inventory items

207

Deterministic lot-sizing for coordinated families of production/inventory items Amiya

K.

CHAKRAVARTY

Department of Management, Washington State University, Pullman, WA 99164, U.S.A. Received April 1982 Revised February 1983

Coordination of order placing points, in a multi-item environment, becomes desirable as it reduces the set-up costs and eases the problem of order-control implementation. Because of the computational complexities most of the analysis in the literature is confined to single families of items. In this paper, we establish the basic property of item consecutiveness for the multi-family situation which facilitates a shortest-path formulation of optimal families. We also develop an efficient algorithm for lot size computations of a single family to be incorporated in the shortest-path model.

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

Coordination of order placing points, in a multi-item environment becomes desirable as it reduces the set-up costs and eases the problems with order control implementation. Goyal [3,4] and Silver [9] have approached this problem by assuming that all the items belong to the same 'family', that is, the order-interval of an item is an integer (>/1) multiple of the order-interval of the family. The single family approach, while reducing the major set-up cost of the family, unnecessarily penalizes the individual items as they are made to deviate too far from their optimum operating points. A multi-family policy permitting localized

groups (families) of items would, therefore, be a more general approach. Such a policy allows formation of several families when the family set-up cost is not high. This approach, however, has n o t received much attention because of its inherent computational complexities, since there are many ways of grouping the items. As in Fig. 1, if the 9 items shown are treated as a single family, the order interval for the ith item is KiT where T is the family's order interval. If the items are divided into three families as in l(b), the order interval for thejth item in ith family is KijTi where T~ is the ith family's order interval. In this paper, we vastly simplify the grouping problem by establishing that the optimal groups are consecutive in the ratio AJh,D,. Using this property, a shortest-path approach to create the optimal families and their lot sizes is developed. We also develope an efficient algorithm for the single family lot sizing to be incorporated in our multi-family shortest path model and report on the computational performance of the multi-family approach. The remainder of this paper is divided into five sections. In the next two sections we develop the single-family lot sizing algorithm. This is followed by the discussion of multi-family approach and the consecutiveness property. The integrated shortest-path model and the computation performance are presented in the next two sections before concluding in the last section.

2. The

I

I K1 K2 K3 K4 K5 K6 K7 K8 K9 I i

2

T1

4

5

6

7

8

T2

Kll

K12 K13 K14

1

2

3

3

4

IK2zK22]

9

(a)

I

T3

I

K31 K32 K33

i (b}

7

I

8

9

Fig. 1. North-Holland European Journal of Operational Research 17 (1984) 207-214

single-family

model

2.1. Assumptions and notations The following assumptions are made: planning horizon is infinite, - demand per period is constant, - no shortages are allowed, - lead time is zero. The last two assumptions can, however, be relaxed with minor modifications to the model. In keeping with the mainstream research [3,9], we have considered the set-up cost to be made up

-

0377-2217/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)

.4. K. Chakravarty / Deterministic lot-sizing for coordinated families

208

of a minor set-up cost for each individual item and a major set-up cost for the family, irrespective of the composition of the family. The following notations are used: Order interval for the family, D, = Demand rate of the ith item, h i = Holding cost per unit time per unit, A i = Set up cost for the ith item, A 0 ---- Set up cost for the family, r, = Order interval for the ith item, and

~ K j ( K j + a ) . hjvj Aj

(4)

From (3) we can easily derive the condition

T

Y'.KihiD i

hJDJ . Kj( Kj _ I ) ~

,

Aj

<.Kj(Kj+,). hjDj Aj

T~ = KiT, K i a positive integer. or

2.2. The cost equations

hjDj

The inventory related cost per unit time for the family of n items can be expressed as F1

K j ( K j - 1)--~--g

2

h/)j

Top,

Aj

or

A o + ~,(Ai/Ki) n C'(T) = i + ~, ½. KiTh,D i. T i

(1)

For fixed values of K / s C'(T) can be minimized by equating OC'(T)/OT= 0 and the minimum cost can be expressed as

Kj( Kj- 1)/,) <_.1 / r L ~ Kj( Kj + ~)/t~

where t j = ~(2Aj/hjDj) is the 'natural' order interval for itemj. The inequalities (4) and (5) can be called the strong and weak optimality conditions, respectively. From (5) it follows that for any value of T = T~, if

Kj( K j - 1)/t 2 <~1 / r 2 ~ Ky( Kj + 1)/t 2 and

(5)

(6)

for all j, then we have a local minimum for T = T~.

2.4. Computation of Kj And the global optimum is obtained by choosing K~'s such that

C=

Ao +

"-~-

KihiDi

(3)

i

is minimized.

2.3. Optimality conditions From(3) the optimality condition for Kj can be written as

C ( r j - 1)> c ( r , ) < C(Kj + 1) which after simplication can be written as

KihiDi hgDJ.Kj(Kj_I) < Aj

i*j (Ao+ ~,_,AjKi)

Computation of Kj using (6) is not straightforward. An easily programmable expression for Kj can be developed as shown below: Letting I be the maximum integer not exceeding tg/T~, we can write tJTo = I + F, where F is a fraction. From expression (6) it can be verified that K j = I or I + 1 depending on the value of F. Specifically, Kj = I if I(I + 1) >/t2/Tj 2, otherwise

Kj=I+I. Substituting ty/T a = I + F, the condition for Kj can be written as

= I

I ( I + I)>~(I+ F) 2 or

F2 + 2 I F - I < O

A.K. Chakravarty / Deterministic lot. sizing for coordinated families

209

Step 2. With N, = max(n j), compute

or

F<~-1+¢(12+I).

[N"+I-¢(IZ+I)+I-8],

K?= I "J Therefore, if the amount 1 - { - 1 + ( ~ + 1) } is added to I + F and the sum is truncated, the integer value obtained would be Kj. Thus, Kj = [tj/T, + 1 + I - ( ~ 5 + I ) ] where [x] denotes the integer part of x. Note that when tj/To= ¢ ( I 2 + I ) , the above expression sets Kj = I + 1. However, from (6) it is clear that both I and I + 1 are feasible values of Kj in such a case. In fact, in the solution procedure adopted, subsequently, we require that Kj be I and not I + 1 in such a situation. Therefore we add a small quantity - 8 of the order of 10 -~° to the above expression before truncation so as to adjust the value of Kj appropriately. Hence, the final expression for Kj can be written as

+1-8+I-¢(12+I)

,

I=

where I = [NJ%], [x] is the integer part of x and 8 very a small number of the order of 10 -z°. Compute N ~ , = (EflC~hjDj)/2(Ao + E j A / K 7) as in Goyal [4]. Step 3. For each j compute aj, -- K ( K - 1)n~, 2 K = 1, 2, 3 . . . . . etc., such that NmZih < a j, < N ~2 x and arrange the %, in ascending order with/3~ = least value of a j, and/3q = highest ajk, for some q. Start the next phase with/3p Step 4. For some /3, compute Kj as in Step 1 with No -- V~," Step 5. Compute

Nz=

E K , h,Dj i

Y

. If N 2 >/3, go to Step 8.

2.5. Optimality test The algorithm is based on determining the values of K / s from (6) for a certain value of T = To and then subjecting T and Kj's to the optimality conditions (5) and (4) in that order. Letting a r = K j ( K g + 1)/t~, Kj= 1, 2, 3 . . . . . etc. For some T~ = ar and T~t, determined from (5) with Kj = Kj(a~) as in (6), if there exists an % * Ot r within the range ar ~ T~t then the weak optimality conditions will be deemed violated. However, if this is not the case the strong optimality condition (3) can be tested for eachj to ensure optimality. In case of violation of condition (4) To is reset to a,, within the range % --* T~t such that a,, is closest to To~t and the tests are repeated.

Step 6. I f ( Y - Kj hj Dj)/( Z - 2. A / K j ) violates the optimality condition (4) for anyj, go to Step 7. Otherwise the current solution is optimal. Step 7. Modify /3, to /3, such that K = I + I (/3, >/3~), and go to Step 5. Step 8. Modify/3, to/3, such that/3,-1 < N2 < fl~ and go to Step 5. 3.2. A n example We apply our algorithm to a variant of Goyal's [3] example, shown in Table 1. The values of n 2, shown in Table 1 are computed as

n2 = 1 = hiD i ' t) 2Aj' 2O

3. Lot-sizing computations for a single family

E h,O, j=l _ 143 000 Nm~m-(2 A 0+i=l~ - - 2) ( 4 5 + 1 7 3 ) = 3 2 8 " 2 1 A ;

3.1. The algorithm Step 1. Compute njz = I / t ) = hjDJ2Aj, ~,h,D,

Nm2in

1

T,~.,,

i

(

2 A o + ~A, i

)

Nm2axcan be computed as in Steps 2 and 3 of the algorithm. However, because of the small size of the problem we chose not to compute Nm2ax and instead set it equal to maxj(n})-- 900. To obtain the values of 13, in ascending order,

A.K. Chakravarty / Deterministic lot-sizing for coordinated families

210

Table 3

Table 1 2

,8,

Itemj

Dj

hj

Aj

nj

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

500 200 900 1200 200 400 1000 2000 600 60 150 250 500 100 1200 1000 800 1000 35 1000

13 16 20 7.5 15 10 7.5 5 10 5 12 10 10 6 10 15 12.5 8 20 20

10 7 10 11 8 10 12 12 6 2 3 5 8 7 11 10 7 10 4 20

325 228.5 900 409.1 187.5 200 312.5 416.66 500 75 300 250 312.5 42.86 545.45 750 714.29 400 87.5 500

Item groups with

375 400 450

N~2

Kj ~ 1

Kj= 2

Kj = 3

1-17 1-16 1-15

18,19 17-19 16-19

20 20 20

377.72 407.8 415.34

328.21 ~
For the first value of 13 = 375, the values of Kj obtained as in Step 4 of the algorithm are shown in the first row of Table 3. The corresponding value of Ni2 computed as in Step 5 is 377.72. Since N~2 = 377.72 is not within the range 328.21 to 375, /3i = 375 is not optimal. Repeating the above steps for fl, = 400, we have N E = 407.8 as shown in row 2 in Table 3. Again, N, 2 = 407.8 is not within the range 375 to 400. Proceeding in this manner, we find that for fl, = 450, N/2 = 415.34 and it is within the range 400 to 450. Hence N,2 = 415.34 with Kj values as in row 3 could be optimal. Now for f l = 4 5 0 , Y-- Ej_IKjhjD 20 j = 152300 20 and Z = 2(A o + Ej,~A~/Kj) = 366.66. Check for optimality as in Step 6, i.e.,

The feasible values, starting from j = 1 to j = 20 are listed in Table 2. Thus, the values of fl~ are

hjDj KI( KJ-') Aj

A 0 = 45

the values of that

Kj(Kj-

1)n ] for a l l j are listed such

152300 - K~hjDj 366.6-2AJKj

375,400,450,457,500,514.32,525,600,625,650, 800, 818.2,833.32,857.2. Table 2 Feasible values of Kj( K~ - l)n~

j 1 2 4 5 6 7 8 10 11 12 13 14 18 19

It can be verified that the above condition is satisfied for allj. Hence K / s corresponding to row 3 in Table 3 are the optimal Ki.

K/Kj-1) 2

6

Aj

12

20

650 457 818.2 375 400 625 833.32

1 optimal T = - ~ = / v

-

-

1

~/415.34

=

0.049.

Thus the lot size for t h e j t h item = 0.049 for items 450

600 500 625 514.32 800 525

1 tol5,

Qj=0.049Dj,

16 to 19,

Qj=0.098Dj,

20,

Qj=0.147 ~.

KjDj, i.e.,

857.2

A comparison of our procedure with that of Goyal [4] is shown in Table 4.

A.K. Chakravarty / Deterministic lot -sizing for coordinated families

211

Table 4 Comparison of single family algorithms ~: items

Goyal's optimal

20 40 80 120 160

Modified algorithm

CPU times (see)

Statements executed

CPU time (see)

Statements executed

0.11 0.35 1.88 4.37 7.34

15 208 52 852 304 926 705 257 1158408

0.02 0.04 0.08 0.08 0.13

731 1502 3210 2984 4983

Thus, for the consecutive partition ( F t, F 2)

4. The multi-family model

PaX2 C( F,, F2) = PtXt + P2X2 + RP2X, "~ R '

4.1. Consecutiveness of families For the multi-family case, the cost to be minimized for fixed values of K, can be expressed as

and for the non-consecutive partition (/:~, F~) P'X:

. . +. P~X~ . . +. R . P~X{ + - - 1~ , 2 C( F~, F~) = PtX~

I(

A,

/'+12

/ 1

'='

(7)

Letting P j - - A o + E,+eA+/K, and Xs E,+FK+h+D, the function to ge minimized in K, s

(10)

where P; + P~ --- Pt + P2 and X~ + X~ = X 1 + X 2. We shall let P(=P1 +SA and X ~ = X I + 6 x where 8.4 and 8x can be assumed to be the exchange values of some items(s) between F l and F 2. Obviously, for the consecutiveness to be optimal

can ge written as

c(~,&)~c(~,~),

C'=~

where FI', F~ can be formed from various combinations of ~x and 8A. It is shown in the Appendix that the optimal families F 1 and F2 are consecutive in x+/A,, i.e.,

P i/2. (,x,) j-I

In what follows consecutiveness for then be generalized For two families

we establish the property of the two-family case which can to m families. we can write

c ' = ( e , x , ) '/2 + (P:X~) t/2.

(8)

K,2h i D J A c It would appear that since Ki's are unknown, consecutiveness by K~(h,Di/A ) cannot be ensured. Consider two families F t and F 2 such that hiDt/A ` ~ hjOj/~j.

i~C

Substituting R 2 = P1 )(2/1'2 Xt in (8): 1 c = ( c ' ) ~ = e, xt + e~x~ + R & x , + - ~ e , x ~ .

Determine p and q such that

Assume that the items are divided into the above two families F~, F 2 such that

K, 2 hiDi •

hjDj A, <~K2 Aj

j~F:

hpD? hiD: , Ap " > ~ Aj

j~F 2

and (9)

hqDq

hiD i A----~>~---~i, i ~ F , .

Assuming K,'s to be fixed and letting x, =

K,2h,D, (9) can be written as x~/A, <~xj/Aj, i ~ F I, j~V2.

Now since the minimum value of K,'s in any family is 1, it is clear, considering the minimum

212

A. K Chakravarty / Deterministic lot- sizing for coordinated families

Let F ( N ) = minimum cost of the allocating N items to an optimal number of families. Then

possible value of K T h j D / A j in F2, that minimum ( K f h j D / A j ) = hpD?/Ap .

j ~ F2

F(O) = O,

p ~ r2

F(1) = (A o + A , / K 1)t/2( K, h t D, ),/2, Goyal [3,4] has shown that the optimum value of family order interval always exceeds the lowest independent order cycle in the family, i.e.,

F(n)=Minimum

Min

Ao+ i=n-r+ l

g

Aq

To~,' >/min t2 = h ~Dq "

×

Now

+ F(n - r ) )

where n = 2, 3 . . . . . N, and

Minimum(Ao+ i/'j2 (

1 Aq K T / t ) = r 2 , <~ hqDq'

K,,K 2..... K,

therefore, for family F I :

i=n-r+ l Ki ]

X

KihiD i i=

max( ri2h, Di/A i) <~h q Oq/hq. Thus, in general, ranking of the items by hiDi/A, will also ensure the family consecutives by Ki2hiDJAi. S i n c e t h e c o s t o f m g r o u p s Y~",~I(PiXi) I/2 i s

+1

is determined as in Section 3 where r items are chosen consecutively from a list of items ordered by h y D / A i values.

separable in m groups, the consecutiveness property of two groups can be easily generalized to m

5. Computationexperience

g r o u p s [1].

The single-family and the multi-family models were programmed on an Amdahl time-sharing computer system and the results are tabulated in Table 5. In all the five single-family problems the solution from our model was the same as that obtained from Goyal's. Superiority of computation time of our model is clearly borne out in Table 4.

4.2. Determination of optimal families by shortest path

For N items, the shortest path problem, to create the optimal number of families, can be set up as follows.

Table 5 Comparison of single-family and multi-family models Standard deviation

a! 5 10 15 20

Multi-family model

Single-family model A0

2

10

Cost Number of groups

19797-8 1

20189.7

Cost Number of groups

19745-6 1

20158-3

Cost Number of groups

19653.5 1

20094.3

Cost Number of groups

19418.3 1

19934.6

1

1

1

1

2

10

19520.3 2

20171.5 2

19394.3 3

20107.8 2

19170.5 3

20003.1 2

18968.7 3

19860" 2 2

A.K. Chakravarty / Deterministic lot- sizingfor coordinatedfamilies Table 6 Computation time for multi-family model items

Computation time

(sec) 50 100 150 200

0.07 0.35 0.97 1.93

In Table 5, the overall cost of grouping for the single-family model is compared with the multifamily model for two values of A 0 and various data pattern. The data pattern is designated by the standard deviation of h D / A ( = f ) : Several conclusions are apparent from Table 5. Firstly, as the standard deviation of the data (o/) increases, the number of optimal families increase, the cost of grouping for both the models reduce and the superiority of multi-family over singlefamily model increases. Secondly, as A 0 increases the superiority of multi-family over single-family model diminishes as the number of optimal families approaches unity. For most production problems A 0, signifying a changeover cost, is relatively high--whereas for most replenishment problems A 0 is fairly small. Finally in Table 6, we show the computational effectiveness of the multi-family model. The order of computation time seem to exceed N: but not as high as N s.

6. Conclusions

We have established the consecutiveness property of an optimal coordinated m-family inventory problem. This consecutive can be extended to most of the application areas such as the multi-echelon assembly system and the warehouse retailer system. Our computational results clearly establish the applicability of our procedure in a real-world environment.

213

that C ( F I " , F2")~< C(FI', F2'). Proof. Let Xr Xj --, r~FI,j~F2, Aj

for alli

and Xr

Xj >/--, Aj

reFl,jeF1,

forallj.

Non-consecutive groups (FI', F2') can be formed by exchanging the rth and kth items between the groups. Assuming that C(F1, F2) > C(FI', F2'), i.e., 8C' = C(FI', F2') - C(F1, F2) < 0. Substituting for P(, P~, X~ and X~ in (10) we have C ( F I ' , F2')-- (V t + S A ) ( X , + S x )

+ ( e2 - ~A )( Xs - 8x ) 1 + T C e , + 8A)(Xs 8~) -

+ R'(es - ~A)(x, + 8x). Therefore, 8 C = ( R ' - 1)[ P, S x - &4( X s - 8x) R' + PsSx - &4(x~ +

8x)} }

,. r P ~ x s


Letting P18x - 8A( X= - 8x) = u I and P28x 8A( X 1 + 8x) = u s we can write ~C=(R'-1)(~, +(R-

+ u2)

R'){ P ' X s - p 2 x ' }

(A2)

From (A2) it is clear that for 8 C < 0 , ( R ' 1 ) ( u J R ' + u2) < 0 since (R - R ' X P I X s / R R'P2X1)> O, or u l / R ' + u s < O, i.e., x , . W - A~.S <~x , W

-

ArS

,

where Appendix

w = ( N"+ P) s

Theorem. I f C ( F l, F s) > C ( F I ' , F2') where (FI', F2') is not consecutive then there must exist at least another consecutive partition ( FI", F2") such

and S = X 2 - xk + x , R' t- X1 + Xk -- X,.

(A3)

A.K. Chakravarty/ Deterministiclot-sizingfor coordinatedfamilies

214

F r o m (A3) the following two inequalities follow. Either

can be argued similarly, i.e., now ( F I " ) ' = F1 - q - r and ( F 2 " ) ' = F2 + q + r.

xj,.W- A , S > 0

R e m a r k 2. It is possible that in the above analysis when the item k is transferred to F 1 instead of exchange of items k and r, the value of R m a y change to R t, i.e., R t is preferred to R' for the case of transfer. Letting

and

x , W - ArS > 0,

(A4)

or

x~W- A,S < 0

ad

X r W - - ArS < 0.

(A5)

N o t e that the possibility of x k W - AkS < 0 and x,W-ArS>O is ruled out because xj,/Ak>_. x , / A r . F r o m (A3) and (A4) it follows that - ( x , W - A , S ) <.

- xr)W- (A, - A,)S. (A6)

Similarly, from (A3) and (A5) it follows that

x,W-A,S~

(x~,-x,)W-(A,-Ar)S.

(A7)

Also since S ' ~ S ~< S", where S' ----( X 2 + x , ) / R ' + X) - x, and S" = (X: - x k ) / R ' + X) + x~ inequality (A3), using (A6) and (A7), can be rewritten as either

(-xr)W-(-A,)S'

~ (x,-

xr)W-(A k - A r ) s , (A8)

or

xkW-A~S"<~(x,-x,)W-(Ak-A,)S.

(A9)

F o r m (A8) and (A9) we can conclude that C ( F I " , F2")~< C(FI', F2'), where F I " = F1 - r or F l + k and F 2 " = F 2 + r or F 2 - k , i.e., ( F I " , F2") is consecutive. R e m a r k 1, If Xp/At, < x k / A k ( p ~ F2, k ~ F2) and if inequality (A5) applies, then it can be shown that x p W - A e S " ~ < O and (x, + x t , ) W - ( A t + Ap)S" ~ x ~ W - A , S " . Since (S")' >1S", where (S")' = (X2 - ( x , + Xp))/R' + Xl + (x~ + x,). we can write ( x , + x e ) W - (A~ + A p ) ( S " ) ' <~x ~ W A , S " , i.e., C ( ( F I " ) ' , ( F 2 " ) ' ) ~< C ( F I " , F2") where ( F I " ) ' = F I " + p = F1 + p + k and ( F 2 " ) ' = F 2 "

-p=F1 -p-k. The case for Xq/Aq~ Xr/A r ( q ~ F1. r e F1)

8Ct=(R,-1)

~-~-+ I"2

where Vt = P l x , - A , ( X 2 - x , ) V 2 = P2x, - A , ( X 1 + x , ) , and since R 1 is preferred over R', it can be easily shown that 8Cj < 8C, i.e., a transfer of item k, will be even m o r e preferable to an exchange. F r o m the above theorem it follows that the optimal families F 1 and F 2 would be consecutive in x i / A i, i.e., K2hiDi/Ai. References

[1] A.K. Chakravarty et al., A partitioning problem with additive objective with an application to optimal inventory grouping for joint replenishments, OperationsRes. 30 (1982) 1018-1022. [2] W.B. Crowston, M. Wagner and J.F. Williams, Economic lot size determination in multi-stage assembly systems, Management Sei. 19 (1973) 517-527. [3] S.K. Goyal, Determination of optimum packaging frequency of items jointly replenished, Management Sci. 21 (4) (1974) 436-443. [4] S.K. Goyak Note on determination of optimum packaging frequency of items jointly replenished, Management Sci. 22 (3) (1975) 386. [5] S. Love, A facilities in series inventory model with nested schedules, Management Sci. 18 (1972) 327-338. [6] E. Page and R.J. Paul, Multi-product inventory situations withone restriction, Operations Res. Quart. 27 (1978) 815. [7] L.B. Schwarz and L. Schrage, Optimal and system myopic policies for multi-echelon production/inventory assembly systems, Management Sci. 21 (11) (1975) 1285. [8] L.B. Schwarz, A simple continuous review deterministic one warehouse N retailer inventory problem, Management Sci. 19 (5) (1973) 555-566. [9] E.A. Silver, A simple method of determining order quantities in .joint replenishments under deterministic demand, Management Sei. 22 (1976) 1351.