(S − 1,S) Perishable systems with stochastic leadtimes

Pergamon

Mathl. Comput. Modelling Vol. 21, No. 6, pp. 95-104, 1995 Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177195 $9.50 + 0.00

0895-7177(95)00026-7

(S - 1, S) Perishable Systems Stochastic Leadtimes

with

S. KALPAKAM AND K. P. SAPNA Department of Mathematics Indian Institute of Technology, Madras, India, 600 036

(Received July 1993; accepted October 1993)

Abstract-This article discusses a one-to-one ordering perishable system, in which reorders are processed in the order of their arrival and the processing times are arbitrarily distributed, and as such, the leadtimes are not independent. The Markov renewal techniques are employed to obtain the various operating characteristics for the case of Poisson demand and exponential lifetimes. The problem of minimizing the steady state expected cost rate is also discussed, and in the special case of exponential processing times, the optima1 stock level is derived explicitly.

1. INTRODUCTION One of the basic assumptions implicit in most of the inventory systems is the infinite lifetime of products while in storage. However, the effect of perishability cannot be discounted in many inventory systems. Food stuff and photographic film are common examples of products with finite lifetimes. In the medical sector as well, almost all drugs have expiry dates, and blood banks can store whole-blood units only for 21 days. In all these situations, the exclusion of deterioration from the model yields an inaccurate performance analysis of the inventory system. The analysis of perishable inventory systems has been receiving considerable attention from many researchers, and the review article by Nahmias [l] provides a comprehensive survey of such systems. Most of these models deal with periodic review systems with fixed lifetimes. An extensive literature on continuously deteriorating systems is cited in the recent survey article of Raafat [2]; much of the work is related to instantaneous supply of orders. The analysis of perishable systems becomes extremely complex once a positive order lead time is introduced. Schmidt and Nahmias [3] were the first researchers to deal with positive lead time for perishable systems. They analyzed a continuous review (S - 1, S) system with Poisson demand and fixed life and lead times. More recently, an (S - 1, S) perishable system with Poisson demand and exponential lifetime was studied by Pal [4]. Further, in the analysis of one-to-one systems not much work has been done in the case of arbitrary lead times, even when the items are nonperishable in nature. Higa et al. [5] and Kruse [6] discussed the waiting time distribution of an (S - 1, S) system with full backlogging, in which the lead times have general distribution. In this article, we present a detailed study of a continuous review (S - 1, S) system with Poisson demand and exponential lifetimes, in which the lead times are nonindependent and non-Markovian in nature. More specifically, orders are processed one after the other, and the processing times are independently and identically distributed random variables with arbitrary distribution. Further, the demands that occur during stock out periods are assumed to be lost. Although the above assumptions make the analysis complex, the same was achieved by identifying Typeset 95

by AM-TEX

36

P.SAPNA

S.KALFAKAM~NDK.

a suitable techniques

Markov renewal process embedded in the inventory level process and employing the of semi-regenerative processes. The problem of optimizing the long run expected

cost rate in the special case of exponential processing times is discussed expression for optimal S is derived. In the general case, the determination level is illustrated

in detail, and explicit of the optimal stock

numerically.

2. ASSUMPTIONS

AND NOTATION

Consider an inventory system with a maximum stock of S units. Unit demands occur according to a Poisson process with parameter X (> 0). The lifetime of each item is exponential with rate 7 (> 0). A one-to-one

ordering

policy is adopted.

According

to this policy, orders

are placed

for

one unit, as and when the inventory level drops either due to a demand or due to a failure. In this model, we deal with the case of correlated lead times, where orders are placed with a single supply

source and are processed

identically during

distributed

stock out periods Inventory

one after the other.

with probability

density

The processing

function

times are independently

g(.) and mean

are lost. We will use the following

m.

Demands

and

occuring

notation.

level at time t which is right continuous;

(0, I, 2,. . . , S); (0,1,2,. . . ) s - 1);

(0, 172,. ‘. h = Jtrn g(t) & Laplace transform purchase, inventory cost/unit cost/unit

of any arbitrary

price/unit; carrying cost/unit/unit shortage; and failure.

function

a(t);

time;

3. ANALYSIS Prom our assumptions, it is clear that the replenishment epochs constitute a sequence of regeneration points for the inventory level process. Let 0 = TO < T’ < Tz < . . be the successive Then, it can be shown that (I,T) = epochs at which the replenishments occur. Let I, = I(T;). (I,,T,;n E No> is a Markov renewal process (MRP) [with state space El x 10, oo)], which is completely determined by its semi-Markov kernel. Let S(i,j, t) denote the derivative of the semiMarkov 0(i,j,

kernel, which is defined as t) = r;‘mo i

To determine

the auxiliary

epochs or part thereof

t) = Pr[l(t

- T, < t f A ] 1, = i],

= j, t < T,+r

Q(i, j, t), we introduce

two replenishment x(i,j,

Pr[T,+i

functions

i,j

E El,

n f No.

x(i, j, t), which are defined

(3.1)

between

as follows

+ r) = j 1I(7) = i]>

o
ieIE;

T,
(3.2)

CASE 1. 0 5 i 5 S-2. In this case, after the replenishment I (T,f) = i + 1.From (3.1), we have B(i,j, t) = irne i

at T,, there will always be at least one order pending

and

Pr[t < T,+l - T, I: t + A 11, = i] Pr[JQ+i = j 1T,+l = t, I, = i]

= g(y) Pr[l,+i

= j 1Tn+l = t, I(T,i)

= i -k 11.

Hence, S(i,j, t) =

g(t)x(i+l,j,t), 0,

otherwise.

o
OIiIS-2,

(3.3)

(S - 1, S) Perishable Systems

97

CASE 2. i=S-1. In this case, after the replenishment at T,, pending at Tz.

= S, and hence, there will be no order

I(T,f)

As such, the inventory level should necessarily drop before i, so as to reach the

state j at t. Hence, we have e(S-l,j,t)

=

s0

‘(XfSy)

exp[-(X+Sy)u]x(S-1,

0 5 j < s. (3.4)

j, t-u)g(t-u)du,

In order to determine x(i, j, t), we note that between two replenishment epochs, {I(t);t > 0) is a pure death process with death rate (A + j y), when in state j(j E J&). Solving the Kolmogorov forward differential equations for x(i, j, t), WCobtain

x(i,j,t)

1, exp[-(A + iy) t], A (3 + 1, i) C&-1)“p9

=

exp [-(X + 127) t] ,

(:I:)

1 1 -A(l,i)yiC~=,(-l)n-l(~~:)~,

i = 0,

j = 0,

j =i, lIj
l

j = 0,

l
lli
where A(j + 1, i) = [l$+,,(~ 3.1. Inventory

k y)]/

+

[#--))

(i - j)!].

Level Process

Define P(i,j,

t) = Pr[l(t) = j 1I, = i],

iEEl>

jEE,

We note that once the inventory level at T, = sup,{Ti < t} is known, the history of I(t) prior to T, loses its predictive value. Hence, {T,; n f No} are stopping times and {I(t); t 2 0) is a semiregenerative process with embedded Markov renewal process (I,T) (see [7]). As such, the functions P(i, j, t) satisfy the following Markov renewal equations: P(i,j,t)

= K(i, j, t) + c /’ Q(i,e, u) P(l, j, t - u) du? eEEl O

i E El,

j E&

(3.6)

where K(i, j, t) = Pr[l(t) = j, T n+~ > t / I, = i]. Using arguments similar to those employed for determining 8(i, j, t), we obtain (A + ~7)

K(i, j, t) =

I

Ji exp [-(X

t Sy) ~1G(t - u> x(S - l,j,t- ~1 da, i-s-1,

o
exp[-(A+S7)4,

i=s-1,

j = S,

G(t) x(i + 1,j,t),

o
(3.7)

o
otherwise. 0, Taking Laplace transform on both sides of (3.6) yield

i

E

El, j E lE.

(3.8)

Tl 7e system of Equations (3.3) and (3.7), on taking the same transform yields _4(j + l,i+

1)Cflt_l(-l)n-3(“,~~‘)g*((I+X+7L~),

05i
l
g*(a) - A(1. i + 1) y(i + 1) C;:I(-l)“-‘(,:,) OlilS-2, 8*(i,j,a)

A(j;:fit(SSyj)Y

= I

I

0,

j=O,

c~=,‘(-l)“-j(S,“51)9*(”

i=s-1,

l
i=s-1,

j = 0,

otherwise;

“(i’;f+;$+ + x + nr),

(3.9)

S. KALPAKAM AND K. P. SAPNA

98

f A(j + l,i + 1) c~~(-l)“-j(in”_:‘)~*(cy o
+ x + nr),

1
G*(a) -A(1,i+1)7(i+1)C~~l,(-l)n-1(n41)~’(Ol+nXy+n7), O
K*(i,j,a)

j=o,

=

(3.10)

, 0,

otherwise.

Failures, Shortages, and Reorders

3.2.

Define the following el: occurence e2: occurence es: occurence Consider

events:

of a failure, of a shortage, of a reorder.

the counting

and

process

{A&(t);

t 2 0) associated

with the ek-events

(k = 1,2,3),

to

determine their mean rates of occurence. Each of the ek-events occur either at failure or demand epochs, and the sequence of such epochs do not form a renewal process. Let hk(i, t) denote the conditional first order product density of the point process corresponding to the ek-events, which is defined

as hk(i, t) = jyo

Then,

$Pr[Mk(t

+ A) - h&(t)

= 1 110 = i],

k = 1,2,3.

we have hl(i,t)

= tzo

A Pr[I(t)

> 0, a failure in (t,t + A)

1 10 =

i]

S =

C

= j, a failure in (t, t + A) 1IO = i]

izo

i

Pr[l(t)

trno

i

Pr[a failure in (t, t + A) ( I(t)

j=l

=

k

= j] Pr[l(t)

= j 110 = i]

j=l

(3.11)

h2(i, t) = trno $ Pr[I(t) --+

= 0, a demand

= jrno k Pr[a demand

in (t, t + A) 1IO = i]

in (t, t + A) ) I(t)

= 0] Pr[I(t)

= 0 110 = i] (3.12)

= XP(i,O,Q h3(i,

t) = jmo i

Pr[l(t) + trno i

= 2 j=l

izo

i

> 0, a demand Pr[I(t)

Pr[a demand

in (t, t + A) 1IO = i]

> 0, a failure in (t, t + A) 110 = i] in (t, t + A) 1I(t)

= j] Pr[I(t)

= j I 10 = i] + hl(C t)

99

(S - 1, S) Perishable Systems

(3.13)

= -&X+jy)P(i,j,t). j=l

4. STEADY In order

to obtain

the steady

{Inn; n E No} embedded bility

functions

state

operating

in the Markov

characteristics,

renewal

process

we consider

(1,T)

the Markov

whose one step transition

chain proba-

are given by

P(i,j) From Equation

STATE RESULTS

=

CO 8(i,j,

t) dt = 0*(&j, 0).

s0

(3.9), we have

p(i,j)=A(j+l,i+l)~(-l)nj(inl:l)g*(~+.y), n=j

o
p(S-l.j)=A(j+l.S-~)s~(-l)“j(Snill)~*(~+n7),

osj
llj
n=j

From the structure

of p(i,

j), it can be seen that {I,;

chain is also finite, the unique

stationary

distribution

n E No} is irreducible. n(j);

Since the Markov

j E E exists and is given by

S-l

n(j) =

c r(i) P(&.d,

and

(4.1)

FO S-l 7r(j) = 1.

c

(4.2)

j=O

The system

of Equations

(4.1) on simplification

reduces S-l

1

r(j + 1)

to

c n(i) P(i>j + I), 7r(j) = g* [A + (j + I) r] - g* [A + (j + 1) r] i=j+l

o
(4.3)

The system of Equations (4.3) can be solved recursively in terms of r(S - l), which in turn can be uniquely determined using (4.2). The Markov renewal process (I,T) is irreducible and positive recurrent, as the underlying It is also aperiodic, since the Markov chain {I,; n E No} is irreducible and positive recurrent. derivative

of the semi-Markov

kernel exists.

Hence, we have from [7]

(44 where m(e) is the mean sojourn

time in state C. Since

m(e) = K’(i,j,

osess-2,

my m + (A+ls+

0) = 0.

we have

P(j) =

ifj

e=s-1,

(4.5)

>i+l,

C&f_, 7(i) K*(i,j,0) m+

and

7r

(S

-

1)/(X + S y) ’

(4.6)

S. KALPAKAM AND K. P. SAPNA

100

As P(j) also represents the long run proportion of the time the system is in state j, the average inventory Ievel r in the steady state is given by

&P(j).

I=

(4.7)

j=o Further, the mean rates ,& of the ek-events (Ic = 1,2,3) t + oo, Hence, we have, from Equations (3.11)-(3.13),

are the limiting values of ~k(~,~) as

(4.10)

w41 f rC3W).

P3 = x [l -

j=l

5. OPTIMAL

COST ANALYSIS

In this section, we obtain the optimum stock level that minimizes the long run total expected cost rate. The total expected cost rate C(S) in the steady state is given by

C(S) = hlfkP1

+g,@z+cp3.

From Equations (4.7) to (4.10), we obtain

(5.1)

Table 1. X = 1, y = 0.01, c = 100, k = 50. hg 5

10

15

20

25

30

35

40

50

100

200

300

400

500

600

700

800

1

1

17

24

31

36

41

46

50

8.61

10.03

11.12

11.62

12.02

12.37

12.69

12.99

13.28

1

1

12

18

22

26

30

33

36

8.76

10.06

11.47

12.12

12.63

13.07

13.46

13.83

14.17

1

1

10

15

18

21

24

27

29

8.88

10.08

11.74

12.51

13.11

13.63

14.08

14.51

14.90

1

1

9

13

16

19

21

23

25

9.00

10.11

11.98

12.85

13.53

14.11

14.62

15.09

15.53

1

1

9.03

10.13

8

, 12.18

11

14

17

19

21

23

13.15

13.89

14.53

15.10

15.62

16.10

1

1

7

10

13

15

17

19

20

9.06

10.16

12.36

13.41

14.22

14.92

15.53

16.09

16.61

1

1

6

10

12

14

16

17

19

9.10

10.18

12.53

13.66

14.53

15.27

15.93

16.53

17.08

1

1

9.15

10.21

1

6

9

11

13

15

16

18

12.68

13.88

14.81

15.60

16.30

16.94

17.52

(S - 1, S) Perishable Systems Table 2. X = 10, h = 10, c = 100, k = 50

300

400

500

600

700

800

18

17

16

15

15

14

14

13

13

12.14

12.27

12.42

12.56

12.69

12.81

12.93

13.04

13.15

22

21

20

20

19

18

18

17

17

12.63

12.84

13.04

13.22

13.40

13.56

13.72

13.87

14.02

26

25

24

23

23

22

21

21

20

13.07

13.33

13.58

13.82

14.04

14.25

14.44

14.64

14.82

30

29

28

27

26

25

25

24

23

13.46

13.78

14.09

14.37

14.64

14.89

15.14

15.37

15.59

33

32

31

30

29

29

28

27

26

13.82

14.21

14.57

14.91

15.22

15.52

15.81

16.08

16.34

36

35

34

33

33

32

31

30

30

14.17

14.61

15.03

15.42

15.79

16.14

16.47

16.79

17.09

In general, when the processing times have arbitrary distribution, it is not possible to determine the optimal stock level (say S*) explicitly. However, one can resort to numerical evaluation of the same, for specific distributions. We have considered several examples, and in all the cases C(S) turned out to be a convex function, thus indicating the existence of a unique minimum S*. Further, we noted that when c 2 g, S’ = 1. Intuitively, we see that with an increase in S, the average inventory level increases and the shortage rate decreases, as a result of which the C(S) given in (5.1) is an increasing function when c > g, thus resulting in S* = 1. Tables 1 and 2 illustrate the effect of varying a given parameter on the optimal values. In each box, the upper entry denotes optimal S and the lower entry the optimal cost rate. Table 1 deals with the case in which g(t) is uniform over the interval (1,5), and Table 2 presents the results when g(t) is Erlangian of order 8 with mean 10, for specific values of the other parameters. We note that whenever c 1 g, S* = 1; and also in Table 1 (Table 2) as g increases beyond c, the increase in S* is larger for smaller values of h (or y).

6. SPECIAL CASE OF TIMES EXPONENTIAL PROCESSING Although,

in general,

due to the complex

form of p(i,j),

it is not possible

to obtain

P(j)

explicitly, the same can be achieved when g(t) is exponential. However, in this case, as the process {I(t); t > 0} is Markovian, it is easier to obtain the limiting probability distribution directly from the balance equations than as a special case of the earlier method. Let g(t) = p exp(-pt), then the steady state balance equations are given by

Pm) = (A+ 7) W), (A + j y + p) P(j) = ,uP(j - 1) + (X + Sy) P(S)

= pLp(S - 1).

[A+ (j + 1) r] w

+ l),

l
S. KALPAKAMAND K. P. SAPNA

102

Solving the above set of equations recursively and using ~~__e P(j)

= 1, we obtain

l
J’(j) = Aj P(O),

(6-l)

(6.2) where

4=2[(X+ny) p 1,

l
la=1

In this case, we can also derive the decision rules for S* explicitly. Substituting for P(j),

from

(6.1) and (6.2), Equation (5.1) reduces to where

C(S) = Cl(S) + CA, Cl(S) =

(h+k7+cy)CjS=ljAj+(g-c)X [I + C;=,

(6.3)

Aj]

’

Since C(S) and Cl(S) differ by an additive constant, the value S’ that minimizes Cr (S) also minimizes the function C(S). To obtain S*, we first prove the following lemmas. LEMMA PROOF.

1.

Cl(S)

is neither decreasing nor possessesan integer maximum.

Let A(S) = Cl(S) - Cr(S - I.). From (6.3), we have

Cl(S)

=(htky+cy)~jAj+(g-c)X,

l+eAj

[

3=1

(6.4)

j=l

1

S-l

=(h+ky+cy)

xjA&g-cc)X.

(6.5)

J=l

Subtracting (6.5) from (6.4) yields

(6.6) If Cr (S) is a decreasing function, A(S)

should be nonpositive for all integers S.

From (6.6),

A(S) < 0 implies Cl(S)

> (h + ky + cy) s.

Multiplying both sides of (6.7) by [l f C,“=, Aj]

(6.7)

and using (6.4), we have

(h+k~+c7)~jAj+(g-c)h,(h+ky+c7)S

(6.8)

j=l

Let v(s) = (h+ky+cy)

[

,I,

S+~(S-Mj

(6.9)

Then, from (6.8), we have (g - c) x 2 v(S),

s E M,

(6.10)

(S - 1, S) Perishable Systems

103

which leads to a contradiction as v(S) is a positive increasing function which is not bounded above. Hence, Cl(S) cannot be a decreasing function. Let there exist an integer maximum S of Ci (S) . Then S must necessarily satisfy A( S + 1) 6 0 and A(S)

2 0. From (6.6), A(S + 1) 5 0 and A(S) 2 0 imply Cl(S + 1) 1 (h + Icy + c-y) (S + 1) 2 (h + Icy + cy) S L Cl(S),

which contradicts the fact that S is the integer maxima of Cl(S). LEMMA 2. Cl(S)

is an increasing function iff ~(2) 2 (g - c) X.

PROOF. From Lemma 1, Cl(S)

iff Ci(2) - Cl(l)

Hence, the lemma.

has no integer maximum. Hence, Cl(S) is an increasing function

> 0. On simplification and appropriate combination of terms, we obtain G(2)

- Cl(l)

=

(c-g)XA:!+(h+ky+cy)(2+AdAz (l+A~)(l+Al+Az)

’

The denominator in the above expression is always positive. As such, G(2)

- Cl(l)

Hence, from (6.7)) Cl(S)

> 0

iff

(h + Icy + cy) (2 + Al) > (g - c) X.

is an increasing function iff ~(2) 2 (g - c) X

The integer minimum S* of Ci (S) is either: (i) unity, or (ii) is the unique S satisfying V(S) i (g - c) x I V(S + 1).

THEOREM. PROOF.

Case (a). (g - c)X 5 v(1): Since v(S) is an increasing function, v(2) > ~(1). Hence, ~(2) > (g - c) X and from Lemma 2, S* = 1. In particular, whenever c > g, S* = 1. Case (b). (g-c) X > ~(1): From Lemma 1, Ci (S) is either an increasing function or possesses an integer minima. In the former case 5” = 1, and in the latter case there exists an S* satisfying

A(S*)

5 0 and A(S* + 1) > 0. From (6.6), A(S*) < 0 implies

(h+ky+cy)S*


Multiplying both sides of the inequality by [l + C,“l,

(h+ky+cy)

on simplification, we have

A’],

S*+F(S*-j)A, j=1

I (g-c)X.

(6.11)

I

Similarly, A(S* + 1) > 0 implies

S*+l+&S*-j+l)A, 3=1

I

2

(g-c)X.

(6.12)

From the definition of v(S), Equations (6.11) and (6.12) imply

v(s*) I (g - c) x I v(s* + 1).

(6.13)

Since v(S) is an unbounded increasing function of S and v(1) 5 (g - c) X, there exists a unique S* satisfying (6.13). We note that since ~(2) > 0, ~(2) > (g-c) X whenever c > g, and hence, from Lemma 2, S* = 1. Thus, our observations based on numerical results are established when g(t) is exponential. A numerical example illustrating the results of the theorem is presented in Table 3.

S. KALPAKAM AND

K. P. SAPNA

Table 3. k = 5, h = 0.1, X = 2, fi = 2, y = 0.01

26.300 20

25

30

35

40

33.423

34.754

35.933

37.070

38.190

39.302

40.412

41.520

1

1

20

26

32

36

40

44

48

31.300

41.325

44.152

45.381

46.534

47.661

48.777

49.888

50.997

1

1

14

23

29

34

38

42

46

36.300

46.325

53.473

54.810

55.991

57.129

58.250

59.363

60.473

1

1

1

19

26

31

36

40

44

41.300

51.325

61.350

64.206

65.437

66.593

67.721

68.838

69.949

1

1

1

14

23

29

33

38

42

46.300

56.325

66.350

73.523

74.865

76.049

77.189

78.310

79.424

1

1

1

1

19

26

31

35

39

51.300

61.325

71.350

81.375

84.259

85.494

86.652

87.781

88.898

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