Abstract With the significant role that warehouse plays in

connection to suppliers, distributors, and clients, and considering the costliness of storage systems, efforts have focused on the reduction of procedural expenses. Among the most significant expenses, one involves the distance traveled by operators or S/R machinery for the selection of ordered items; the common solution to which has been through the classification of stored items. Previous articles have focused on linear solutions to the abovementioned problem, such as P-Median. This article shall focus on the problems arising from the said solution, and instead a non-linear method proposed. Further, a heuristic algorithm is produced for the solution that will be compared to similar solutions through Lingo, proving acceptable results. Keywords; Clustering, Heuristic, P-Median, Supply Chain

1. INTRODUCTION

In the competitive markets of today, client satisfaction plays major role, at times altering the direction of market strategies entirely. With the evolution of the market economy, the demands of the clientele have diversified, conversely building up minimum standard requirements. This has further led to the diversification of demands and orders, where in turn shorten order lists of singular items. On the other hand, due to a tighter competition in the market, client satisfaction of product and service proves vital to businesses. Among service quality factors, one can mention

delivery time. Delay in the delivery of services or goods may result in the loss of market share. For this, warehouse systems have developed in line with supply chains, in order to strengthen the bond between clientele, supplier, and distributor. This has lead business managers to seek organizations based on diversified low quantity demand lists, with short delivery time spans. This is where warehouse systems can provide major assistance to reduce overall costs and increase market share through the reduction of In the competitive markets of today, client satisfaction plays major role, at times altering the direction of market strategies entirely. With the evolution of the market economy, the demands of the clientele have diversified, conversely building up minimum standard requirements. This has further led to the diversification of demands and orders, where in turn shorten order lists of singular items. On the other hand, due to a tighter competition in the market, client satisfaction of product and service proves vital to businesses. Among service quality factors, one can mention delivery time. Delay in the delivery of services or goods may result in the loss of market share. For this, warehouse systems have developed in line with supply chains, in order to strengthen the bond between clientele, supplier, and distributor. This has lead business managers to seek organizations based on diversified low quantity demand lists, with short delivery time spans. This is where warehouse systems can provide major assistance to reduce overall costs and increase market share

Items in Supply Chain Management A Non-Linear Model for the Classification of Stored

Faculty of Computer Science and Faculty of Industrial Faculty of Computer Science and Seyed Yaser Bozorgi Rad Mir Abbas Bozorgi Rad Mohammad Ishak Desa

Information Systems Engineering Information Systems Universiti Teknologi Malaysia Sharif university of Universiti Teknologi Malaysia 81310 Skudai,Johor, Malaysia technology,Tehran, Iran 81310 Skudai,Johor, Malaysia

mishak@utm.m abbas.bozorgirad@gmail.comy.bozorgi.r@gmail.com

Sarah Behnam Sina Lessanibahri Faculty of Industrial Engineering

Sharif university of technology Tehran, Iran

sarah.behnam gmail.com

Faculty of Industrial Engineering Iran University of Science and Technology

Tehran, Iran sinal essanyy ahoo.com

1636978-1-4244-6716-7/10/$26.00©2010 IEEE

through the reduction of delivery time, existing quantity, and costs, and improvement of service quality. Yet among the concerns of modern managers is that the warehouse system itself can incur detrimental costs upon an organization. Warehouse management includes all transportations of goods between warehouses and distribution outlets, which in turn comprise of receiving of goods, storage, order selection, arrangement, and delivery. Indeed the task handles a significant portion of all goods transportation activities, resulting in costly operations. One way to reduce operation costs is to reduce goods transportations. To obtain this goal, an analysis of the individual activities in the warehouse must be conducted, in order to discover efficient ways to reduce costs with particular focus on transportation. To reduce warehouse costs –including transportation expenses- the following basic methods have been in use:

1. customer order invoice classification 2. identification of routs in the selection of goods 3. zoning of the warehouse 4. classification of existing stocks [6,7]

The purpose of this paper is to analyze the first item of the above. The next subtopic shall delve into the issue classification of stocks, reviewing a number of proposed methods and offering some enhancement techniques to the pertinent ways.

1.1 CLASSIFICATION OF EXISTING STOCKS

With the most recent developments in industry and market, organizations have moved from indirect to direct, focused distribution. This forced business institutions to pay attention to warehouse stockpiling, leading to organized arrangement of warehouse stocks. Hausman et al have proposed the following three methods for space allocation to stocks:

1. random storage 2. specialized storage 3. storage by class [6,8]

The first method comprises of storage of goods on random basis, with no predetermined plan; the second method allocates particular spaces to each item, and finally the third method classifies items before allocating storage spaces to them. Indeed the first two approaches are mere fragments of the third. In the first method, a single class is considered for all items, while in the second, a different class is attributed to each. In the third approach, after classifying the items they will be sent to their specified locations in the warehouse. Among the most applicable measures for this, one can name COI (Cube per Index), defined through the ratio of required item space to the volume of market demand. The smaller the ratio, the closer the item will be stored to the entry ports of the warehouse. The significance of COI is in the fact that it helps minimize required space per item, in addition to the fact that a solution will be reached more easily. Nevertheless the result may not be the best, as this index proves to have a number of limitations as to the resultant space. [6, 9]

Problems relating to classification and space allocation can be divided into two groups, namely problems pertaining purely to the classification of items, and those pertaining to the classification and then space allocation of items. Yet all of these problems are of the NP-hard type, the exact solution of which is not possible through reasonable time. Therefore to solve them a number of meta-heuristic approaches have been offered such as Simulated Annealing methods, and Genetic Algorithms. [4, 10] In the second group of problems pertaining to classification and space allocation, the space in the warehouse will be considered as a grid, where the P number of products will be allocated to these spaces depending on the number of orders, and their distance from the entry/exit ports. [4] Among the novel methods in arranging items in the warehouse, is to classify them based on similarities. [1] Here, all those items with similar demand procedures will be classified together. For example if two items are usually ordered together, they will be grouped together as well in the warehouse. By far, most articles delving into the problem of storing similar items have proposed solutions based on the P-Median method, requiring heuristic solutions, since the former is of the NP-hard type. Among these heuristic methods, one can name the Tietz and Bart [1, 2] solution.

1.2 P-MEDIAN PROBLEM In using P-Median to solve the problem of classification, it is presumed that there are “n” numbers of items in the warehouse, which are to be classified into “p” number of classes. The number of “p” is given. The classification of items will follow a distance matrix of “D”. In such method, a number of “p”s will be considered as medians between which, the items will be classified in fixed arrangements. Therefore the median number of each class will define the class, and the aim in such a method is to minimize the total distance of items in a class with the pertinent median. [3] Now we shall introduce the model in mathematical terms:

Xij= 1 if item “I” is allocated to class median “j”

0 otherwise

Yij= 1 if item “I” is allocated to class median “j”

0 otherwise

dij = distance between items “i” and “j” n = total number of existing items in the warehouse for classification j = total of items capable of being median

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n

ik jkk 1

,

( p , p )(7)

( )

ij

ik jk

d

p p

=

= Γ

Γ =

With regards to the definitions of parameters and variables we have: The target function of problem (1) will minimize the distance of all items from the pertinent class median. Limitation (2) will ensure that each item will be allocated only to one median. Limitation (3) ensures that the number of classes is equal to “P”. Limitation (4) ensures the existence of the classes to which items will be allocated, and limitation (5) has applications in finding binary answers. The most important factor in defining the classification problem through P-Median lies in the definition of the pertinent distance matrix. As previously mentioned, warehouse classification

occurs by virtue of the similarities of the items. The definition itself comprises of the number of times that two items will be ordered together. Based on this, it is possible to offer a Product-Order matrix, which is binary, and then change the matrix into one applicable to the P-Median model. Since defining the distance plays a major role in the formulation and solution of the problem, it is important to analyze in detail, the techniques used in the preparation of this matrix.

• The prefix “non” is not a word; it should be joined to the word it modifies, usually without a hyphen.

• There is no period after the “et” in the Latin abbreviation “et al.”.

• The abbreviation “i.e.” means “that is”, and the abbreviation “e.g.” means “for example”.

An excellent style manual for science writers is [7].

2. DISTANCE MATRIX As clear in the P-Median model, the distance between each item and the median of each class will be considered as multiplications of the target function. Therefore the target function in this model will minimize the total distance. Yet, what is meant by distance, in the classification of items? As a matter of question, what particular anomaly exists for us to minimize, in the classification of items? It is possible to use the term “dissimilarity in demand procedures” of two items rather than “distance”. This is to presume that items more similar in demand procedures are considered closer to each other. In this way, we classify items together when they are more similar in terms of demand procedures. To find the

distances or dissimilarities, it is possible to use Product-Order matrix, as defined below: m= number of existing stock n= number of total order invoices Pij = 1 if item “I” is allocated to median “j”

0 otherwise ( i =1..m , j=1..n)

ij m n [P ] (6)m nP× ×

=

As observed, the above mentioned matrix is binary, converted into a distance matrix through a distance function. Consider each of the lines in matrix P.

presuming )p ,...,p ,(p ini2i1=ip to be line “i” of matrix P.

The distance between the unknowns “i” and “j” will be defined as follows:

1 if ik jk( p p )≠

0 if ik jk( p = p )

With this definition matrix nmij] [d ××

=nmd will form the

distance matrix, which is square and symmetrical [1]. As seen above, in defining the matrix of distance, we consider the total number of orders to be “n”. Yet this number is not predetermined. In this form of solution it is possible to refer to the existing data from the warehouse, and in view of the gathered data based on order invoices, it will be possible to presume the distance matrix as above. V. Propose the “Other Objective Function” in Classifying Stocks So far the target function for classification was defined by the target function of P-Median. In the latter model, the target function will minimize the distances between stocks based on the class median. However we require stocks to be most similar to each other in pairs. In other words, to define dissimilarity in terms of distance, we intend the stocks to be closest to each other in pairs. To achieve this we formulate the problem in the following manner: C=1,2,3,…,p index of classes i=1,2,3…,m index of existing stocks Xic= 1 if item “I” is allocated to class “c”

0 otherwise Dij= distance (dissimilarity) of “i” and “j” In view of the definition of functions and parameters we have:

ij1

ijj J

jj J

ij j

ij j

Min Z d (1)

s.t.

X 1 i 1,...n (2)

Y (3)

X Y i 1,...n , j J (4)

X , Y {0,1} i 1,...n , j J (5)

n

iji j J

X

P

= ∈

∈

∈

=

= =

=

≤ = ∈

∈ = ∈

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1

ij1 1

ij1

ij

(1) M in Z d

s.t.

(2) X 1 i 1,...,m

(3) X {0,1} i 1,...,m

1,...,p

p n n

ic jcc i j i

p

c

X X

c

−

= = =

=

=

= ∀ =

∈ =

=

Target function (1) minimizes the total distance between each pair of items in one class. Limitation (2) guarantees each item to be allocated to one class, and limitation (3) ensures binary answers. As evident from the above, the problem is non-linear, and requires heuristic solutions. For this problem the following solution is proposed: First a variable

<

=

jiijd D will be defined with zero

initial value, representative of the total distance present in one class (this is the target function in the problem). We assign each product to a different class. Since dii=0 then the sum of all distances (D) will be 0. However the number of classes is equal to n, which is undesirable, and our aim is to achieve the value “p” for them. For this, we will combine each two items together each time. The decision on which two items to combine will be made based on the least distance added to D. In fact we intend to discover which combination of two items adds the minimum dii to D. Step 1: D=0, all items are classified separately Step 2: all pairs will be examined as to the minimum dii added to the D. for this, the sum of all distances between items in each two groups will be calculated. Step 3: the two groups which add the least to D will be combined and the new value of D, measured. Step 4: is the number of classes equal to P? If not, go to Step 5; if yes, process is complete and D shows the total distance (dissimilarity) between items in “p” number of classes. Step 5: update the distance between classes. For this the distance between other classes with the new class, comprised of the combination of two old classes will be equaled to the distance of the two old classes with those other classes. Then move back to step 2. Finish. For a better understanding of the algorithm, we will explain it using an example: Presume that we have 5 products we want to classify into three classes. The Product-Order matrix and the subsequent distance matrix will be as follows:

1 0 0 1 0 0

0 0 0 1 0 0

P = 1 0 0 1 0 1

1 0 1 1 1 1

1 1 0 0 0 1

0 1 1 3 3

1 0 2 4 4

D = 1 2 0 2 2

3 4 2 0 4

3 4 2 4 0

Step1: D=0 (all items are in separate classes) “d” is the distance between items in one class. Therefore D=d1+d2+d3+… Step 2: we examine the combination of pairs of groups, and choose the pair with the smallest d.

d(1,2)=1 d(2,3)=2 d(3,4)=2 d(4,5)=4

d(1,3)=1 d(2,4)=4 d(3,5)=2

d(1,4)=3 d(2,5)=4

d(1,5)=3 Step 3: we combine products 1 and 2, and D=1 Step 4: is the number of classes equal to 3? A: no, go to step 5. Step 5: update distances and go to Step 2.

d((1,2),3)=1+2=3 d(3,4)=2 d(4,5)=4

d((1,2),4)=3+4=7 d(3,5)=2

d((1,2),5)=3+4=7 Step 5-2: we compare the combination of each pair of groups and select the combination of 3 and 4. Step 5-3: we combine classes 3 and 4, and D=3. Step 5-4: is the number of classes equal to 3? A: yes. Process is complete and D=3 is the total of distances. The End However as discovered from the above example, two problems are inherent of this method: 1-The question remains as to what solution to adopt when we confront a multitude of minimum values. in this case we can choose the group with the smallest element among the column. Otherwise a larger value will be added to D. the only other solution is to randomly select one answer. 2-The second question is on the acceptability of the answer we get from this method, since it may well be based on random numbers. For examples situations occur when to add the minimum value to D, we will have to compromise on the next items with larger added values. To find a relatively enhancing solution, we will need to add other steps to the procedure as follows:

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Step 6: we analyze the replacement of each product in the group with other items in the column. If the displacement helps reduce D, it will be adopted. The sum of distance of each two alternatives in their own group will be compared to the sum of distance of the same items in their newly adopted groups. If the adoption will reduce the distance, it will take place. Step 7: the previous step will be repeated until no further enhancement to D is possible. The End

Step 6 can be divided into two parts, and other than the pair replacement in the classes, it is also possible to take individual items and examine them in new groups. The above algorithm will provide an optimum answer in an efficient time. The algorithm is written using VBA in Excel, and performed in a computer with 2.8MHz CPU. The results of this algorithm are written using the results of the model (table1). Since even Lingo software is incapable of providing optimum answers to non-linear problems, it becomes evident that this algorithm provides at times a better result, especially when the number of items varies between 10 and 30. In order to achieve the distance matrix in the above table, we have first drawn the Product-Order matrix, and then converted

it into the former. To form the product-order distance, we presumed the number of orders to be 20, then for each element in the matrix we assigned a random number. If the number was smaller than 0.5, we presumed it as 0 and if larger, as 1. The codes pertaining to the formation of product-order matrix and distance matrix have also been provided in the appendix. As explained before, this non-linear model has also been drawn in Lingo, the codes of which are also attached. Further, in one of the occasions where the method has proven to be better than lingo, the grouping results have come together with the distance matrix, in order to help the reader in a better assessment of the algorithm. 3. CONCLUSION This article has discussed one of the major techniques in warehouse cost reduction, comprising of the classification of existing stocks. As we discovered, the linear models for this problem such as the P-Median do not prove efficient; the problem requiring non-linear solutions. On the other hand, since the problem cannot be solved in rational time, a heuristic algorithm was proposed. Because we could not compare the efficiency of the algorithm with any available optimum answers, Lingo software was used whereby the results of the algorithm and the software were compared. The comparison proved that the algorithm can result in not only acceptable

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answers but at times when the number of items is low; it will provide better results than Lingo. Among proposals for future research, we can consider the division of customer orders into groups, and since this presents a limitation, its elimination can ensure better results. Furthermore, since the number of future orders cannot be predicted, through speculative methods or by data mining into the warehouse data sources, a more accurate distance matrix maybe offered. Steps 6 and 7 (singular and pair replacements) of the provided algorithm only offer minimal improvements. Other steps can be explored for triplet replacements or more. Most articles, including this, have considered the distance traveled within the warehouse as the most important factor of cost. Future researches may as well focus on the costs of space occupation in the warehouse. REFERENCES [1] J. Ashayeri, R. Heuts, B. Tammel. A modified simple heuristic for the p-median problem, facilities design applications. Robotics and Computer-Integrated Manufacturing 21 (2005) 451-464. [2] Tietz MB, Baret P. Heuristic methods for estimating the generalized vertex median of a weighted graph. Oper Res 16 (1968) 955-61 [3] Francis RL, McGinnis LF, White JA. Facility layout and location: an analytical approach. 2nd ed., Englewood Cliffs, NJ, USA: Prentice-Hall; 1992. [4] Chih-Ming Hsu, Kai-Ying Chen, Mu-Chen Chen. Batching orders in warehouses by minimizing travel distance with genetic algorithms. Computers in Industry 56 (2005) 169-178. [5] Deutsch SJ, Freeman SF, Helander M. Manufacturing cell formation using an improved p-median model. Computers ind. Engng 34 (1998) 135-146. [6] Venkata Reddy Muppani, Gajendra Kumar Adil. Efficient formation of storage classes for warehouse storage location assignment: A simulated annealing approach. Omega 36 (2008) 609-618. [7] Mu-Chen Chen, Hsiao-Pin Wu .An association-based clustering approach to order batching considering customer demand patterns. Omega 33 (2005) 333-343. [8] Hausman WH, Schwarz LB, Graves SC. Optimal storage assignment in automatic warehousing systems. Management Science 1976; 22(6):629–38. [9] Heskett JL.Cube-per-order index—a key to warehouse stock location. Transportation and Distribution Management 1963;4:27–31. [10] J.P. van den Berg, A literature survey on planning and control of warehousing systems, IIE Transactions 31 (1999) 751–762.

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