integrated multi-item production-inventory systems

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

ELSEVIER European Journal of Operational Research 89 (1996) 326-340

Theory and Methodology

Integrated multi-item production-inventory systems

Antonio Arreola-Risa 1

Department of Management Science, School of Business Administration, University of Washington, Seattle, WA 98195, USA

Received December 1992; revised May 1994

Abstract

We study an integrated multi-item production-inventory system with stochastic demands and capacJ,~ated production. The problem is to find the base stock levels which minimize expected inventory costs per unit time, for an infinite time horizon. When unit manufacturing times are deterministic or exponentially distributed, we derive analytical expressions that lead to the optimal base stock levels. Our results provide several implications about the interaction of inventories, capacity utilization, and variation in the production environment.

Keywords: Inventory; Production; Queues; Stochastic processes

1. Introduction

Increasing evidence suggests that the success of most world-class companies is due in grand part to a skillful management of inventories, capacity utilization, and randomness in the production environment (Womack et al. [13]). Not surprisingly, these three areas have become the target of major continuous improvement efforts in a great number of western business organizations. However, what is surprising is the fact that operations research studies dealing with the interaction of these three factors seem to be more the exception than the rule. In this paper we explore such interaction. We study an integrated product ion-inventory system with stochastic demands, in which items share a capacitated production process, and where the items are produced under either deterministic or stochastic circumstances. The problem is to find the base stock levels which minimize expected inventory costs per unit time, for an infinite time horizon.

The connection between capacity utilization and variability in production/service systems has been successfully modelled using queueing theory (see, for instance, McClain and Thomas [6] and Murdick et al. [7]). Nevertheless, it is rarely the case that papers in this field go beyond proposing performance characteristics of a system (as opposed to optimization results), or go beyond modelling an isolated production system (as opposed to integrated product ion-inventory systems).

1 Presently at Department of Business Analysis and Research, College of Business Administration and Graduate School of Business, Texas A & M University, College Station, TX 77843-4217, USA

0377-2217/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 4 ) 0 0 2 4 8 - 7

A. Arreola-Risa / European Journal of Operational Research 89 (1996) 326-340 327

Inventory theory has also been victorious in conquering, for the most part, the relationship between inventories and random environments (see, for example, Porteus [8]). In most inventory theory papers, however, it has been assumed implicitly or explicitly that: 1) The stocked items are replenished from an exogenous source, or equivalently, the stocked items are

produced at a manufacturing facility whose capacity per period is infinite; and /or

2) the items are 'independent' in the sense that they do not utilize the same production resources. The fundamental approach behind our paper is to combine insights from queueing theory and

inventory theory, to model the system under consideration. This approach has been independently pioneered in Karmarkar [4], Williams [12], and Zipkin [15]. Related work is given in Ha [2], Wein [11], and Zheng and Zipkin [14]. The paper makes two major contributions. From a theoretical point of view, this paper represents a novel effort in modeling exactly (prior work relies heavily on approximations) an inventory problem with the following characteristics: inventory replenishment leadtimes are not independent and are endogenous to the system; the source of inventories is capacitated; and different items share the same production resources. From a practical point of view, the paper explores the crucial interaction of inventories, capacity utilization, and randomness in the manufacturing environment. To our knowledge, this has not been done before,

Our results conduce to two major counterintuitive implications. First, given our set of assumptions, the decision to hold stock for an item may or may not depend on the capacity utilization of the system. If the manufacturing environment is deterministic, capacity utilization plays a role; if the manufacturing environment is stochastic, capacity utilization does not necessarily play a role. Second, for production-inventory systems with high capacity utilization, the f o rm of the optimal base stock levels seems to be insensitive to whether or not there is variation in the production environment.

The contents of the paper are organized as follows. The next section presents the assumptions, notation, and problem definition. The two following sections study the optimal base stock levels when the manufacturing environment is deterministic and stochastic, respectively. The final section summarizes several managerial and theoretical implications of our results, and offers further research directions.

2. Problem statement

We consider a multi-item production-inventory system with stationary stochastic demands. The items are stocked according to order-for-order or base stock policies, and unsatisfied demands are backo- rdered. The same production process is used by all items, and it operates at a fixed and finite production rate. The system's inventory management costs are stationary, and consist of inventory holding and stockout costs.

We will assume the following: 1) Demand for item i, D i, can be modelled as a Poisson process with rate A i. 2) Orders for all items will be served first-come-first-serve (FCFS) at the manufacturing facility.

Required raw materials, if any, will be available in a just-in-time fashion. 3) The manufacturing process operates at a rate/z for all items. For deterministic manufacturing times,

1//z represents the unit manufacturing time; for stochastic manufacturing times, 1/z/z represents the expected unit manufacturing time. Setup times are negligible or nonexistent.

Notice that assumption 3) would typically be satisfied for systems where the different items are distinct in physical properties such as color, flavor, size, or quality of raw materials, but are identical for manufacturing purposes. Assumption 3) could also be satisfied by production systems in which 'internal' setup times have been transformed into 'external' setup times. Robinson [9,10] and Inman et al. [3] provide illustrations of these systems.

328 A. Arreola-Risa /European Journal of Operational Research 89 (1996) 326-340

Let: n ---

M =

Pi p =

O H i =

B O i =

SOi = OO = OO i = h i =

Pi = K = S i+ 1 = E ( . ) = L ( ) =

Number of different items. Unit manufacturing time. Load offered by item i = Ai/tz. Capacity utilization = Ei~ 1 Pi. (Steady-state) on-hand inventory level for item i. (Steady-state) backorders level for item i. (Steady-state) stockouts per unit time for item i. (Steady-state) outstanding orders for all items. (Steady-state) outstanding orders for item i. Inventory holding cost rate for item i. Inventory stockout cost per unit of item i. Expected total cost per unit time. Base stock level for item i (hence S i is the ' reorder point'). Expected value operator. Probability function of random variable A.

The problem is then to find S i, for i = 1, 2 , . . . , n, such that

K = ~ { h iE ( OHi ) -~PiE(SOi)} (1) i = 1

is minimized. Notice that even though the constraint p ~< 1 should be included in the problem statement, given that for p >~ 1 the value of K would be infinite, the constraint will always be satisfied and hence it will be omitted.

3. Optimal base stock levels in a deterministic manufacturing environment

As a first step in our analysis, we will assume that M is constant and thus equal to 1//~. This assumption will allow us to concentrate on the relationship between inventory levels and capacity utilization.

3.1. Operating characteristics of the production-inventory system

Since inventories are managed by base stock policies and demand for each item i is Poisson distributed with parameter Ai, orders for item i will arrive at the manufacturing facility according to a Poisson process with parameter Ai, and at the same time, orders for all items will arrive according to a Poisson process with parameter A -= Ei~=IAi. Consequently, when manufacturing times are deterministic and equal to 1//z, the manufacturing process can be modelled as a simple M / D / 1 queueing system, whose arrival and service rates are A and /z, respectively. Let Ox(') denote the generating function of random variable X. Thus,

4 , o o ( Z ) = (1 - p ) ( 1 - z ) / ( 1 - z eo(1-z ).

Furthermore, conditioned on a fixed number of total orders outstanding, say O 0 = N, it is easy to show that O 0 i is binomially distributed with parameters N and Yi =- Ai/A. Hence,

~Ooo,(Z ) = ~Ooo((1 - 7,) + y i z ) ,

A. Arreola-Risa /European Journal of Operational Research 89 (1996) 326-340 329

or,

Pi e -Pi (1-z ) ( 1 -- p)(1 -- Z)

I~00, ( Z ) = Pi (1 __ Z) -- p [ 1 -- e-Pi(1-z)] ' ( 2 )

where {(1 - %) + yi z} is the generating function of a Bernoulli random variable with parameter %. Using (2) along with

Pr{BOi = 0} = Pr{OOi ~< S i + 1} (3a)

and

Pr(BOi=x ) =Pr{OOi=Si+ 1 +x} , x > 1, (3b)

the distribution of BOi, fBO~('), can be obtained. The resulting expression for fBO~('), unfortunately, would render an intractable cost function. Thus our modeling strategy will be to develop and use a simplified form for fBo,('), based on its asymptotic behavior as P -+ 1.

Proposition 1. I f M is deterministic, then as p -+ 1, the distribution of BO i converges asymptotically to

s,+l+x [Pi+ 2 ( 1 / p _ 1)] n f B ° ' ( x ) = e - m ( 1 - ~ ' ) f l s ' + l + x ~" n! , x>~l ,

n=O

fuo~(x) =P[Pi; Si + 1] - e20/P-1)~&+2p[p i + 2(1 /p - 1); S i + 1], x = O,

where

fli = Pi and P[y; x ] = ~ e-yYn Pi + 2(1 /p - 1) n=O n!

(4a)

(4b)

Proof. Utilizing arguments similar to those in Arreola-Risa [1], we can show that if in our production-inventory system M is deterministic, then as p ~ 1, the distribution of OO i converges asymptotically to the following distribution:

with

[Pi + 2 ( 1 / p - 1 ) ] " f oo,( X ) e-m(1 ~ ~i ~ ~ ~ 2., x = O, 1, (5)

n = 0 n ! ' " ' "

E(OOi) = pi(2 - p ) / [2(1 - p) ] .

From (3a) and (5), we find that

Si+ l Si+ l

f a o , ( x = 0 ) = E foo,(Y) = E y=O y=O

Si+l Si+I-Y [Pi + 2(1 /0 - 1)]" = e - P i ( 1 - [ ~ i ) [ ~ s i i+l E ~ z y E n!

y = 0 n = 0

s,+lV, [pi+ 2(1 /p - n ! 1)1 n ( /~ -- /~/SI+ 2) = e - p i

n=O

=P[Pi; Si + 1] - em/O-1)~&+2p[p i+ 2 ( l / p - 1); S i+ 1].

The other result is obtained from (3b) and (5). []

(6)

s ,+l -y [Pi ÷ 2(1 /p - 1)]n E e - O , ( 1 _ /~ i ) /~ / s i+ 1 -y n !

n ~ 0

330 A. Arreola-Risa / European Journal of Operational Research 89 (1996) 326-340

Using Proposition 1, we have that E(BO i) thus converges asymptotically to the following value:

~ s,+l+x [Pi + 2 ( l i p - 1)] ~ E(BO, ) = E XfBo,(X) = E x e-P~(1 -/3,) /3 s'+s+x E n]

x = O x = l n = O

/3s~+2 e2O/,-1) e [p i + 2 ( 1 / p - 1); S i + 1]

1 - / 3 i

-t- 1--j~i

But given that by definition

(si+ 1)]{1-e[p,; si+ ill +o,{1-e[o,; s,]}.

E ( O H i ) - E(BO~) = S i + 1 - E ( O O i ) ,

we obtain from (6), (7), and (8) that

~i ~ 3&+2 e2(1/°- 1) E ( O H i ) - Si + 1 Pi "+" P[Pi + 2 ( 1 / p - 1); Si + 11

1 - / 3 i 1 - /3 i

+ [ t - f l i jSi ( S i + l ) ] { 1 - P [ p i ; S i + l ] } + p i { l - P [ p i ; Si]}.

Let 6 i denote the fill rate for item i. Then from basic principles

E ( S O i ) = / ~ i ( l - ~ i ) ,

and observing that

( 1 - ~i) = Pr{BOi > 1} + Pr{OOi = S i + 1}

oo Si+l+x [Pi-Jr" 2 ( 1 / 0 -- 1 ) l n

= E e-P'( 1 -t[3,~)1~7 +l+x E n! x = 0 n ~ 0

s,+l+x [Pi + 2 ( 1 / p - 1)}" = e - O ~ ( 1 - r4 ~tqs~ +1 * r'i/r-i E [~i E n!

x = 0 n=:0

[pi+ 2 ( 1 / 0 - 1)]" (¢l,[O, + 2 ( 1 / o - 1)]}" = e-' 'eg '+ 'r* '+ ' + E [ 5 0 n, ,7.+'... n~Si+2

=- e~ l / ' - l )~S '+lp[p , + 2(1/p - 1); S; + a] + 1 - P [ P i ; Si + 1], we find

E(SOi) = ai{e2(1/p-1)fisi+lP[p i + 2 ( 1 / p - 1); S i + 1] + i - P [ P i ; Si + 1]}.

(7)

(8)

(9)

(lo)

3.2. Optimization results

Continuing with our modeling strategy, we now propose to substitute (9) and (10) into (1), and then find the base stock levels which minimize the resulting cost function. These minimizing levels will be called, for obvious reasons, asymptotically optimal base stock levels (AOBSL). The closer p is to 1, the closer the AOBSL will be to the truly optimal base stock levels (TOBSL). To study the robustness of the AOBSL, we will consider 225 test problems obtained from all different combinations of the following


Table 1 Comparison of the truly optimal base stock levels (TOBSL) and the asymptotically optimal base stock levels (AOBSL), for the first 75 test problems. Bold numbers indicate a difference, and the number in parentheses is the cost penalty (Zi = 10, for all i)

n p~/h~ p = 50% p = 70% p = 90%

TOBSL AOBSL TOBSL AOBSL TOBSL AOBSL

10

25

50

100

2 3 3 5 5 9 9 5 4 4 6 6 12 12

10 5 4 (1%) 7 7 16 15 (2%) 25 5 5 9 8 (2%) 19 19

100 7 6 (3%) 10 9 (3%) 25 25

2 2 2 2 2 3 3 5 2 2 2 2 4 4

10 2 2 3 3 4 4 25 2 2 3 3 5 5

100 3 3 4 3 (4%) 6 6

2 1 1 2 2 2 2 5 2 2 2 2 3 3

10 2 2 2 2 3 3 25 2 2 2 2 3 3

100 2 2 3 3 4 4

2 1 1 1 1 2 2 5 1 1 2 2 2 2

10 2 2 2 2 2 2 25 2 2 2 2 3 3

100 2 2 2 2 3 3

2 1 1 1 1 1 1 5 1 1 1 1 2 2

10 1 1 2 2 2 2 25 2 2 2 2 2 2

100 2 2 2 2 3 3

pa r ame te r values: A i = 10, 100, 1000, for all i; p = 0.5, 0.7, 0.9; n = 1, 10, 25, 50, 100; and p i / h i -~ 2, 5, 10, 25, 100, for all i. Notice that in this p rob lem set

P i = p / n for a l l i .

For each test problem, we assumed that since all i tems are homogeneous in terms of cost and opera t ing characterist ics (but pe rhaps they are different in color, flavor, etc.) their opt imal base stock levels would also be identical . W e found the T O B S L by exhaustively s imulat ing the system for different base stock level values, i.e., s tar t ing with S / = 0 for all i, we s imula ted the system for increas ing values of S i unt i l E ( S O i ) = 0. The value of S i in tha t range which provided the m i n i m u m s imulated K was selected as the TOBSL. The A O B S L were de t e rmined by per forming a l ine search on (1) us ing (9) and (10). The results of our exper iment are p resen ted in Tables 1-3 . Table 1 conta ins the first set of 75 p rob lems w h e n A i = 10 for all i; Tab le 2 conta ins the second set of 75 problems when h i = 100 for all i; and Tab le 3 conta ins the third set of 75 problems when Ai = 1000 for all i.

The robus tness of the A O B S L is impressive. Even when p is as low as 50%, the average cost pena l ty for all cases is 1.36%, and in 88% of the cases, the A O B S L are ident ical to the TOBSL. As expected, for h igher values of p the average cost penal t ies are lower: for p = 70% the average cost pena l ty is 0.73%,

332 A. Arreola-Risa ~European Journal of Operational Research 89 (1996) 326-340

Table 2 Comparison of the truly optimal base stock levels (TOBSL) and the asymptotically optimal base stock levels (AOBSL), for the second 75 test problems. Bold numbers indicate a difference, and the number in parentheses is the cost penalty (A i = 100, for all i)

n pi/hi p = 50% p = 70% p = 90%

TOBSL AOBSL TOBSL AOBSL TOBSL AOBSL

1 2 5 5 5 6 5

10 7 6 25 7 6

100 9 7

10 2 2 2 5 3 3

10 3 3 25 3 3

100 3 3

25 2 2 2 5 2 2

10 2 2 25 3 3

100 3 3

50 2 2 2 5 2 2

10 2 2 25 2 2

100 3 3

100 2 2 2 5 2 2

10 2 2 25 2 2

100 2 2

8 7 (5%) 18 18 (11%) 9 9 22 22 (3%) 10 9 (3%) 25 25 (22%) 11 11 30 30 (12%) 14 12 (7%) 36 36

3 3 5 5 3 3 6 5 (3%) 4 3 (5%) 6 6 4 4 7 7 4 4 8 8

2 2 3 3 3 3 4 4 3 3 4 4 3 3 5 5 3 3 5 5

2 2 3 3 2 2 3 3 2 2 3 3 3 3 4 4 3 3 4 4

2 2 2 2 2 2 2 2 2 2 3 3 2 2 3 3 3 3 3 3

and for p = 90% the average cost pena l ty is 0.14%. In summary, our s imula t ion results indicate that for

systems with p > /50%, the A O B S L are excel lent es t imators o f the T O B S L .

Severa l impor t an t insights about the impact of capaci ty ut i l izat ion on the opt imal base stock levels can

be der ived f rom Tables 1-3 . First, w h e n P/Pi <~ 10, the op t imal inventory levels are very sensit ive to

changes in capaci ty ut i l izat ion; but as P/Pi increases, the sensitivity to p decreases greatly. Second, the

impact of p on the op t imal levels is a u g m e n t e d or d imin ished by the average d e m a n d ra te A i. Such

impact , however , is not p ropo r t i ona l to the m a g n i t u d e of the changes in A i. Also of in teres t is the fact

that for large values o f P/Pi, the op t imal base stock levels b e c o m e almost invar iant to the rat io pi /h i , regardless of the u t i l iza t ion level.

3.3. Starting base stock levels

U s e of the A O B S L to es t imate the T O B S L avoids the painful task of exhaust ively s imulat ing the

p r o d u c t i o n - i n v e n t o r y system. Never the less , f inding the A O B S L entails a much less but still s ignificant t ime-consuming task: a l ine search on the cost funct ion for each and every i t em in the system. To

expedi te this job, we next looked at ways to ident i fy base stock level va lues that could be used as good


Table 3 Comparison of the truly optimal base stock levels (TOBSL) and the asymptotically optimal base stock levels (AOBSL), for the third 75 test problems. Bold numbers indicate a difference, and the number in parentheses is the cost penalty (Ai = 1000, for all i)

n Pi/hi p = 50% p = 70% p = 90%

TOBSL A O B S L TOBSL AOBSL TOBSL AOBSL

10

25

50

100

2 7 6 (20%) 11 10 (2%) 28 29 (1%) 5 8 7 (0%) 14 11 (9%) 35 33 (3%)

10 9 7 (12%) 14 12 (5%) 36 36 25 9 8 (18%) 15 13 (10%) 42 40 (2%)

100 9 9 15 15 46 46

2 3 3 4 4 7 7 5 3 3 4 4 7 7

10 3 3 4 4 8 8 25 4 4 5 5 9 9

100 4 4 5 5 10 10

2 2 2 3 3 4 4 5 3 3 3 3 5 5

10 3 3 3 3 5 5 25 3 3 4 4 6 6

100 3 3 4 4 6 6

2 2 2 3 3 3 3 5 2 2 3 3 4 4

10 3 3 3 3 4 4 25 3 3 3 3 4 4

100 3 3 3 3 5 5

2 2 2 2 2 3 3 5 2 2 2 2 3 3

10 2 2 3 3 3 3 25 2 2 3 3 4 4

100 3 3 3 3 4 4

starting points in the line search. One possibility, observing that in most cases several terms in (9) and (10) would be negligible, is to use as starting points the base stock level values which minimize (1) but where now

/3 i ]~/Si + 2 e 2 ( 1 / p - 1)

E ( O H i ) = S i @ 1 Pi d- 1 - - [~i 1 - - [3 i

E ( S O i ) = l~i{e2(l/p-1)[~Si+l}.

(9')

(10')

These new minimizing values will be called starting base s tock levels (SBSL), and will be denoted by S%. It is not hard to show that for each i, Si is the largest Si such that

Si >/h i e -2 (1 /p -1 ) / / ( [3ihi + ( 1 - [3i) PiAi) , (11a)

o r

l{ hie lJ l ) 1 S i = I n ~ihi.~ ( 1 -[~i)Piai / In ~i , ( 1 1 b )


Table 4 Comparison of the truly optimal base stock levels (TOBSL) and the starting base stock levels (SBSL), for the 75 test problems in Table 1. Bold numbers indicate a difference, and the number in parentheses is the difference in expected total cost

n Pi/hi p = 50% p = 70% p = 90%

TOBSL SBSL TOBSL SBSL TOBSL SBSL

10

25

50

100

2 3 3 5 5 9 9 5 4 4 6 6 12 12

10 5 4 (1%) 7 7 16 15 (2%) 25 5 5 9 8 (2%) 19 19

100 7 6 (3%) 10 9 (3%) 25 25

2 2 2 2 2 3 3 5 2 2 2 2 4 4

10 2 2 3 3 4 4 25 2 3 (7%) 3 3 5 5

100 3 3 4 3 (4%) 6 6

2 1 2 (27%) 2 2 2 2 5 2 2 2 2 3 3

10 2 2 2 2 3 3 25 2 2 2 2 3 3

100 2 2 3 3 4 4

2 1 1 1 1 2 2 5 1 2 (16%) 2 2 2 2

10 2 2 2 2 2 2 25 2 2 2 2 3 3

100 2 2 2 2 3 3

2 1 1 1 1 1 1 5 1 1 1 1 2 2

10 1 2 (14%) 2 2 2 2 25 2 2 2 2 2 2

100 2 2 2 2 3 3

w h e r e [x ] r e p r e s e n t s t h e l a rges t i n t e g e r s m a l l e r t h a n o r e q u a l to x. I n T a b l e s 4 - 6 we c o m p a r e t h e S B S L

to t h e T O B S L for t h e 225 tes t p r o b l e m s c o n s i d e r e d in T a b l e s 1 -3 .

R e m a r k a b l y , t h e S B S L a r e n o t on ly v e r y g o o d s t a r t i ng v a l u e s fo r t h e l ine sea rch , b u t fo r sys tems wi th

p >~ 5 0 % , o u r resu l t s : fo r p = 5 0 % t h e a v e r a g e cost p e n a l t y is 2 .55%; fo r p = 7 0 % t h e a v e r a g e cos t

p e n a l t y is 0 .73%; a n d fo r p = 9 0 % t h e a v e r a g e cos t p e n a l t y is 0 .15%, sugges t t h a t t h e S B S L c o u l d e v e n

b e u s e d to e s t i m a t e t h e T O B S L , t h u s a v o i d i n g e x h a u s t i v e s i m u l a t i o n s a n d l ine sea rches .

4. Optimal base stock levels in a stochastic manufacturing environment

I n th is s ec t i on w e i n t r o d u c e m a n u f a c t u r i n g p r o c e s s va r i ab i l i ty i n to o u r p r o d u c t i o n - i n v e n t o r y sys tem. In an e f f o r t to focus o u r a t t e n t i o n on t h e i ssues b e i n g e x p l o r e d , a n d n o t on t h e m a t h e m a t i c a l i n t r i cac ie s

t h a t m a y ar i se by us ing s o p h i s t i c a t e d p robab i l i s t i c m o d e l i n g , w e wil l a s s u m e t h a t t h e r a n d o m n e s s in t h e m a n u f a c t u r i n g e n v i r o n m e n t c a n b e a d e q u a t e l y c a p t u r e d by un i t m a n u f a c t u r i n g t i m e s ( M ) b e i n g s tochas -

tic. A d d i t i o n a l l y , w e wil l a s s u m e t h a t M is e x p o n e n t i a l l y d i s t r i bu t ed .

A. Arreola-Risa ~European Journal of Operational Research 89 (1996) 326-340 335


n pi/hi p = 50% p = 70% p = 90%


10

25

50

100

2 5 5 8 7 (5%) 18 18 5 6 5 (11%) 9 9 22 22

10 7 6 (3%) 10 9 (3%) 25 25 25 7 6 (22%) 11 11 30 30

100 9 7 (12%) 14 12 (7%) 36 36

2 2 2 3 3 5 5 5 3 3 3 3 6 5 (3%)

10 3 3 4 3 (5%) 6 6 25 3 3 4 4 7 7

100 3 4 (5%) 4 4 8 8

2 2 2 2 2 3 3 5 2 2 3 3 4 4

10 2 2 3 3 4 4 25 3 3 3 3 5 5

100 3 3 3 3 5 5

2 2 2 2 2 3 3 5 2 2 2 2 3 3

10 2 2 2 2 3 3 25 2 2 3 3 4 4

100 3 3 3 3 4 4

2 2 2 2 2 2 2 5 2 2 2 2 2 2

10 2 2 2 2 3 3 25 2 2 2 2 3 3

100 2 2 3 3 3 3

4.1. Operating characteristics of the production-inventory system

Paralleling the deterministic manufacturing times case, if now M is exponentially distributed, then the manufacturing process can be model led as a basic M / M / 1 queueing system. Thus the distribution of O O will be geometric with parameter p. Recalling that if OO = N, then O O i is binomially distributed with parameters N and Yi = hi /A, we find that

f o o i ( k ) =" E ( ' ~ i / * ) k ( l - - * i / l ~ ) n - k p n ( 1 - - P ) n=k

= ( l - p ) *--t~i} r~ 1 k X i

r+k-1

+z/p) = (1 -p ) *



n pi/hi p = 50% p = 70% p = 90%


10

25

50

100

2 7 6 (20%) 11 10 (2%) 28 29 (1%) 5 8 7 (0%) 14 11 (9%) 35 33 (3%)

10 9 7 (12%) 14 12 (5%) 36 36 25 9 8 (18%) 15 13 (10%) 42 40 (2%)

100 9 9 15 15 46 46

2 3 3 4 4 7 7 5 3 3 4 4 7 7

10 3 4 (0%) 4 4 8 8 25 4 4 5 5 9 9

100 4 4 5 5 10 10

2 2 3 (0%) 3 3 4 4 5 3 3 3 3 5 5

10 3 3 3 3 5 5 25 3 3 4 4 6 6

100 3 3 4 4 6 6

2 2 2 3 3 3 3 5 2 2 3 3 4 4

10 3 3 3 3 4 4 25 3 3 3 3 4 4

100 3 3 3 3 5 5

2 2 2 2 2 3 3 5 2 2 2 2 3 3

10 2 2 3 3 3 3 25 2 3 (20%) 3 3 4 4

100 3 3 3 3 4 4

B u t p = A / g , so

1 f o o , ( k ) = ( l ~ p ) . + p i p i + ( 1 - - O ) = ( 1 - - a i ) ak

w h e r e a i = Pi/(Pi + 1 - p). S u r p r i s i n g l y , f o r t h e p r e s e n t c a s e t h e d i s t r i b u t i o n o f O O i , is n o t o n l y t r a c t a b l e b u t e l e m e n t a r y as wel l . I n

t h e f o l l o w i n g p r o p o s i t i o n w e d e r i v e t h e d i s t r i b u t i o n o f B O i.

P r o p o s i t i o n 2. I f M is exponentially distributed, then

[ ( 1 - a i ) a s'+l+~ forx>~ l

f n ° ' ( X ) = [ 1 - - a si+2 f o r x = O .

P r o o f . Eqs . (3a ) a n d (12) y i e l d Si+ l Si+1

f , o i ( x = 0 ) = E foo , (Y) = (1 - - a , ) a s~+I E O/Z y = 1 - a s ' + z y=0 y=0

T h e o t h e r r e s u l t is a d i r e c t c o n s e q u e n c e o f ( 3b ) a n d (12). []


We find from Proposition 2 that

E ( B O i ) = E XfBoi(X) = 0 ( 1 - - a S'+2) -}- E X(1 --ai)OlSi i+l+x x=0 x ~ l

-~- a S i + 2 / / ( 1 - - a i ) .

Since OOi is geometrically distributed and hence

E ( O 0 3 = a i / ( 1 - a , ) ,

we obtain

E ( O H i ) = S i + 1 - E ( O O i ) + E(BOi) = S i + 1 + a:~ +2 -- a i

1 m a i

From Proposition 2 and (12) we also derive

E(SOe) =,~i[Pr{BO~/> 1} + Pr{OOi = Si + 1}]

=h i ( 1 - a i ) a s , + l + x + ( 1 - a i ) a s i +1 = h i ( a / S / + l ) .

(13)

(14)

4.2. Optimization results

Substituting (13) and (14) into (1), the expected total cost per unit time becomes

[ K = ~ h i S i + l + i=1

We can redefine K as

K = ~ Ki(Si ) , i=1

where

Ki(Si) -~ hi( 1--1----~i ) + aSi(PiAiai-4- hia2 )

Let Si* be the smallest S i that minimizes Ki(Si). Then Si* must satisfy AKi(Si* ) < 0 and AKi(Si* + 1) >/0, where

AKi( Si) = Ki( S,) - Ki( S i - 1).

If AKi(Si* + 1) = 0, then both Si* and Si* + 1 minimize Ki(S ) . Since

A Ki ( Si) = - a s , + % - as i (1 - a i ) Pihi + hi,

it follows that Si* is the largest S i such that

a s` >~ h i / ( a i h i + (1 - ai)Pihi), (15a)

or

h i

where again [x] represents the largest integer smaller than or equal to x.

(15b)


The behavior of Si* in Eq. (15b) is not surprising in that decreases in hi, o r increases in Pi o r hi, would lead to an increase in Si*. However, what is not necessarily trivial is to predict how Si* would change as a result of changes in a i (due, for example, to changes in the load offered by item i, or in the capacity utilization of the system, or in both). As a starting point, in the following proposition we derive conditions under which the optimal base stock level will be equal to zero, will be equal to 1, or will be greater than or equal to 1.

Proposition 3. I f M is exponentially distributed, then

Si* + 1 = 0 i f A i < hi /Pi , (16a)

Si* + 1 = 1 i f A i = hi /Pi , (16b)

S ; +1>~1 if A i > h i / p i. (16c)

Proof. From (15b) we know that Si* = 0 if

In{hi~ ( olih i %- ( 1 - ogi) PiAi)} --- 0,

o r h i =piai . Eqs. (16a) and (16c) can be obtained using a similar argument. Notice that in (16c) an equal sign has been added even though A i > h i / p i. This is to account for the fact that Si* may still be zero even when

ln{h i / (o t ih i + ( 1 - ot,i) PiAi) } < O. []

Astonishingly, the results in Proposition 3 are independent of a i, and hence of Pi and p. That is, the decision to hold stock for an item is independent of the item's offered load and the capacity utilization of the system. (Li [5] obtains a similar result for a discounted model.) Combining Proposition 3 and (15b) we can also deduce that the optimal base stock level is nondecreasing on oti if A i > hi /Pi , and is independent of O/i otherwise.

For comparison purposes with the deterministic manufacturing environment case, we provide in Table 7 the optimal base stock levels when manufacturing times are exponentially distributed, for the 225 test problems in Tables 1-3. These optimal levels were obtained directly from (15b). No simulation was required because (13) and (14) hold exactly for any value of p.

5. Summary

We have studied an integrated multi-item product ion-inventory system with stochastic demands and capacitated manufacturing. For deterministic and exponential manufacturing times, we derived expressions that lead to the base stock levels which minimize expected inventory costs per unit time. Equally important, the various results presented in the paper, conduce to several theoretical and managerial implications about the interaction of inventories, capacity utilization, and randomness in the manufacturing environment. Some of the implications are intuitive, but others are not.

• Looking at Tables 1, 2, 3, and 7, it is clear that, as expected, higher utilization levels translate into higher optimal inventory levels. However, according to Proposition 3 this is not always the case.

• Contrasting Tables 1-3 to Table 7, it is also clear (and expected) that variation in the production environment increases the optimal inventory levels. Comparing these tables though, we observe that with or without variation in the manufacturing environment, as P/Pi increases: 1) The sensitivity of the optimal base stock levels to p and p i / h i decreases; and 2) the impact of p in the optimal base stock levels is a nonlinear function of A i,

A. Arreola-Risa / European Journal of Operational Research 89 (1996) 326-340

Table 7 Optimal base stock levels for the 225 test problems in Tables 1-3 , when manufacturing times are exponentially distributed

339

Pi/hi A i = 10, for all i h i = 100, for all i h i = 1000, for all i

p = 50% p = 70% p = 90% O = 50% p = 70% p = 90% p = 50% p = 70% O = 90%

10

25

5O

i00

2 4 6 11 7 12 29 10 18 51 5 5 8 17 8 15 38 12 21 60

10 6 10 23 9 16 44 13 23 66 25 7 13 31 11 19 53 14 26 75

100 9 16 44 13 23 66 16 29 88

2 2 2 4 3 4 7 4 5 10 5 2 3 5 3 4 8 4 5 11

10 2 3 6 3 5 9 4 6 12 25 3 4 7 4 5 10 5 6 13

100 3 5 9 4 6 12 5 7 15

2 1 2 3 2 3 4 3 4 6

5 2 2 3 2 3 5 3 4 7 10 2 2 4 3 3 5 3 4 7 25 2 3 4 3 4 6 4 5 8

100 3 3 5 3 4 7 4 5 9

2 1 1 2 2 2 3 2 3 4 5 1 2 2 2 2 4 3 3 5

10 2 2 3 2 3 4 3 3 5 25 2 2 3 2 3 5 3 4 6

100 2 3 4 3 3 5 3 4 7

2 1 1 2 2 2 3 2 3 4 5 1 2 2 2 2 3 2 3 4

10 1 2 2 2 2 3 2 3 4 25 2 2 3 2 3 4 3 3 5

100 2 2 3 2 3 4 3 4 5

• From Proposition 3 we know that the decision to hold stock is not necessarily dependent on the capacity utilization of the system. Interestingly enough, ( l lb ) would lead us to infer that for deterministic manufacturing times, capacity utilization does play a role on stocking decisions.

• Comparing (11b) and (15b), we may conclude that for systems with a high capacity utilization, the form of the optimal base stock levels seems to be insensitive to whether or not there is randomness in the manufacturing environment.

• The results in Tables 1-3 suggest that the use of asymptotic behavior may be a promising modeling alternative for inventory problems with intractable distributions.

• Similarly, the results in Tables 4 -6 indicate that the development of approximate but parsimonious expressions for the variables of interest (like those in ( l l a ) - ( l l b ) ) may be a worthwhile research direction. Such expressions, besides facilitating the search for optimal solutions, provide invaluable insight about the fundamental trade-offs of the problem being studied.

The results in this paper could be extended in several fruitful directions. Consideration of lotsizing issues by looking at (Q, R) policies is one possibility. Two other major generalizations would be to include multi-stage production processes and to allow for multi-echelon inventory structures. Addition of manufacturing issues such as sequencing and dispatching may also be worth exploring.


Acknowledgments

This research was part ial ly suppor ted by a grant f rom the School of Business Admin i s t r a t ion at the Univers i ty of Washing ton .

References

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