multi-echelon inventory production system solution

4
Pergamon Computers ind. Engng Vol. 27, Nos 1--4,pp. 201-204, 1994 Copyright © 1994 Elsevier Science Ltd 034S0-8352(94)00163"4 Printed in Great Britain. All rights reserved 0360-8352/94 $7.00 + 0.00 Multi-Echelon Inventory Production System Solution Toshio ODANAKA*, Kciichi YAMAGUCHI*,Tadayuki MASUI** *Hokkaido Information University **Musashi Institute of Technology Abstract The multi-echelon inventory problem can be examined from the points of view of dynamic progrmnming. The difficulty is that the number of possible states will be exceptioztally large. The authors avoid this difficulty by using some formulation. An introductory section surveys what Clark and Clark-Scarf formulated, the problem of a single product carried at an arbi- trary number of activities arranged in a series structure. Sub- sequent sections provide the researches on analytic considera- tion and numerical consideration. The final section discusses the production planning in a multi-echelon inventory produc- tion system. Key Words: multi-echelon, dynamic programming, lead time, inventory, production. 1. Introduction The dynamic, multi-installationinventory problem can be ex- amined from the point of view of dynamic programming. As usual, a sequence of cost functions is defined, whose indepen- dent variables indicate the state in which the system may be. The cost functions willsatisfya recursive equation, which may, at least in theory, be solved so as to obtain optimal purchas- ing and transshipment rules. The difficultyis, however, that the number of possible states will be exceptionally large. The disposition of stock at all of the installations will have to be indicated, along with a description of quantities on order and being shipped. With such a large number of possible states, dynamic programming calculations take a long time. The same type of dii~culty appeared in discussion on op- timal policies in the presence of a lead time in delivery. The authors overcame the delivery lead time problem by assum- ing that excess demand is backlogged, and, on the basis of this assumption, transformed the functional equation to one involving a single wxiable. For certain rather extreme types of multi-echelon problems, a similar type of approach is possibles) (see Clark l) and Clark and Scan~)). Williams dis- cussed the stochastic multi-echelon production and inventory problem 4),5). Consider the case of N installations arranged in series, so that installation 1 receives stock form installation 2, 2 from 3, and so on. Stock enters the system through the highest installation, with a fixed delivery lead time, and is shipped through the subsequent installations, stock being removed at each step to satisfy random demands. If the following assump- tions are made, then the optimal policies may be calculated easily4) 's).(Fig.1) 1. The cost of shipping from a single installation to the next one is proportional to the quantity shipped. 2. Excess demand is backlogged. 3. The expected holding and shortage cost to be charged to each installation during a single step are functions of the stock at that installation plus all stock in transit and on hand at lower installations. Of the three assumptions, the first is the crucial one; the third may easily be satisfied by an appropriate accounting of holding and shortage costs. Supplier mpienJflmum " ~ ' i ruplenistuneaa ~lwk~ orders ' ' t Demand ~plea~lune~,. , Iwk~torder. ~ / t I l / Cus~n'~m Fig.1 Information and Stock Flow in a Base Stock System 201

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Pergamon Computers ind. Engng Vol. 27, Nos 1--4, pp. 201-204, 1994

Copyright © 1994 Elsevier Science Ltd 034S0-8352(94)00163"4 Printed in Great Britain. All rights reserved

0360-8352/94 $7.00 + 0.00

Multi-Echelon Inventory Production System Solution

Toshio ODANAKA*, Kciichi YAMAGUCHI*, Tadayuki MASUI**

*Hokkaido Information University **Musashi Inst i tute of Technology

A b s t r a c t

The multi-echelon inventory problem can be examined from the points of view of dynamic progrmnming. The difficulty is that the number of possible states will be exceptioztally large. The authors avoid this difficulty by using some formulation. An introductory section surveys what Clark and Clark-Scarf formulated, the problem of a single product carried at an arbi- trary number of activities arranged in a series structure. Sub- sequent sections provide the researches on analytic considera- tion and numerical consideration. The final section discusses the production planning in a multi-echelon inventory produc- tion system.

Key Words : multi-echelon, dynamic programming, lead time, inventory, production.

1. I n t r o d u c t i o n

The dynamic, multi-installation inventory problem can be ex- amined from the point of view of dynamic programming. As usual, a sequence of cost functions is defined, whose indepen- dent variables indicate the state in which the system may be. The cost functions will satisfy a recursive equation, which may, at least in theory, be solved so as to obtain optimal purchas- ing and transshipment rules. The difficulty is, however, that

the number of possible states will be exceptionally large. The disposition of stock at all of the installations will have to be indicated, along with a description of quantities on order and being shipped. With such a large number of possible states, dynamic programming calculations take a long time.

The same type of dii~culty appeared in discussion on op- timal policies in the presence of a lead time in delivery. The authors overcame the delivery lead time problem by assum- ing that excess demand is backlogged, and, on the basis of this assumption, transformed the functional equation to one involving a single wxiable. For certain rather extreme

types of multi-echelon problems, a similar type of approach is possible s) (see Clark l) and Clark and Scan~)). Williams dis- cussed the stochastic multi-echelon production and inventory problem 4),5).

Consider the case of N installations arranged in series, so that installation 1 receives stock form installation 2, 2 from 3, and so on. Stock enters the system through the highest installation, with a fixed delivery lead time, and is shipped through the subsequent installations, stock being removed at each step to satisfy random demands. If the following assump- tions are made, then the optimal policies may be calculated easily 4) 's).(Fig.1)

1. The cost of shipping from a single installation to the next one is proportional to the quantity shipped. 2. Excess demand is backlogged. 3. The expected holding and shortage cost to be charged to each installation during a single step are functions of the stock at that installation plus all stock in transit and on hand at lower installations.

Of the three assumptions, the first is the crucial one; the third may easily be satisfied by an appropriate accounting of holding and shortage costs.

Supplier

mpienJflmum " ~ ' i

ruplenistuneaa ~l wk~ orders ' '

t Demand

~plea~lune~,. , Iwk~torder. ~ / t I l /

Cus~n'~m

Fig.1 Information and Stock Flow in a Base Stock System

201

202 Selected papers f rom the I6th Annual Conference on Computers and Industrial Engineering

Consider a special ease in order to indicate how the method works. There will be two installations, with installation I re- ceiving stock from installation 2. Both the shipment time from 2 and the lead t ime to 2 will be assumed to be a single period. Demand for the i tem occurs only at the installation with a den- sity ~(~). The expected holding and shortage cost functions will be Ll(xl) and La(x2), respectively, with xa representing stock at both installations, where hi and pi are the marginal holding and shortage costs of installation i (i = 1, 2).

Li(xi) = fo hi(xi-~)~(~)d~+ f f p;(~-xi)~(~)d~, xi >_ 0

f o p,(~-x,)~o(~)d~, z, < 0

At any moment of t ime the s tate of the system is described by the pair (xl,x2), and we may then define a sequence of min imum cost functions C~(xl,x2) tha t satisfy

C,,(zl, z2) = rain ~e(z)+cr.(y-zl)+L1(z])+L2(x~) "'-<~-<~'L

0_<*

i ° } +~ c ,_~(~ - ( , ~ + z - O~(Od( O)

where c(z) is the cost of purchasing z units, c r is the unit transportation cost and ct is the discount ratio. Analysis of this equation is begun by considering the lower installation by itself, and the functional equation that would apply if there were no limitation on available stock, i.e.,

rain /c~ .(y - z l ) + LI (x l ) C,(z~) = ~,>--"~

If L, is convex, the optimal policy for this equation, which as yet has no relevance to the original problem, is determined by a sequence of critical numbers x~, x ~ , x s , . . . , with the in- terpretation that , at the beginning of each period, stock is raised the corresponding level. The problem, of course, is that there may not be adequate stock at the second installation to satisfy these requests. It may be shown, however, that the optimal system policy is to satisfy as many of these requests as possible ~). The optimal system policy is then calculated on the basis of the functional equation

qn(z2) = min~c(z)+L2(za).4-An(z~) • _>o L

+a fo°°q,,-l(z, + z-~)~o(~)d~} (3)

where A,(x~) is the extra cost. In the ease of several installations linked in series, the proce-

dure may be repeated with an additional shortage cost added at each step because of the inability to satisfy requests from below. It is impor tant to realize tha t we are, in fact obtaining optimal policies. Although the argument given above has a fanciful quality, it may be made quite rigorous by means of the result

C.(~,,z~) = C.(~) + qn(~) (4)

2. A n a l y t i c a l a n d N u m e r i c a l C o n s i d e r a t i o n

At first, consider the two installations and one lead time cases. The first functional equation to consider is tha t derived in (I),(2), namely

C.(~) = ~>--~min{c~'(Y-zl)+L*(zl)

+ a Joo °° C, . - , (~ - D~(~)d~ } (1)

The authors impose tile assumpt ion tha t

p > - - c, (2)

This is an exceedingly mild restriction. Let x , represent the optimal inventory level for the n-stage

problem. It will be established that x . converges to an optimal

inventory level x for the full dynamic problem, with x explicitly determined. The general theory asserts tha t C,(z) --* C(x). The proof is by induction on the number of periods n.

Allowing n to tend toward infinity, we deduce that existence of x, necessarily finite, such that the optimal policy is

0 Xl > :~ (3)

aml such tha t x is the unique solution of the equation

c, + C'(e - ~)~(~)d~ = 0 (4)

where C is convex.

Now a transcendental equation is derived from which x may be computed. F.or x < 5:,

[ c(~) = ~.(e - ~i) + L~(~) + C'(e - ~)~(()d~ (5)

Hence

C ' ( z l ) = --Cr + L~(zl) for the interval xl < ~1 (6)

Consequently, x is the unique solution of the equation

[ cr(1 - - a ) + a L~(~: -- ()~(~)d~ = 0 (7)

The uniqueness is assured. To sum up,

T h e o r e m 1. If the assumptions tha t h(x) and p(x) are con- vex increasing and CT(Z) = CT • Z satisfied and that there is one installation, then the optimal policy is in the form

~ - x , x l < ~ , z0(xl) = 0 xl _> ~. (8)

In (3) of 1, let us assume that the ordering cost is linear - i.e., c ( z ) = c . z .

Theorem 2. If h(x) and p(x) are convex increasing and c(z) = c. z, the optimal policy is the form

z(z) = O. z2 >_ ~" (9)

Selected papers from the 16th Annual Conference on Computers and Industrial Engineering 203

N u m e r i c a l C o n s i d e r a t i o n

(1) General figure of model: Illustrated in Fig.1 This is the single i tem multi-edtelon process which consists of m producing stages and m inventory transport ing states from each stage to the succeeding process. (2) Demand for final product ~ with the probability density

function ¢(~).

~ - ~ " + ~ ) (0 _< ~ _< 500)

= 0 (~ < 0 , 5 0 0 < ~)

(3) T iming for production and transportation: Production and t ransportat ion s tar t at the beginning of each period. (4) Input and parameter values

(a) n; 1 , 2 , 3 , . . . , 1 0 0 . (b) ~; 0 < ~ < 500. (c) m = 2

(5) Optimal inventory level 4) (a) The optimal inventory level in the retailer x is the unique solution of the equation

In other words, x is the unique root of

1 - e - ' x i - g . ~ e - 'x /~ = c t p - e ~ ( 1 - o) a(p + h)

(b) The optimal inventory level in the brmlches x r are the unique solution of the equation

5{ e ~ ( l - c~) + a L ' (£ ' , - - 0

/) } + L ' ( : r r - ( l - ~ ) ~ ( ~ 1 ) d ( t ~a (~ )ds = 0

In other words, XT is the unique root of

1 -- e - A ~ -- A ~ e - A ~

+c~ (1 _ e-A~ Ai.e-X. A25:~ -A ' )

= ap(a + 1) - (1 - c0c - c~(1 - a)Cr o ( p + h )

T a b l e 1 values in I t , I~, Is and T of the simulation 100 weeks in A = 0, 1, 2 where T = alIl + a212 + asia

at the retailer

A = I A m 2 A = 3 It 105.1 148.9 182.8 /2 17.3 20.0 24.0 /3 106.1 106.6 107.0 T 210.1 271.0 328.9

3. P r o d u c t i o n I n v e n t o r y S y s t e m

At any moment of time, the s tate of the system is described by the pair (Yt, Q*-I ), and we may define a sequence of min imum cost functions tha t satisfy

n

C,(X,,Q*-I) = m n ~ " ' M , lq, ,-qi, ,- t l ¢ ( X O L *-. . ¢

CAI[ 27:1/4-0

where

n - - I d m a z

+ E Fi'yiJ+ E P(d, l . r , , . (y , , , -d , ) i m l d t ~ O

dme tz

E P(d,) .G . (d , - J , ) %

d¢=Xn,¢+i

drtt a z

+ ~ e(~,).c,+,(x,+,,o,)} d t = O

yi.t : inventory of stage i at moment of t ime t. di : demand at moment of t ime t. P(Jj) : probability of demand d~. d~ : dt if yi j >_ dt and Yi,t if yl,t < dt Xt : the echelon stock vector, Xt ~ 0

X, = {:r, . . . . . . z . , ,} ,~:~. , < X . , . ~ Qt : production vector at moment of t ime t.

Q, = {q,,, . . . . q. , , }, Q, > o O(X,) = {Qd:~-l,~ - z~,~ _> q~,, > 0} CT+I(XT+I, QT) = 0

Table 2 indicates the assumptions for the demand distribution, holding costs, and production cost.

T a b l e 2 Demand distribution and cost conditions

Demand 0 Probability 0.5

Process Production Cost Mi

1 1 0

2 10 3 10

1 2 0.3 0.2

Holding Cost Penalty Cost F~ G 10 20 I00 3O

Through a stages series processes for n stages, success Is achieved. The lag for each process is 1. Excess demand is lost sales. Unit penalty cost is G and unit holding cost is Fi. Production cost is Mdqi,t -qi,t-1 [.

Fig.2 Production - Inventory System

Table 3 shows the optimal production schedule of the results in computat ion 12).

4. C o n c l u s i o n

The production planning in a multi-echelon production- iuventory system was discussed. The demand is stohastic, and the costs for production, holding stocks and shortage are con- sidered. Then, tile optimal production plans for each period and each production process are obtained by D.P.algorithm. These optimal solutions are decided by the states of the present stock levels and of the production levels in the last period. These will be stable over a long term. These are regarded as a Markov Process. The authors propose a method for making the optimal production plans and of calculating the expected cost by the properties of Markov Process.

204 Selected papers from the 16th Annual Conference on Computers and Industrial Engineering

Table Echc io~ Stock Prev iou i Period

Product ion

. . . . o O . ~ . o o . V

~ . ~ . 1

o

o o

. . . . . oo o o

I 0

0 0

0 I

? ' . o '

o o . ~ . ~.. ~:

,, .. ~

4 , ~ , o ~: o ~ ' ,

:

. . . . . o O . ~ . , o

~ . o . o o oo

. . . . . o 2 . ° ,

. . . . . o o °

o . . o . o ' °o

?

? 0 ~

? ? ? ?

I

i : o

,o

'o

o o

References

1) Claxk,A.J.,emd ScaxLH.;"Optimal Policies f o r a Multi- Echelon Inventory Problem," Manage,Sci.,(1960). 2) Monden,Y.;"Toyota Production System," Industrial Engi- neering and Management Press,(1983). 3) Odanaka,T.;"Optimal Inventory Processes," Katakura Li- brexy,(1985). 4) Odanaka,T.;'On Approximation Multi-Echelon Inventory Theory," Memoirs of Tokyo Metropolitan Institute of Tech- nology, No.3,(1980). 5) ZangwilI,W.I.;"Inconsistencies and Parax]oxes in ~apanese Production Theory," TIMS XXIX,Osaka,Japan,July 2 3 ,

(Z98Q). 6) Rosling,K.;'Optimal Inventory Policies for Assembly Sys- tems under Random Demand," Opns.Res.,Vol.3T,No.4,(1989). 7) P~Qewski,L.J.et al.;"Kemban, MRP, and Shaping the Man- ufacturing Environment," Manage.Sei.,VoL33,No.l,(1987), pp.39-57.

OptimM Production Scheduling Period Period Period Period Pcrmd Period Period ~ d o d Period Period

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0Ol o01 OOI OOI 001 001 o01 O0 - - I - ~ 000 001 001 nO[ 0(11 001 nol OOl 0(11 ~Ol - -0]~[ 011 01] U l l O i l 011 F -0-0 [ - - - o~ V - - ~TO~ - - -O"~T - " ~Tt~ 0]1 O i l Ot l O l l O l l I r)o[ 00~ Not o4~i O i l OlZ Ot i O[J O i l O i l I OOl OOL OOI 001 010

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o01 001 o01 oOt OOi - d o t . . . . . . . ~ t - ", oco 0Ol co l OOI 00[ OOI 0{11 00[ OOI nOi ~q 010 001 OOl OOt 001 OOI OOt OOt OCt OOt 010 go1 001 OOl 001 OOt OCt OOI 001 001 ' 001 CO1 OOt 00] 001 001 0Ol 001 001 i 010 riO0 OOI 001 oOI 00] 091 o01 001 001 0111 i 010 OOl 001 OOl 001 O01 00] 001 HOI OOl I ~10

i i i ! : i ~ : ! ! Ol1 011 Ol l O i l 011 o l l o i l L Col OCt COo O i l O i l Ot L _ _ O I t _ _ _01L_ _ 91J _ _ OLI _ -~1"~_ ~ oct n l o ]J ) [11 ~-O~J O I t (111 011 0]1 00[ 0o i Ioo I l l I l l L 0 I_1 _ _0..[[_ _ _0_[ L _ _o.J_l _ _0 i t O l l I l l I lO 010 oi0 ~ I0 Oln oi0 010 OlO ~ oco noll ooo 01o o to Or0 ~10 otO 010 01O i 000 Ooo 10O

01o 010 OlO 0 [0 r) ln o10 L--q9 - 0 oo0 000 COO 0o0 000 00o 0oo ooc ooO ooo O.901 001 OOI 001 001 OOi 001 001 001 001 ~ ' 00 ] " OOt COl OOL Q_pt .OOJ QPL ooJ O,~L .0._ol , ooo l l o E '~Fo - - - o~ ' o - - . oLo - - 0 . J9 - - - o l o - - o l o - - - 0OO- -oo ' 0 - coo I10 ' [ r o - - -~ l~J - - i Jo - - - i iO - l l . . OlO OlO ooo ooo ioo 000 uo0 000 000 OOO ~0"0- - -0~0" - - Oh*O - - ~ iTO'- - -~d~T

8) Muramatsu,R.et ed.;"Some ways to Locrcase Flexibility in Manufacturing Systems," Int,J.Produc.Res.,Vol.23,No.4, ( 1 9 8 5 ) , p p . 6 9 1 - 7 0 3 . 9 ) Oe~shwin,S.B.;'An Efficient Decomposition Method for the Approximation Evaluation of Tandem Queues with Finite Storage Space and Blocking," Opns.Res.,Vol.35,No.2,(1987), pp.354-380. 10) Deleernyder,J.L.,Hodgson,T.J.,Muller,H. & O'Grady, P.L.; "Kanban Controlled Pull Systemms: An AnMytic Approach," Manage.Sci.,Vol.35,No.9,(1989). 11) Williams,J.F.;"A Hybrid Algorithm for Simultaneous Scheduling of Prodnction and Distribution in Multi-echelon Structure," Manage.Sci.,Vol.29,No. 1,(1983),pp.77-92. 12) Masui,T.;'Stability Analysis of Optimal Production Planning for Multi-Echelon Production-Inventory System," 31MA,VoI.41,N o.4,(1990).