[ieee sixth international conference on intelligent systems design and applications] - jian, china...

Classifying ABC Inventory with Multicriteria Using a Data Envelopment

Analysis Approach Qing Liu Dao Huang

Research Institute of Automation East China University of Science and Technology

,Shanghai 200237, China (E-mail: [email protected])

Abstract

This paper presents a modified Data Envelopment Analysis (DEA) model to address ABC inventory classification. The new model is derived from reduced DEA model, and some restricts are added to make classification results more reasonable. The evaluating process has two steps. Firstly, all criteria data for each item are normalized between [0, 1]. Then, the prior scores for all inventory items are computed using the proposed Model. A simulation example is employed to verify the efficiency of the model. The results show that this model is an effective ABC classification tool and the prior score of each item make classification credible. Furthermore, the classification result is compared with that of traditional Analytic Hierarchy Process (AHP) methodology. 1. Introduction

An inventory may have thousands of items, and the resources needed in managing inventory, such as: time and money, is limited. To gain efficient inventory management, the logical action is to try to use the available resources in the best way. In other words, focus on the most important items in inventory. To get around this problem, the ABC inventory classification is widely employed. The ABC classification derives from Villefredo Pareto,

who thought that the few having the greatest importance and the many having little importance, it divides all inventory items into three groups by annual dollar usage: A items constitute roughly the top 15 percent of the items, B items the next 35 percent, and C items the last 50 percent.

ABC classification is simple and effective when the main difference among the inventory items is in its annual dollar usage. However, when other criteria, for example: inventory cost and lead time, become important [1] in establish the appropriate degree of control over each item, traditional ABC method may not be able to provide a good classification. This problem of multicriteria inventory classification has been addressed by many studies [2][3][4][5].

Many methods are used in multicriteria inventory classification。Flores et al [4.] use the joint matrix in two criteria case. There are also approaches in the Artificial intelligence field. In [3], authors used two methods in ANN learning, namely the back propagation (BP) and genetic algorithm (GA), the result indicated the ANN classifier model is effective, but many new important qualitative variables might be difficult to incorporate into the model [5]. The Analytic Hierarchy Process (AHP) [6] has been successfully employed in many studies [7]

[8], the significant drawback of AHP is that decision makers’ preference was too much involved in evaluating progress.

The data envelopment analysis (DEA) is firstly proposed by Charnes [9]. Traditionally, DEA is used

Proceedings of the Sixth International Conference on Intelligent Systems Design and Applications (ISDA'06)0-7695-2528-8/06 $20.00 © 2006Proceedings of the Sixth International Conference on Intelligent Systems Design and Applications (ISDA'06)0-7695-2528-8/06 $20.00 © 2006

as decision making tool rather than evaluating method. But the decision making progress involves evaluating, so DEA methodology could be used to evaluating [10]. This paper presents a modified DEA model to address multicriteria inventory classification.

The rest of this paper is organized as follows: Section 2 gives an illustration of the proposed model. Section 3 tests the model with a real example, and we conclude the model in Section 4.

2. Model Development 2.1. DEA model

Data Envelopment Analysis is becoming an increasingly popular management tool. It was developed by Rhodes [11]and initially detailed and publicized by Charnes et al [9]as a nonparametric method of measuring the efficiency of decision making units (DMUs) based upon multiple criteria. DEA was built on the theoretical foundations provided by Farrell [12] and continues to be popular for a wide variety of applications [13]. DEA had virtually instant success in management science.

DEA uses linear programming (LP) to arrive more directly at the best set of weights for each DMU. The representative models of DEA are CCR and BCC DEA models, these two models are named after their authors. CCR was first introduced by Charnes Cooper, and Rhodes (CCR) [9], and Banker, Charnes, and Cooper (BCC) [14] extended the CCR model to accommodate technologies that exhibit variable returns to scale.

Suppose that there are a number n of DMUs, each one making use of m different inputs x to produce t different outputs y. Using the standard notation,

xij denotes the amount of input i used by the jth

DMU and yrj the amount of output r produced

by the jth DMU. v={ V 1 , V 2 ,….., V m }T

represents the input weight vector, u={ U 1 ,

U 2 ,…..,U s }T represents output weight vector.

In order to measure the efficiency of a specific DMU, labeled as “0”, the following output oriented model is solved:

Max h0 =∑∑

=

=m

i ii

s

r rr

xVyU

1 0

1 0

Subject to:

∑∑

=

=

m

i iji

s

r rjr

xVyU

1

1<=1, j=1,

2,……,n

U r >=ε >0, r=1, 2,……, s

V i >=ε >0, i=1, 2,……,m

Where ε is non-archimedean quantity. Model(1)

Model(1) can convert to LP model:

Max yU r

s

r r 01∑ =

Subject to:

∑ ==m

i ii xV1 0 1

∑ =

s

r rjr yU1 - xV ij

m

i i∑ =1<=0, j=1,

2,……,n

U r >=ε >0, r=1, 2,……, s

V i >=ε >0, i=1, 2,……, m

Model(2)

2.2. Modified DEA model

The LP DEA model(2) will reduce to model(3).

Max yU r

s

r r 01∑ =

Subject to


∑ =

s

r rjr yU1 <=1, j=1, 2,……,k

U r >=0, r=1, 2,……,s

Model (3)

In this paper, the reduced DEA model (3) is used to aggregate the performance of an inventory item in terms of different criteria to a single prior score with some modification. First, weights of criteria are sum to 1 , further more, constraints have been suggested for weights for the various criteria [15] and those guidelines are expressed as a set of constraints added to model. In this way, the optimized weight can be more reasonable.

Table 1 Guidelines for various criteria Criteria Min Max Unit cost 0.04 0.36 Procurement/Downtime 0.16 0.47 Demand 0.04 0.36 Lead time 0.36 0.04 Consider that there are k items in an inventory, and each item is characterized by n criteria. For each item, its prior score can be calculated with the following model:

Max ∑=

n

iijij sz

1

Subject to

∑=

n

iijij sz

1<=1, j=1, 2, 3,………….,k

∑=

n

iijz

1=1, j=1, 2,……,k

a ≤≤ zij b, ∈∀i 1,2,……,n; ∈∀j 1, 2,……,k

1 ≤ a ≤ 0; 1 ≤ b ≤ 0.

Where zij denotes score for the jth items on the ith

criterion, sijthe jth item data on the ith criterion.

Model (4)

2.3. Scoring items

In DEA model, original criteria data for each item are used without modification. It is a simple way, but some criteria data may be very large, such as: unit price and ordering costs, and some criteria data are relative small. In this article, a normalizing function is addressed to make all criteria data for each item between [0, 1]. It worth pointing out that part of original data didn’t have determinate value, so the between value are addressed when normalizing.

Y=1-xx

x x

minmax

max

−−

Function (1)

3. Simulation 3.1. Result

The data in this paper come from Dyckhoff and Allen (2001) [13]. Note that all the four criteria are positively related to the importance level of inventory items. The optimal inventory scores for all the inventory items are computed using Model (2) repeatedly by changing the objective function for each item. As mentioned earlier, A items constituted approximately roughly the top 15 percent of the items, B items approximately the next 35 percent, and C items approximately the last 50 percent.

Table 2 Rating by prior score

Item Prior score Sorting Sorting by AHP cg 0.7594 A A bb 0.7084 A A df 0.6801 A A cj 0.6801 A B cd 0.6801 A A cn 0.6509 A A ab 0.5702 A A cr 0.5662 A A ds 0.5636 A A cb 0.5636 A A


at 0.5571 A A cc 0.5437 A B bk 0.5182 A A du 0.5127 A A cf 0.5127 A B ce 0.5127 A B ao 0.5127 A A by 0.4844 B B cy 0.4743 B A ad 0.4696 B B cz 0.4663 B A dr 0.4588 B B bm 0.4588 B B bl 0.4588 B B dv 0.4453 B A ac 0.4427 B A dl 0.4168 B B dk 0.4168 B B ax 0.4168 B C ba 0.409 B A bp 0.4081 B B an 0.4078 B A dp 0.3908 B B ay 0.3903 B B ar 0.3852 B B do 0.383 B B au 0.3817 B B dd 0.3773 B B cx 0.3733 B B bz 0.3717 B C bx 0.3717 B C ai 0.3717 B C dn 0.3612 B C ci 0.3612 B C ch 0.3612 B C bv 0.3612 B C bs 0.3612 B C bj 0.3612 B B bc 0.3612 B C ap 0.3612 B B am 0.3612 B C al 0.3612 B C ag 0.3612 B C

br 0.3392 C C ae 0.3288 C C bf 0.2872 C B dj 0.2782 C B aj 0.2635 C B dq 0.2521 C C ca 0.2437 C B ck 0.2386 C B bo 0.2366 C B dt 0.2128 C C cq 0.2078 C B db 0.2042 C C dg 0.2034 C C cm 0.2002 C C cv 0.1904 C C cu 0.1904 C C di 0.1859 C C cs 0.1859 C C bu 0.1859 C C bq 0.1859 C C aq 0.1859 C C ak 0.1816 C C cw 0.1785 C B dm 0.1664 C C ct 0.1549 C C af 0.1362 C C dc 0.1312 C C az 0.1269 C B be 0.1166 C B bh 0.1132 C B bi 0.1111 C B de 0.1103 C C aa 0.1095 C B bn 0.1086 C B cl 0.1078 C C as 0.1078 C C ah 0.1078 C C bd 0.099 C C da 0.0904 C C aw 0.0904 C C av 0.0859 C C bg 0.0664 C C bt 0.0505 C C


3.2. Comparison and illustration

As shown in table2, the ABC classification using Model (2) provides different results compared with the traditional AHP classification. The difference could be because of two reasons. The first is scoring method, a normalize function is addressed to determine criteria data for each item in this paper. The second is weights for each criterion. AHP classification used determinate criteria weights, and a LP function is addressed in model 1. For example: item cj is classified as class B by AHP method because of the weights for criteria adopted in the method, and this item is classified as A class by model(2) because it has one of the high values in terms of average unit cost, procurement and lead time,

Table 3 Comparison of two methods

AHP Modified DEA Sum of A items 18 17 Sum of B items 35 35 Sum of C items 43 44

consistency 39.5% 4. Conclusion Multicriteria classification is a complex process, for all the criteria must be considered systematically. Many methods have been used to this problem. This paper presents a new approach using modified DEA model to multicriteria inventory classification. The new approach is based on reduced DEA model (Model 3), and some constraints are proposed to make the results more reasonable. The evaluating process has two steps. Firstly, a normalizing function is addressed to make all criteria data for each item between [0, 1]. Secondly, The optimal inventory scores for all the inventory items are computed using Model (4) repeatedly by changing the objective function for each item. At last, classification results using Model (4) are compared the AHP classification.

The model proposed in this paper is a simple ABC classification tool and the prior score of each item make classification credible , it also has several limitations: (1)other kinds of normalize function could be addressed to get more balanced result. (2) this model can not process new items effectively. So, further research will need to be addressed on this issue. References [1]Guvenir HA, Erel E, “Multicriteria inventory classification using a genetic algorithm”, European Journal of Operational Research, Volume105,1998, pp. 29–37. [2]Flores, B.E., Whybark, D.C., “Implementing Multiple Criteria ABC Analysis”, Journal of Operations Management, Volume 7,1987, pp. 79-84. [3]Partovi, F.Y., Anandarajan. M, “Classifying Inventory Using An Artificial Neural Network Approach”, Computers and Industrial Engineering, Volume41, 2002, pp. 389-404. [4]Flores BE, Olson DL, Dorai VK, “Management of multicriteria inventory classification”, Mathematical and Computer Modeling, Volume 16, Issue 12, 1992, pp. 71–82. [5]Guvenir HA, Erel E, “Multicriteria inventory classification using a genetic algorithm”, European Journal of Operational Research, Volume 105, 1998, pp. 29–37. [6]Saaty TL, The Analytic Hierarchy Process, McGraw-Hill: New York; 1980. [7]Gajpal PP, Ganesh LS, Rajendran C, “Criticality analysis of spare parts using the analytic hierarchy process”, International Journal of Production Economics, Volume 35, Issue 1-3, 1994, pp. 293–297. [8]Partovi FY, Hopton WE, “The analytic hierarchy process as applied to two types of inventory problems”, Production and Inventory Management Journal, Volume 35, Issue 1,pp.13–9. [9]Charnes, A., Cooper, W., & Rhodes, E,


“Measuring the efficiency of decision making units”, European Journal of Operational Research, Volume 2, 1978, pp. 429-444. [10]Schonberger, R., Knod, E, Operations Management: Continuous Improvement (6ed), Chicago: Irwin, 1997.

[11]Rhodes, E, “Data Envelopment Analysis And Related Approaches For Measuring The Efficiency Of Decision Making Units With An Application To Program Follow Through In U.S. Education (Ph.D. dissertation) ”, Pittsburgh, PA: Carnegie-Mellon University School of Urban and Public Affairs, 1978. [12]Farrell, M, “The measurement of productive efficiency”, Journal of the Royal Statistical Society, Volume 3, 1957, pp. 253-290.

[13]Dyckhoff, H., Allen, K, “Measuring ecological efficiency with data envelopment analysis (DEA)”, European Journal of Operational Research, Volume 132,Issue 2, 2001, pp. 312-325. [14]Banker, R.D., Charnes, A., and Cooper, W.W., “Some model for estimating technical and scale in efficiencies in data envelopment analysis”, Management Science, Volume 30, 1984, pp. 1078-1092. [15]Fariborz Y. Partovi, Jonathan Burton, “Using the Analytic Hierarchy Process for ABC Analysis”, International Journal of Operations & Production Management, Volume 13, Issue 9, 1993, pp. 29-44.


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