Feb 28, 2009 - The objectives of the present investigations are to develop the Modular Structure of CAPP. ... Manual process planning is very tedious, time consuming and subjective. ..... source performance and customer satisfaction. ...... cities through Identified Banks SBI, ICICI, HDFC and then e-Payments were made to ...
Proceedings of National Conference on Advances in Mechanical Engineering (NCAME 2009) th st February 28 and March 1 2009 Department of Mechanical Engineering Moradabad Institute of Technology, Moradabad
Goal Programming Approach For Supplier Selection P.K.Chaudharya, Sarita Munagalab, Dr. A.K.Agrawalc , Priyank Varmab a Research Scholar, Department of Mechanical Engineering, I.T.-B.H.U., Varanasi (U.P.) b M.Tech.Student, Department of Mechanical Engineering, I.T.-B.H.U., Varanasi (U.P.) c Professor, Department of Mechanical Engineering, I.T.-B.H.U., Varanasi (U.P.) ABSTRACT One of the most important challenges faced in the management of supply chains is related to supplier selection. This problem deals with the selection of right number of suppliers for supplying raw materials, parts and components to the purchaser. In doing so, generally, distribution of order size is also decided. Selection is based on multiple criteria ranging from quality, cost, reliability, etc. In the present work, a critical analysis of the work of Lokesh Vijayvargy (2008) has been made. The author had claimed his formulation to have goal programming framework, which is not. He used column dropping rule of multiple criteria decision making framework for his formulation. In the present work, the same supplier selection problem has been addressed and solved through true goal programming approach from many perspectives, numerical examples are taken to illustrate the use of the formulation. Some of the improvements and additions have also been made to the considerations made by Lokesh Vijayvargy (2008). 1. INTRODUCTION In present day competitive business environment, it becomes essential for every industry to manage the stocks of the items properly. This is the mostly the function of production strategy and supply policies established to respond to customer demands. On the other hand, raw materials and other items as parts and components are also to be properly procured from the vendors. The first issue mostly gets affected by the uncontrollable customer’s behavior. The later issue is of much greater importance as the companies can exercise their discretion in opting for right group of vendors for receiving selected items and raw materials from them in right quantities. Needless to mention that the size of the order would ultimately depend upon the customer’s order volume. Basically there are two kinds of supplier selection problems: • Supplier selection when there is no constraint. In other words, all suppliers can satisfy the buyer’s requirements of demand, quality, delivery, etc. • Supplier selection when there are some limitations in suppliers’ capacity, quality, etc. In other words, no one supplier can satisfy the buyer’s total requirements, thus the buyer needs to meet part of his/her demand from one supplier and similarly remaining ones from suppliers to compensate for the shortage of capacity or low quality of the first supplier. In the first kind of supplier selection, there is definitely a supplier who can meet all the buyer’s needs (single sourcing) and the management needs to identify the best supplier; NCAME 2009
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whereas in the second supplier selection process, as no supplier can satisfy all the buyer’s requirements, more than one supplier are to be selected (multiple sourcing). In this case, management needs to decide as how much should be purchased from a set of selected suppliers. Lokesh Vijayvargy (2008) has addressed the problem of multiple sourcing and presented a mathematical programming formulation of the problem. He used column dropping rule for the problem formulated as multiple criteria decision making problem. Therefore, his claim of the problem being formulated as goal programming does not seem to be correct. His method can only be used to optimize objectives rather than to satisfy goals. Here the author has explicitly put the goals as constraints and at the same time the objective function is expressed as an overall function to be minimized. This is only to be done in a situation when all the goals are considered to be of equal importance and priority wise. In this work, contrary to the work done by Lokesh Vijayvargy (2008), two versions of goal programming have been proposed in order to select the suppliers by maximizing their total additive utility for a situation. Also, the proposed methodology helps in determining the optimum order quantity. The power of the proposed methodology is depicted with the help of a hypothetical numerical example. This numerical problem is solved using both the methods: (i) as suggested by the Lokesh Vijayvargy (2008) and (ii) two proposed methods of goal programming, i.e., weighting method and preemptive method by using LINDO software on PC/AT. The results obtained in this way are compared with each other to show the effectiveness of the formulation. The paper is organized as follows. Section 2 deals with reviews of pertinent literature where as Sections 3 and 4 accommodate the methodology in details and its illustration respectively. At last, after a detailed discussion in Section 5, the whole work is concluded in Section 6. 2. PERTINENT LITERATURE Over the years, supplier selection has gained great attention in managing the business efficiently and effectively. Suppliers are evaluated on several criteria such as pricing structure, delivery, quality, services and many more, depending upon the situation and objective of a company. In this regard, Weber and Ellram (1993) presented a detailed list of supplier selection criteria and suggested multi objective-programming for volume allocation to the supplier. It has been observed that only a limited number of supplier gets business frequently in most of the traditional organizations though the number of registered suppliers is large (Kauffman and Leszczyc, 2005). Many methods are available in the literature for the selection of supplier. De Boer et al. (2001) have categorized the distinguished supplier selection methodologies into five headings such as Linear Weighting models (De Boer et al., 1998; Narasimhan, 1983; Petroni and Braglia, 2000; Timmerman, 1986), Total Cost of Ownership models (Monczka and Trecha, 1988; Smytka and Clemens, 1993; Timmerman, 1986), Mathematical Programming models (Degraeve and Roodhooft, 1998; Degraeve and Roodhooft, 1999; Degraeve and Roodhooft, 2000; Karpak et al., 1999; Weber and Desai, 1996; Weber et al., 1998), Statistical models (Soukup, 1987; Ronen and Trietsch, 1988), and Artificial Intelligence (AI) based models (Vokurka et al., 1996). According to Korhonen (1992), in multiple criteria decision situations it is very difficult, if not impossible, to specify at the operational level what the “best” (most preferred) decision really means precisely. The author has also reported that the most-preferred solution is the one that accommodates all the dimensions of the problem. In this regard, Charnes and Cooper (1961) proposed a goal programming framework for the multi- objective nature of supplier selection
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problem. Following this is the work of Keeney and Raiffa (1976) on multi-attribute utility assessment. Cohon (1978) defined a non-dominated solution as a feasible solution to a multi-objective programming problem that yields an improvement in one objective without causing a degradation in at least one other objective. Buffa and Jackson (1983) have approached a schedule purchase from mix of vendors through goal programming. Although literature review reveals that many works have already been done to handle the multicriteria multi-objective nature of supplier selection problem with the goal programming, it still remains an important area of research. 3. METHODOLOGY −
+
Goal programming (GP) minimizes the achievement function f k (d i , d i ) one term at a time in the order of any stated preemptive priorities. The function f k is typically a linear function of −
+
−
+
deviational variables, for example, f k = di , f k = di , f k = di + di and so on. If we assume, K preemptive priority levels, the model can be described in general notation as follows: Find X= ( x1 , x 2 ,....x j ......xn ) so as to K
Lexicographically minimize Z= ∑ Pk . f k (d i , d i ) −
+
k =1
n
Subject to
∑a j =1
−
ij
+
x j + d i − d i = bi −
for i = 1, 2,…,m.
+
x j , d i , d i >= 0 where x j = decision variable (unknown)
aij = technological coefficient associated with variable j of constraint i
bi = aspiration level or target value for goal i n = total number of decision variables m = total number of constraints or goals to be achieved − d i = negative deviation (or slack variable) for constraint i (underachievement) +
d i = positive deviation (or surplus variable) for constraint i (overachievement) Deviational variables on the same priority level must be commensurable although deviations that are on different goal levels need not be commensurable. Any rigid constraints (i.e., absolute goals) are ranked at priority 1. Unlike general goal constraints, these constraints do not generate any deviational variables in the solution procedure. In the goal formulation, the relationship between the original goal form and the deviational variables to create equalities in constraint equations is as follows. Original goal GP format Deviation variables to be formulation minimized − + f(x)=b d − (underachievement) f(x)+ d − − d + = b f(x)=b f(x)+ d − − d + = b d− +d+
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3.1 Goal Programming Algorithms: There are two methods (i) Weighting method and (ii) Preemptive method for solving the goal programming problem. Both methods are based on representing the multiple goals by single objective function. The two proposed methods are distinct in the sense that they will not generally produce the same solution. Neither method however can be claimed to be superior to the other because each technique is designed to satisfy certain decision-making preferences. 3.1.1 Weighting method: A single objective function is formed as the weighting sum of the functions representing the goals of the problem. In this model, the decision maker gives different weights to goal deviations, based on their relative importance. The greater the weight is, the greater is the importance assigned to a given deviation. In this model, the objective function tries to minimize the sum of al1 the weighted deviations between the goals and their target levels. The general form of this model is: Minimize: Z= ∑ ( wi + d i + + wi − d i − ) i∈m
Subject to: n
∑a j =1
+
−
ij
+
x j + d i − d i = bi
−
where wi and wi are positive constants representing relative weights to be attached to positive and negative deviation variables, respectively. 3.1.2 Preemptive method: The preemptive method starts by prioritizing the goals in order of importance as judged by the decision maker. The model is then optimized using one goal at a time, and in a manner such that the optimum value of a higher-priority goal is not degraded by a lower priority goal. Thus, solving any GP model involves achieving the highest priority goal before any of the lower priority goal are considered. Once the highest priority goal is attained to the fullest extent possible, the GP model proceeds to find a satisfactory level for the next-highest priority goal, and so on. The lexicographic model differs from the weighted model only in the representation of the objective function. In this structure, the objective function can be stated in the general form as: lex min a = [ g1 (d + , d _ ), g 2 (d + , d − ),...., g p (d + , d − )] This model has P priority levels, and a is an ordered vector of these P priority levels. A priority level is assigned to each relevant goal in terms of its rank. (i.e., p1 > p 2 > .... > pn ). In mathematical terms, the preemptive priority relation implies that multiplication by a number m > 0, however large m may be, cannot make a lower priority goal as important as a higher priority, goal (i.e., p j > p j +1 ). 4. ILLUSTRATION THROUGH NUMERICAL EXAMPLE For the fair comparison, numerical problem of Lokesh Vijayvargy (2008) has been taken.
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4.1 The Problem: ABC Company wants to purchase 1,00.000 unit of each of five different items to make a final product. Quality of product, quality of percent rejection, cost and reliability of each item according to different vendors is as shown in table 1. Table1: Vendor Assessment Data Vendor
Item No.
Cost (Rs. per item)
Quality of Product
Quality of Percent Rejection 0.2 0.3 0.1
Reliability
Vendor1
1 3 5
5,800 4,000 20,200
8.0 8.0 9.0
Vendor2
1 2 3 4
5,500 10,000 4,100 4,150
8.0 9.0 9.0 8.0
0.4 0.1 0.2 0.2
0.950 0.960 0.980 0.980
Vendor3
2 3 4 1 2 4 5
9,800 3,900 4,100 6,000 9,700 4,200 20,200
8.0 7.5 9.0 9.0 8.5 8.0 8.0
0.2 0.5 0.2 0.1 0.3 0.1 0.1
0.945 0.970 0.970 0.970 0.940 0.990 0.980
Vendor 4
0.980 0.980 0.970
The other characteristics of the problem are as follows. Capacity Limitations of Different Vendors 1. Vendor1 can supply at most 50,000 units of item 1. 2. Vendor 1 can supply items 1, 3, and 5 but at most in total equal to 25,0000 units. 3.
Vendor 2 can supply item 2 in quantity at most equal to 30,000 units.
4. Vendor 2 can supply items 1, 2, 3 and 4, but combined together at most equal to 270,000 units. 5. Vendor 3 can supply at most 40,000 units of item 4. 6. Vendor 3 can supply items 2, 3 and 4 but combined together at most equal to 150,000 units. 7. Vendor 4 can supply at most 40,000 units of item 5. 8. Vendor 4 can supply items 1, 2, 4 and 5, but combined together at most 30,0000 units.
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Priority 1: Product cost (cost): total cost function is formulated to be minimized because our goal is to minimize the purchasing cost. Priority 2: Maximize order supply to vendor 1(in terms of price), because of his being the customer also. Priority 3: Quality of product: maximize the quality of products. Priority 4: Quality of percent rejection: minimize the quality of percent rejection. Priority 5: Reliability: maximize delivery reliability of each item. 4.2. Model Formulations: To formulate a goal programming model, the numbers of items supplied by four different vendors are defined as choice variables. Here ni and pi are the negative deviation (underachievement) and positive deviation (overachievement) respectively associated with ith goal. Taking xij as the decision variable denoting quantity of item i to be supplied by supplier j the constraint equations with deviational variables are: Total cost constraint 5.8X11+5.5X12+6X14+10X22+9.8X23+9.7X24+4X31+4.1X32+3.9X33+4.15X42+ 4.1X43+4.2X44+20.2X51+20.2X54+n1-p1=4,400,000 (1) Vendor 1 order constraint 5.8X11+4X31+20.2X51+n2-p2=2,750,000 (2) Quality of product constraint 8X11+8X12+9X14+9X22+8X23+8.5X24+8X31+9X32+7.5X33+8X42+9X43+8X44 +9X51+8X54+n3-p3=4,250,000 (3) Quality of percent rejection constraint 0.2X11+0.4X12+0.1X14+0.1X22+0.2X23+0.3X24+0.3X31+0.2X32+0.5X33+0.2X42 +0.2X43+0.1X44+0.1X51+0.1X54+n4-p4=1,00,000 (4) Unreliability constraint 0.02X11+0.05X12+0.03X14+0.04X22+0.055X23+0.06X24+0.02X31+0.02X32+0.03X33+0.02X42 +0.03X43+0.01X44+0.03X51+0.02X54+n5-p5=14,500 (5) Quantity requirement constraint X11+X12+X14=1,00,000 (6) X22+X23+X24=1,00,000 (7) X31+X32+X33=1,00,000 (8) X42+X43+X44=1,00,000 (9) X51+X54=1,00,000 (10) Vendor’s capacity limitation constraint X11
