this setting, we aim to minimize the total inventory holding and penalty cost and we distinguish between multiitem and different classes by introducing critical levels. A demand for any value for a certain class is only fulfilled if the physical stock is above the base stock level for that class.

The problem of multiple demand classes has been introduced by Veinott in1965 [1]. He also introduced the concept of critical level policies. After that, the problem has been studied in a number of mathematician and researcher and these papers can be distinguished in two different stream of research.

The first stream studies the structure of the optimal policy, within this stream there are interesting studies that derives the optimality of critical level policy of single item with multiple customer classes. Topkis in 1968 [2] consider a periodic review model with generally distributed demand and zero lead time. In that situation the optimal critical level are dependent on the remaining time period.

In 1997 A.Y. Ha [4] has studied a continuous review model with poisson demand process, a single exponential server for replenishment and lost sales. He derives the optimality of critical level policy and this situation both the base stock level and critical levels are time independent. In 2002 Devericourt et.al. [9] studied the same model as Ha but with back ordering of unsatisfied demand.

The second stream consists of studies that consider evaluation and optimization within a given class of policy within this stream there are interesting contribution in 2000 given by Melchiors et.al. [8], In 2002 by Dekker et.al. [5], in 2003 Deshpande et.al. [6] and Dekker et. al. [5] derived exact and heuristics procedures for the generation of an optimal critical level policy for a continuous review model with multiple customer classes, poisson demands, ample supply and lost sales. For the case with two customer classes, Melchiors et.al. [8] extend this work for fixed quantity ordering. In this model fixed ordering cost, the base stock level, and single critical level are optimized in order to minimize the sum of fixed ordering , inventory holding and lost sales cost. Deshpande et. al. [6] consider the similar model but with back ordering of unsatisfied demand. For detail see the papers given in the reference.

In this paper we study a multi item, continuous review model with multiple demand classes, poisson demand process, lost sales and ample supply. The ample supply represents that the supplier can deliver as much as desired within a given replenishment lead time. We limit ourselves to class of critical level policies with time independent critical and base stock levels. Critical levels are easy to explain in practice and the results on optimal policy in the first stream of research suggest that this is at least close to optimal. Under the given values and class of critical level policies, our problem is to optimize

Ý

P

I

P

,

P

J

P

Þ

critical levels and one base stock level simultaneously, where

I

is the set of items and

J

is the set of demand classes. By a projection of

Ý

P

I

P

+1

,

P

J

P

+1

Þ

dimensional total cost function on the dimension of base stock level and the definition of appropriate convex lower and upper bound function: an exact solution method is obtained for the full problem. Enumeration, however, is expensive from computational points of view, especially when number of items and demand classes greater than two.

The main contribution of this paper is generalization of cost optimization in

Ý

S

?

1,

S

Þ

lost sale inventory model and formulation of three efficient heuristics algorithms. The computational times that take the heuristics algorithms are small and far less than of explicit enumeration. So these heuristics are accurate and efficient.

2. Mathematical Model

.Consider multi item

á

i

: 1, 2, . . . .

m

â

that is demanded by a number of demand classes

Ý

j

: 1, 2, . . .

n

Þ

or customer groups. For each class

j

5

J

demands are assumed to occur according to a poission process with constant rate

m

ij

Ý

>

0Þ.

If any item is not delivered to class

J

on request, the demand is lost and a penalty cost

pij

Ý

>

0Þ

is to be paid in such a way that

pi

,1

/

pi

,2. . . .

/

pi

,PJP

Ý

i

: 1, 2, . . .

m

Þ.

The items is controlled by a continuous -review critical level policy. this means that total stock is controlled by a base stock policy with base stock level.

S

i

5

ß

N

0

=

N

W

á0âà

, and that there is a critical level

cij

Ý

5

N

0Þ _{per class}

j

5

J

,

_with

ci

1

.

ci

2. . . .

M

ci

PJP

M

Si

.

The ordering for the critical levels is assumed because of the opposite ordering in the penalty cost parameters, we call this ordering of the critical level the monotonicity constraint. A critical level policy is denoted by vector

Ý

c

i

,

S

i

Þ

with

c

:

=

Ý

ci

1,

ci

2

. . . ,

c

PiPPjP

Þ

If a class

j

demand arrive at a moment that the physical stock is larger than

c

ij

,

then this demand is satisfied otherwise the demand is lost. At and below level than

c

ij

,

physical stock

can be seen as stock that is reserved for more important class. For easy of notation we define

ci

PjP+1

=

S

i

,

replenishment lead time are iid with mean time

t

i and holding cast per unit time is

h

i .

Let

Ki

,j

Ý

ci

,

Si

Þ

_{denotes the fill rate for class}

j

5

J

_and

i

5

I

_{under the critical level policy}

Ý

ci

,

Si

Þ

_an

expression for

Ki

,j

Ý

c

i

,

S

i

Þ

can be derived as follows:

If the number of parts of sku

i

in the pipe line is

K

5

Ý0, 1, 2. . . .

Si

Þ

and thus the number of parts in

the physical stock is

Si

?

K

then the demand rate

Wi

,k

=

>

_j/k<Ýs

i?cijÞ

mi

,j

,

K

5

Ý0, 1, . . . . .

Si

?

1Þ

Our inventory model can be described by a closed queuing network with

Si

customers and two stations: (i) An ample server with mean service time

t

i

,

which represents the pipe line stocks.

(ii) A load dependent, exponential, single server with first come first serve service discipline, which represent the physical stock.

The service rates of the load dependent server are given by the

Wi

,k

.

by applying the theory of (Baskett et.al, 1975[12]). We find that the steady state probabilities

q

i,k

Ý

K

5

0, 1. . .

S

i

Þ

for having

K

items in the pipe lines is given by

q

i,k

=

K?1

h=0

E

Wi

,h tik

K!

qi

,0

,

K

5

Ý0, 1, . . . . .

S

i

Þ

q

i,0

=

K=0

S

>

á

K?1 h=0

E

W

i,h

â

tik

K! ?1

With the convention that

K?1

h=0

E

W

=

1

for

k

=

0

(This result also follows from Gendenko and Kovalenko [13] p.p. 252,252). By this steady state probability we obtain the fill rate

K

i,j

Ý

c

i

,

S

i

Þ

=

Si?Ci,j?1

K=0

>

q

i,k

,

i

5P

I

P

and j

5P

J

P

with the convention that this sum is empty if

Si

?

ci

,j

?

1

<

0Ý

i

.

e

.

Ki

,j

Ý

ci

,

Si

Þ

=

0

if ci

,j

=

S

i

Þ

Our objective is to minimize the average inventory holding and penalty cost per unit time. The average cost of a policy

Ý

c

i

,

Si

Þ

is

C

Ý

c

i

,

S

i

Þ

=

h

i

S

i

+

j5J

>

p

i.j

m

i,j

á

1

?

K

i,j

Ý

c

i

,

S

i

Þâ

,

i

5P

I

P

and

j

5P

J

P

Our optimization problem is a non linear integer programming problem and is stated as follows:

(

P

)

Min

Ý

Ci

,

Si

Þ

i

.

e

Min

i5I

>

h

i

S

i

+

i5I

>

j5J

>

p

i,.j

m

i,j

á

1

?

K

i,j

Ý

c

i

,

S

i

Þâ

subject

to

ci

,1

²

ci

,2. . . .

²

ci

,PJP

²

Si

ci

,j

5

N

0

and Si

5

N

0

Ý

i

5

I

and j

5

J

Þ

An optimal policy for problem

P

is denoted by

Ý

c

iD

,

S

iD

Þ

and the corresponding cost

C

i

Ý

c

iD

,

S

iD

Þ

For the situation with a fixed base stock level

S

i

5

N

0

,

let the problem

Ý

P

Ý

S

ÞÞ

denote the problem of finding the critical levels such that average cost

Ý

C

i

Ý

ci

,

Si

ÞÞ

is minimized. Problem

Ý

P

Ý

S

ÞÞ

is stated as

Ý

P

Ý

S

ÞÞ

Min

Ý

c

i

,

Si

Þ

i

.

e

Min

i5I

>

hiSi

+

i5I

>

j5J

>

pi

,.jmi,j

á1

?

Ki

,j

Ý

ci

,

Si

Þâ

subject

to

ci

,1

²

ci

,2. . . .

²

ci

,PJP

²

Si

ci

,j

5

N

0

and

Ý

i

5

I

and j

5

J

Þ

An optimal policy for problem

Ý

P

Ý

S

ÞÞ

is denoted by

á

c

iD

Ý

S

i

Þ,

S

i

â

and the corresponding optimal cost is

C

i

á

c

iD

Ý

S

i

Þ,

S

i

â.

Obviously

á

c

iD

Ý

S

iD

Þ,

S

iD

â

=

Ý

c

iD

,

S

iD

Þ.

Note that in problem

Ý

P

Ý

S

ÞÞ

the holding cost

term

>

hiS

i constitute a constant factor. It is included in the formulation of problem

Ý

P

Ý

S

ÞÞ

. 4. Analytical Solution

4.1. Exact Method for Problem

Ý

P

Þ

A method to solve problem

Ý

P

Þ

exactly has been described by Dekker et al [5]. This method exploits convex lower bound and upper bound functions for the function

C

i

á

c

iD

Ý

Si

Þ,

Si

â.

First, a lower bound

Si

,j

is determined for the optimal base stock level. Next, starting at this lower bound, Problem

Ý

P

Ý

S

ÞÞ

is solved by enumeration for increasing values of

Si

until a stopping criterion is met that implies that an optimal policy has been found.

An upper bound for the function

Ci

á

c

iD

Ý

Si

Þ,

Si

â

is obtained by taking all critical levels equal to

0

for each

Si

. This gives the upper bound function

ci

,u

Ý

Si

Þ

=

C

i

Ý0,

Si

Þ.

This function is convex which follows from the convex behavior of the Erlang loss probability for all

S

i

5

N

0 { for proof see smith [3]}

A lower bound function for

Ci

á

c

iD

Ý

Si

Þ,

Si

â

is obtained by replacing all penalty cost parameter

p

i,j _{by the lowest penalty cost parameter}

p

PiP,PjP in problem

Ý

P

Ý

S

ÞÞ.

Under an optimal policy for this

modified problem all critical levels are zero. the resulting cost are denoted by

C

i,l

Ý

S

i

Þ,

and also for this function the convexity follows from the convexity of the Erlang loss probability.

The exact algorithm for problem

P

is as follows: First define

Si

as a minimizing point for the upper bound function

C

i,u

Ý

S

i

Þ.

Next

C

i,u

Ý

S

i

Þ.

Next

S

i,l is defined as the lowest

S

5

N

0 for which

C

i,l

Ý

Si

Þ

²

C

i,u

Ý

Si

,l

Þ.

This

Si

,l is a lower bound for the optimal base stock level

S

iD

.

Then, for

Si

=

Si

,l . Problem

Ý

P

Ý

S

ÞÞ

is solved using explicit enumeration and the current best solution is set equal to

á

c

_iD

Ý

S

i

Þ,

S

i

â.

Here after

S

i is increased by one,

Ý

P

Ý

S

ÞÞ

is solved using explicit enumeration, and the

current best solution is adapted if

á

c

iD

Ý

Si

Þ,

Si

â

provide a better one. This is done repeatedly unit

condition imply that for any base stock level greater than the current base stock level no better solution can be found). At this point, the current best solution constitutes an optimal solution for problem

P

.

4.2. Algorithm for problem

Ý

P

Ý

S

ÞÞ

The exact method uses enumeration to solve multiple values of

Si

and thus is time consuming in particular for problem with two or more demand classes. Therefore, in this section, we described and fast heuristics for problem

Ý

P

Ý

S

ÞÞ.

The heuristics that we consider are local search algorithms. We test the accuracy in an extensive computational experiment and we find that the heuristics produce an optimal solution in all instances.

Before we formulate the heuristics, we show the typical behavior of the function

C

i

Ý

c

i

,

S

i

Þ

for a fixed

S

i

.

The behavior of cost function are shown in the table for the example with

P

I

P

=

2

items and

P

J

P

=

3

demand classes. In this example the critical level for class

1

is fixed at

0,

that is known to be optimal

We see in the following table that

Ci

Ý

ci

,

Si

Þ

is unimodel in

c

13 and in

c

23. for any fixed

c

12 and

c

22 vice versa.This means that the sign of the first order of difference of the cost term changes at most once. If it changes from minus into plus intuitvely, the observed unimodality increases the chance for local search type algorithm to find an optimal solution, but a guarantee can not be given. the property that a local minimum is a global minimum is not obtained for direct extensions of the concept of convexity to discrete spaces, but it is obtained for multimodular functions as introduced see by Altman et. al [11] for a whole theory used on multimodularity.

C

11

/

C

12 ₀

C

22

/

C

23 0 1 2 3 4 5

0 34.4320 30.8605 28.8197 27.9954 27.3506 27.2526

1 28.7878 27.2605 26.5242 26.2552 26.2066

0 2 28.2682 27.2482 26.8618 26.7803

3 29.6231 28.8786 28.6705

4 31.8338 31.3362

5 33.5227

C

11

/

C

12 1

C

22

/

C

23 0 1 2 3 4 5

0 30.8605 27.2890 25.2482 24.4239 23.7791 23.6811

1 25.2163 23.6890 22.9527 22.6837 22.6351

0 2 24.6967 23.6767 23.2903 23.2088

3 26.0516 25.3071 25.0990

4 28.2623 27.7647

5 29.9512

C

11/

C

12 2

0 28.8197 25.2482 23.2074 22.3831 21.7383 21.6403

1 23.1755 21.6482 20.9119 20.6429 20.5943

0 2 22.6559 21.6359 21.2495 21.1680

3 24.0108 23.2663 23.0582

4 26.2215 25.7293

5 27.9104

C

11

/

C

12 3

C

22

/

C

23 0 1 2 3 4 5

0 27.9954 24.4239 22.3831 21.5588 20.9140 20.8160

1 22.3512 20.8239 20.0876 19.8186 19.7700

0 2 21.8316 20.8116 20.4352 20.3437

3 23.1865 22.4420 22.2339

4 25.3972 24.8996

5 27.0861

C

11/

C

12 4

C

22

/

C

23 0 1 2 3 4 5

0 27.3506 23.7791 21.7383 20.9140 20.2692 20.1712

1 21.7064 20.1791 19.4428 19.1738 19.1252

0 2 21.1868 20.1668 19.7804 19.6989

3 22.5417 21.7972 21.5891

4 24.7524 24.2548

5 26.4413

C

11

/

C

12 5

C

22

/

C

23 0 1 2 3 4 5

0 27.2526 23.6811 21.6403 20.8160 20.1712 20.0732

1 21.6084 20.0811 19.3448 19.0758 19.0272

0 2 21.0888 20.0688 19.6824 19.6009

3 22.4437 21.6992 21.4911

4 24.6544 24.1568

5 26.3433

Similarly we can prepare for all possible values of (c 11

|

c12 ) & (c 22

|

_c 23 )

Parameter setting for cost function

C

Ý

C

i

,

S

i

Þ

as function of

C

12

P

C

13 and

C

22

P

C

23

M

11

=

M

12

=

M

13

=

M

21

=

M

22

=

M

23

=

1

P

J

P

=

Ý1, 2, 3Þ

,P

I

P

=

Ý1, 2Þ

_,

t

1

=

t

2

=

1

,

C

11

=

C

21

=

0

,

h

1

=

h

2

=

1

,

p

11

=

100,

p

12

=

10,

p

13

=

1

4.3. Description of Heuristics

In this section, we formulate the heuristics algorithms for problem

Ý

P

Ý

S

ÞÞ,

Which is first introduced by Dekker et. al [5] for problem

P

after that A.A.Kraneburg, G.J. Van Houtum [14], for single item. The difference in their heuristics is that the Dekker incorporate optimization of base stock level while Kranenburg assume constant base stock level. Both are them explain the heuristic for single item with multiple demand classes and here we explain the heuristic algorithm for multi item with multiple demand classes at constant base stock level.

Algorithm 1.Start with an arbitrary choice

c

i,j _,

i

5

I

_,

j

5

J

,

J

>

k

_,

where

k

=

max

á

J5J

i5I

P

pi

,j

=

p

11

â

then define the neighborhood of this current policy

Ý

c

i

,

S

i

Þ

as all policies that still satisfy the monotonicity

constraint and that have critical levels that differ at most one from the corresponding critical level in the original policy. If the cost of the cheapest neighbour is smaller then the cost of current solution, then select this neighbour and set this policy as current solution and repeat the process of evaluating all neighbors for this new policy . Otherwise stop and take the current solution as the solution found by the algorithm.

Algorithm 2.Start with an arbitrary choice for

ci

,j _,

i

5

I

_,

j

5

J

_,

J

>

k

_{that satisfy the}

monotonicity constraint, for

M

=

P

J

P

,

find

ci

,M

5

Ý

ci

,M?1

. . . .

ci

,M+1Þ with the lowest cost, at

fixed values of the other critical levels and change

ci

.M , accordingly. Repeat this optimization for one critical

level at a time for

M

=

P

J

P ?

1

down to

K

+

1.

After that, optimize again for

M

=

P

J

P

.

Continue

this iterative process until for of the

M

values an improvement is found . This is the solution found by the

algorithm.

Algorithm3.Start with all critical levels equal to zero. We first consider increasing

ci

,j by one and accept

this increase if it has lower cost than current solution. then, the critical levels

c

PiP,PJP?1 down to

c

PiP,K+1

are increased with 1 (one) critical level at a time and each time an increase of a critical level is accepted if

lower cost are obtained if

ci

,j _{was increased with one in this first iteration, then we execute another iteration}

and so one. The process stops as soon as

c

PiP,PJP has not been increased in an iteration. The policy found at

the end of last iteration is the solution found by the algorithm.

4.4. Algorithm for problem

P

Problem

P

an exact method was presented in previous section. In the exact method, problem

Ý

P

Ý

S

ÞÞ

has to be solved multiple times and this is to done by explicit enumeration. In previous section we solve the problem

Ý

P

Ý

S

ÞÞ

_{efficiently by one of the proposed heuristics algorithms. In this section the algorithms (1), (2) and (3)}

for problem

P

Ý

S

Þ

can be plugged into the exact method for problem

P

, thus replacing the explicit enumeration. The resulting algorithms are called algorithm (4), (5) and (6). For algorithm (4), (5) and (6) the choice for the starting point is fixed at

c

=

Ý0, 0, . . . . 0Þ.

5. Conclusion

References

[1] A.A.Kranenburg,G.J.van Houtum,Cost Optimization in the

(

S

−

1, )

S

lost sales inventory model with multiple demand classes ,Technische Universiteit Eindhoven (2005).

[2] A.F.Veinott, Optimal policy in a dynamic, single product, non stationary inventory model with several demand classes, Operation research,13 (1965), pp. 761-778.

[3] A.Y.Ha, Inventory rationing in a make to stock production system with several demand classes and lost sales, Management Science, 43 (1997), pp. 1093-1103.

[4] B.V.Gnedenko,I.N.Kovalenko,Introduction of queueing theory,Israel Programe of scientific Translation,1968.

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[9] Hadley, G., Whitin, T., (1963). Analysis of Inventory Systems. Prentice Hall, Englewood Cliffs. [10] L.Dowdy, D.Eager, K.Gourdon, L.Saxton, Throughput concavity and response time convexity,

Information processing Letter, 19(1984), pp. 209-212.

[11] M.A.Cohen, P.R.Kleindorfera and H.L.Lee, Service constrained

Ý

s

,

S

Þ

inventory system with priority demand classes and lost sales,Management Science, 34 (4) (1988), pp. 482-499.

[12] Naddor, E., (1966), Inventory System, Willey New York.

[13] P.Melchiors, R.Dekker, M.J.Kleijn, Inventory rationing in an

Ý

S

,

Q

Þ

inventory model with lost sales and to demand classes, Journal of the operation research society, 51 (2000), pp.111-122.

[14] R.Dekker, R.M.Hill, M.J.Kleijn, R.H.Teunter, On the

Ý

S

?

1,

S

Þ

lost sales inventory model with priority demand classes, Naval Research Logistic, 49 (2002), pp. 593-610.

[15] S.A. Smith, Optimal inventory for an

Ý

S

?

1,

S

Þ

system with no back orders, Management Science, 23 (1977), pp. 522-528.