Color spreading algorithm for the retail industry

Computers and Industrial Engineering Vol. 23, Nos I-4, pp. 463-4456, 1992

0360-8352192 $5.00+0.00 Copyright © 1992 Pergamon Press l.,td

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~olor

Spreading

AlgorltJm for

the R ~ . a i l

Ind~

Marilyn K. Pelosi, Theresa M. Sandifer and C. Edward Sandlfer Department of Quantitative Methods and Computer Information Systems Western New England College, Springfield, MA 01119 Department of Mathematics Southern Connecticut State University, New Haven, CT 06515 Department of Mathematics and Computer Science Western Connecticut State University, Danbury, CT 06810 are costly to the company. Thus, the problem is not merely to allocate items but to spread those items across a spectrum of color and size.

One of the problems which many companies face is the distribution of inventory from one or more centrally located distribution centers. The retail industry, and in particular the women's clothing industry, faces the additional problem that not only does the merchandise have to b e d i s t r i b u t e d but it must be distributed in such a way that every store gets a reasonable spread of colors and sizes.

This paper will only address the problem of allocating the merchandise in such a way that a single store receives merchandise in a variety of different colors. The size spread allocation problem b e d e s c r l b e d at the end of the paper as a second stage to this algorithm.

This research attempts to optimize the multi-criteria allocation objective by developing a computer algorithm. In order to be implemented any algorithm developed must be computationally efficient due to the size of the industry and other system related programming constraints. The algorithm which was developed not only provided a solution which was a marked improvement regarding the color spread of the merchandise but was also efficient enough to be immediately implemented on a national basis.

The following t e r ~ i n o l o g y w i l l b e u s e d to d e s c r l b e t h e existing a l g o r l t h m a n d the new algorithm:

~altlple: a bundle of 6 identical (in terms of color and style) items Store

D ~ : the total number of multlples of a given style that a given store is to receive.

Per~

o ~ p in a ~ l o ~ : p e r c e n t of the total number of multiples in a given color

L

oolor quantity: the color which is in largest supply (i.e. the color with largest percent ownership)

l~r~blmaSt.a~

The problem facing this particular company is one of allocating merchandise to all of their stores. The stores are located all across the country and the merchandise is shipped from a single location. Since the merchandise is clothing, there are two important characteristics which must be considered when allocating items to stores, size and color. The problem at hand is to distribute the merchandise from the central location in such a way that each individual store receives a good selection of colors. For example, suppose the item to be allocated is a particular style dress which comes in 6 different colors. It would not be desirable to ship an individual store 12 of these dresses all in black. Clearly it is also u~desirable to ship 12 black dresses all in size 5. In this case the store would not have enough variety in terms of size or color to adequately meet customer demand. This may lead to unsold inventory or Inter-store shipments, both of which

~

Need:

A measure of the current unsatisfied style allocatlon for a given store

The restrictions on the problem and its solution below:

allooatlon are given

Total number of multiples Color distribution (at the corporate level) Store demand

The existing algorithm focused on supplying the larger stores with a good spread of colors, thus leaving little color variation f o r t he smaller stores. Furthermore, from a retail

463

464

Proceedings of the 14th Annual Conference on Computers and Industrial Engineering

desirable than others and these colors were allocated first to the larger stores. The end result of this procedure was that the smaller stores in the chain had small inventory resulting in poor customer selection leading to lower sales volume. This could cause a downward spiral ultimately ending in store closure. Step

1.

Starting with the largest color quantity, multiply each store's total demand by the percent of ownership and subtract the number of multiples which have already been allocated. This is defined as the store's need.

and the store demands are as follows: Store

Demand (multiples)

1 2 3 4 5 6

6 3 3 4 5 3

Start:

Store

#

Need 2.74 1.37 1.37 1.83 2,29 1.37

Percent ownership is calculated by adding the ownership for the current color to the ownership of all previously allocated colors. Step 2:

Sort the stores on the basis of need, largest to smallest. Then, starting with the largest need allocate one multiple to each store working down to the smallest need

Step 3:

If there are more multiples in a given color than stores, then after a pass through the list the needs are recalculated and the list resorted. Allocation continues is this manner until the total color allocation for the region has been depleted.

Step 4:

If there are more colors to be allocated then go to Step #1. If this is the last color, stop.

Note:

If all stores have received color #1, subtract their allocation from the new need.

Allocate first color (Red) Percent Ownership = 48.53% The final allocation for this problem is shown in Table 1.

Store # 1 5 4 2 3 6 Table 1.

Step 5:

Allocate multiples until the total color allocation for the region has been depleted.

A small example will serve to illustrate the basis of the existing algorithm and a typical solution. In a certain region with six stores, a total of 24 multiples are to be allocated. The color distribution for the multiples is: Color Red Blue Green Black

Multiples

Available 11 5 5 3

Oemand 6 5 4 3 3 3

# Red 2 2 2 2 2 1

# Blue 1 1 1 0 1 1

# Green 1 1 1 1 0 1

# Black 2 1 0 0 0 0

Final A l l o c a t i o n for Sample Problem Using Exlsting Algorithm

As the results from the previous example illustrate, the algorithm concentrates too much on matching percent of store demand to percent ownership in the various colors. This tends to "clump" the allocations and hinders the spread. In addition, the stores with the larger demands control the algorithm, causing smaller stores to be shipped the undesirable colors.

An algorithm was developed which allocates colors to stores based on currant unfilled demand without trying to match this demand to the percent ownership of the colors. The algorithm was based on the work done by Balinski and Young (1) on apportionment methods for the House of Representatives. Particularly the algorithm is developed from the point of view of the desirable properties of the solution.

PELOSI et al.: Color Spreading Algorithm

The new a l g o r i t h m is o u t l i n e d in the f o l l o w i n g steps: Step 1:

Sort the stores in o r d e r of d e c r e a s i n g demand. Then, s t a r t i n g w i t h the largest color quantity, a l l o c a t e one m u l t i p l e to each store. If the n u m b e r of stores exceeds the c o l o r allocation, then go to the next largest color q u a n t i t y and c o n t i n u e in this m a n n e r until each store has been allocated e x a c t l y one multiple.

Step 2:

For each store calculate the proportion, p, of the total demand w h i c h is c u r r e n t l y filled (Need = l-p). Choose the store with the minimum value of p and a l l o c a t e one m u l t i p l e of the c u r r e n t c o l o r to that store. When all m u l t i p l e s of the current color have been allocated, go to the next largest c o l o r quantity.

Step

3:

If all m u l t i p l e s have not been allocated, r e c a l c u l a t e the v a l u e s of p and go to step 2. If all m u l t i p l e s have been allocated, stop. In the case of a tie in the value of p, allocate in the following order:

Note 1:

1. To the store with a zero (or alternatively, minimum) a l l o c a t i o n of the current color) 2. To the store with largest total demand Note

2:

the

If a g r o u p of stores are e q u i v a l e n t on all criteria, the next store to be given a multiple is chosen at r a n d o m from among them.

The n e w a l g o r i t h m is i l l u s t r a t e d u s i n g the same sample problem. Several i n t e r m e d i a t e steps are given with the final a l l o c a t i o n shown in Table 2. Start: All stores have r e c e i v e d m u l t i p l e of the c u r r e n t color (red)

1

Next:

Store #

1

5

4

6

2

3

6

5

4

3

3

3

Allocate 1 multiple of current color (red) to Store #1

Store # p

465

1

5

4

6

2

3

2_ 6

1 5

1 4

1 3

! 3

1 3

Alocate 1 multiple of current color (red) to Store #5 Store #

1

5

4

6

2

3

P

2 6

2 5

1 4

1 3

1 3

1 3

Allocate 1 multiple of current color (red) to Store #4 Store #

1

5

4

6

2

3

P

~ 6

1

!

!

!

I

5

4

3

3

3

:

Allocate one multiple of current color (red) to Store #1 Note: There is a tie among Stores 1, 6, 2 and 3,but store 1 has the largest total demand. Store # 1 5 4 2 3 6

Demand 6 5 4 3 3 3

T a b l e 2:

# Red 3 2 2 2 1 1

# Blue 1 1 1 0 1 1

# Green 1 1 0 1 1 1

# Black 1 1 1 0 0 0

Final A l l o c a t i o n £o¢ S a m p l e

~ I ~ H i o h ,w~,ints o f

using I w the

N~

Al~Itba

Alcmri~m

The new algorithm first g u a r a n t e e s that every store gets one m u l t i p l e of a d e s i r a b l e c o l o r before c o n s i d e r i n g store demands. This is the b e g i n n i n g of e n s u r i n g a spread of color rather than a c l u m p i n g effect. In addition, in the case of a tie, the a l g o r i t h m "hits" the s m a l l e r stores before c o n t i n u i n g to a l l o c a t e on the basis of demand. Thus, small stores have a chance to receive colors w h i c h are a l l o c a t e d early in the process. ie, the desirable colors.

~t~wm

of ml~mri~IMm

Both algorithms were run on e v a r i e t y of scenarios based on actual data. If one c o n s i d e r s the final a l l o c a t i o n as a m a t r i x w i t h stores as the rows and colors as columns, then a m e a s u r e of spread is g i v e n b y the number of n o n - z e r o entries in the matrix. Further, p l a c e m e n t of zeroes provides an additional measure.

466

Proceedings of the 14th Annual Conference on Computers and Industrial Engineering

In all but two cases the new a l g o r l t h m w a s superior to the existing algorithm. The type of problem for which the number of zero entires was the same was one in which the allocations were sparse, that is, certain colors were in extremely tight suppy. Table 3 gives a comparison of the two algorithms for a variety of problems. The entries in the table indicate the total number of 0 entries resulting from use with each algorithm. The size of each problem is also indicated as (# of stores, # of colors, total multiples)

Problem Size (#s,#cr#m) ( 6~ 4 , 24)

(18~4, 36) (10, 6 ,300) (15~ 7 r2901 (24, 6 ,144) (20r101400) (20,10,400) (24,10,249) T a b l e 3:

Old Algorithm

New Algorithm

6 5 29 11 16 19 23 129

5 5 17 5 16 0 2 111

Comparison o f A l g o r i t h n s

The algorithm developed was implemented by a retail chain of women's clothing stores. The algorithm not only provided better results, but was more efficient when coded in COBOL which was a requirement of the project.

size

a~PEc~blem

In addition to the problem of allocatlng a good color spread to each of the retail stores, it is necessary to consider the sizes associated with the items. The problem of spreading the sizes is one which involves allocating sizes both across the stores in general and within each store across the existing color spread. That is, not only is it required that each store get a good representation of available sizes, but the sizes must be allocated across the colors so that all of the size tens are not blue, etc.

Sizes are available in curves within a multiple. Each size curve has a particular distribution of sizes associated with it, and the number of available curves for any given item can range from 2 to 4. The algorithm for allocating sizes to the stores is used after the color spread problem has been solved and is developing to be conslderably more complex than the color spread algorithm.

i. Balinski, M.L. and Young, H.P., "Apportionment Schemes and the Quota Method", ~ A m a E i ~ M K ~ h M Q m ~ l y , 84

(1977) 450-45s. 2.

Balinski,

Fair ~

t

i

M.L.

~

:

and

Young,

H.P.,

Hoetlng the Ideal

of O n e N s n , O n o V o t e , Yale University Press, New Haven, 1982. 3. Garfunkel, S., et a•, FOr A11 Prm~.~l ~ , W.H. Freeman and Company, New York, 1991.