Evaluation of a conveyor belt material based on multi-criteria decision making

Transcription

1 Indian Journal of Engineering & Materials Sciences Vol. 11, October 2004, pp Evaluation of a conveyor belt material based on multi-criteria decision making S Bhattacharya a, B Sarkar b & R N Mukherjee c a Department of Mechanical Engineering, b Department of Production Engineering, Jadavpur University, Kolkata , India c Department of Mathematics, Burdwan University, Burdwan , India Received 12 February 2004; accepted 2 June 2004 Conveyor belts are required to transport different types of materials in industries under varying operating conditions. A number of materials for such belts are available in the market. However, it is impossible to identify one such material, which will perfectly respond to all such requirements in a given environment. Hence arises the question of optimization. The present paper is based on one such optimization technique known as analytic hierarchy process (AHP). It considers a number of alternative materials as well as design criteria and arrives at the most suitable material. The algorithm calculates the priority vector (PV) values by eigen-vector method of pair-wise comparison. These PV values are calculated through an iterative process, and are accepted if the concerned matrices yield consistency ratios (CR) below a specific limit. Finally, composite priority (CP) values are calculated, and the alternatives are ranked according to their CP values. The algorithm is coded in FORTRAN language. The program is general, and can be used to handle smaller or larger number of criteria and alternatives as well. IPC Code: Int. Cl. 7 B29D 29/06 The belt conveyor system is the most common method of transportation used in the industry for conveying bulk materials 1. Now-a-days, belt conveyors are utilized for material handling purposes for practically all kinds of pulverized, granular and lumpy materials. The essential parts of a belt conveyor are: an endless belt to carry the material and transmit the pulling force, drive arrangements for the belt, supporting rollers or idlers, take-up devices to maintain belt tension, loading and unloading arrangements, and means for providing cleanliness as well as protection to the belt. Figure 1 shows the schematic diagram of a simple belt conveyor system. The conveyor carries material on the upper run of the belt and discharges the load over the head or drive pulley, as the belt bends round the pulley. The empty belt returns below. The drive pulley or drum is powered by a motor through a suitable reduction unit. The driven pulley or drum acts as the take-up or tail pulley. A snub pulley is used to increase the arc of contact at the drive side, thereby proportionately increasing the belt traction. The conveyor belt runs on rollers called idle rollers or simply idlers. Some of the popular types of conveyor layouts are shown in Fig. 2. There are mainly four types of belts used for power transmission: flat, V, ribbed and toothed 2. Of these, the first two are most widely used. Incidentally, the selection of a belt drive depends upon (i) power to be transmitted, (ii) speeds and orientations of the driver and driven shafts, (iii) speed ratio, (iv) centre distance and (v) space available. The different types of flat belts are (i) open, (ii) crossed, (iii) quarter twist, (iv) right angled, (v) stepped, (vi) fast and loose, (vii) compound and (viii) reverse. The advantages of flat belt drives include flexibility, shock absorption and efficiency at high speeds; resistance to abrasive and other harmful environments; simplicity, low cost, smoothness of operation, low cost of maintenance, long life and low noise. Further, such drives allow long distance between shafts, and the belts can be spliced or connected for endless operations. They give excellent performance in serpentine drives as well. Some of the disadvantages of flat belt drives are that in these applications the velocity ratio does not always remain a constant; the drives are of comparatively large size; Fig. 1 A simple belt conveyor system

2 402 INDIAN J. ENG. MATER. SCI., OCTOBER 2004 speeds are moderate, and the tension in the belt should be kept high in order to reduce slip. Choosing an appropriate material for a conveyor belt to suit some particular application is a problem with no definite or unique solution. Usually the demands on the belt performance are many, and the choice of materials wide-ranging. Among the most important characteristics of belt materials are 3 : (i) strength (modulus of elasticity, ultimate tensile strength and flexibility), (ii) friction, (iii) slip, (iv) creep, (v) density, (vi) durability and (vii) cost. No belt-material caters to all the above requirements equally well. The demands are often conflicting too. The requirement of high flexibility of the belt material (to absorb shocks and damp out vibrating forces so as to increase the life of the driving machine), for example, is often countered by the high cost of such materials. In older designs, the cost-reduction element used to be recognized as the most important factor in the selection of belt materials 4. Presently, the approach is different. In cases of conflicting demands and varying choices, the designer usually assigns relative priority values to the concerned parameters and choices, and tries to arrive at a good and not the best solution. Such design procedures are generally called multi-criteria decision making (MCDM) process. As regards the materials, some of the most popular belt materials are leather, fabric and canvas, rubber, balata and woven cotton. Of these, leather is quite common with many pulley drives. This material has high flexibility and can be used for drives carrying considerably varying loads. However, its frictionproperties are not quite of a very high grade. The elastic and friction properties of some common belt materials 2 are given in Table 1. Fig. 2 Different types of conveyor layouts Belt material Density (kg/m 3 ) Problem Formation Table 1 Properties of belt materials Modulus of elasticity (MPa) Multi-criteria decision making (MCDM) Process MCDM may have as its input several kinds of problems. However, there are some common characteristics of such problems. Firstly, each problem has multiple objectives or criteria, which are often conflicting (the objectives, however, must be precisely defined). Secondly, each objective or attribute has its own unit of measurement. The aim of MCDM is to finally select one alternative which fits Ultimate tensile strength (MPa) Coefficient of friction (for CI/steel pulley) Leather Canvas Rubber Balata

3 BHATTACHARYA et al.: EVALUATION OF A CONVEYOR BELT MATERIAL 403 best with the criteria requirements. In fact, a criterion is a measure of effectiveness. These criteria emerge as forms of attributes or objectives in the actual problem setting. A MCDM problem involves at least two conflicting criteria and at least two alternative solutions to the problem. There are several types of methods for MCDM processes. One of them is analytic hierarchy process (AHP), which has been adopted in the present case. It may be mentioned here that some of the other multiattribute decision making networks relevant here are TOPSIS (technique for order preference by similarity to ideal solution), ANP (analytic network process), LINMAP (Linear Programming in Multi-dimensional Attribute Process). Analytic hierarchy process (AHP) AHP is a powerful tool in formulating and handling complex, multi-person and multi-period problems hierarchically. This process is used in a large number of applications including engineering design, economic planning, energy policy, project selection and budget allocation. It can provide useful insight into the trade-offs embedded in a decision-making problem. The three fundamental steps of AHP are (i) defining a multi-criteria problem hierarchically, (ii) assigning relative priorities to the various elements using pair-wise comparison techniques and (iii) integrating these priorities to converge at an overall evaluation of decision alternatives. For assigning weights to the alternatives as well as criteria for constructing the decision matrix and pairwise comparison matrices, the concept of relative importance is used to arrive at the decision-maker s preferences. The intensity scale of importance (Table 2) introduced by Saaty 5 has been used here. The decision matrix and the pair-wise comparison matrices used in AHP are all square matrices. The consistency of the judgment values assigned to the decision alternatives and criteria are checked using eigen values and eigen vectors. Based on these Table 2 The nine-point scale of pair-wise comparison Intensity of relative importance Definition checks, the decision-maker revises and modifies the judgment values, if required. Evaluation of a Conveyor Belt Material Using AHP Five criteria, e.g., friction, strength, slip, density and durability, have been considered and coded as C 1, C 2, C 3, C 4 and C 5. Four materials are considered and coded as M 1, M 2, M 3 and M 4. It is presumed that the behaviour/performance of these four alternatives (materials) with respect to each of the five criteria are known. Methodology The proposed methodology is the integration of Brown and Gibson model and analytic hierarchy process. This methodology is applied to calculate the priority weights for functional, design factors and other important attributes by eigen vector method for each pair-wise comparison matrix. Next, global priorities of various attributes are found by using AHP. The pair-wise comparison matrices for the alternatives and five different factors (Tables 3-8) are constructed on the basis of Saaty s nine-point scale (Table 2). Algorithm of the methodology Step 1: Listing of the set of alternative materials (M 1 to M 4 ). Table 3 Decision matrix I II III IV V I 1 1/3 1/2 1/3 2 II 3 1 1/3 3 5 III IV 3 1/3 1/4 1 3 V 1/2 1/5 1/7 1/3 1 Table 4 Pair-wise comparison matrix for criterion 1 M 1 1 1/5 1/5 1/3 M M 3 5 1/5 1 1/3 M 4 3 1/3 3 1 Table 5 Pair-wise comparison matrix for criterion ,4,6,8 Equally important Moderately preferred Essentially preferred Very strongly preferred Extremely preferred Intermediate importance between two adjacent judgments M /3 3 M 2 1/5 1 1/5 1/3 M M 4 1/3 3 1/3 1

4 404 INDIAN J. ENG. MATER. SCI., OCTOBER 2004 Step 2: Identification of the design criteria (C 1 to C 5 ). Step 3: Assigning weights to each of the criteria based on the relative importance of its contribution according to the nine-point scale. This is the decision matrix. [as an example, if Criterion I stands for friction of the belt material (with the rollers), while Criterion II stands for its strength, then in Table 3 the element a 12 stands for the relative importance of friction as compared to strength (for the present design). If the designer feels that consideration of strength is 3 times as important as that of friction he assigns a value (weight) of 1/3 to a 12.] Step 4: For each criterion, assigning weights to each of the alternatives, based on its relative importance, according to the nine-point scale. These are called the pair-wise comparison matrices. [As with the decision matrix, here also the weights are assigned by the designers. For example, considering Table 4 (for Criterion I, i.e. friction), if material M 3 is known to be about 5 times more efficient than M 1 (i.e. μ M3 = 5μ M1 ), then a 31 = 5.] Step 5: Determination of the priority vectors (PV) for decision matrix and for each of the pair-wise comparison matrices. Multiplication of the sum of each column with the corresponding PV value. Calculation of the sum of these products, i.e., the principal eigen value (λ max ). Table 6 Pair-wise comparison matrix for criterion 3 M /3 3 M 2 1/3 1 1/5 3 M M 4 1/3 1/3 1/5 1 Table 7 Pair-wise comparison matrix for criterion 4 M 1 1 1/3 1/7 1/5 M /5 1/3 M M /3 1 Table 8 Pair-wise comparison matrix for criterion 5 M 1 1 1/3 1/7 2 M /2 3 M M 4 1/2 1/3 1/3 1 Step 6: Calculation of the consistency index (CI) for each of the matrices, according to the formula CI λmax n =, where n is the order of each matrix. n 1 Step 7: Determination of the random consistency index (RI) for each of the matrices, as per the formula 1.98( 2) RI = n. n Step 8: Determination of the consistency ratio (CR) for each of the matrices, using the relation CR = CI RI. If the value of CR is within the acceptable value of 10% (i.e. 0.1), then the process is accepted. Else, the steps 3-8 given above are re-iterated until a CR < 0.1 is reached. Step 9: Calculating the composite priority (CP) value for each alternative. This is done for each alternative by multiplying its PV values for a particular criterion with the corresponding PV value of that criterion, and adding up these products. Step 10: Ranking of the alternatives according to the descending order of their CP values. Step 11: Selection of the best alternative according to the highest value of CP. Results and Discussion The above algorithm is coded as a FORTRAN program that operates under MS DOS. Tables 3-8 show the matrices for the criteria and the alternatives. Figs 3-8 show the PV values for each of these matrices. Finally, Table 9 shows the CP values. The Consistency Ratio values obtained for the pair-wise Fig. 3 PV values for decision matrix Fig. 4 PV values for pair-wise comparison matrix for criterion 1

5 BHATTACHARYA et al.: EVALUATION OF A CONVEYOR BELT MATERIAL 405 Table 9 Composite priorities Fig. 5 PV values for pair-wise comparison matrix for criterion 2 Material M 1 M 2 M 3 M 4 Composite priority(cp) Fig. 6 PV values for pair-wise comparison matrix for criterion 3 Fig. 7 PV values for pair-wise comparison matrix for criterion 4 Fig. 8 PV values for pair-wise comparison matrix for criterion 5 comparison matrices given in Tables 3-8 are, respectively, , , , , and The selection is made based on the highest of the CP values. Thus the selected alternative is M 3. The number of criteria or alternatives should be reasonably small to allow consistent pair-wise comparisons. A maximum of seven criteria or alternatives can be used 5. The methodology presented here can be further integrated with the Brown and Gibson model 6 based on the multi-attribute preference theory. For more than seven criteria or alternatives, one can group the criteria with respect to a common property and add another level to incorporate the groupings 7,8. Conclusions Selection of a belt material under sensitive operating environment is a very complex problem, when looked at from the point of optimization. The present methodology is a very useful tool first to quantify the apparently intangible factor in a systematic manner and then to find out the best alternative. References 1 Maitra & Prasad, Handbook of mechanical design (Tata McGraw-Hill Publishing Company Ltd., New Delhi), 1995, Sharma & Aggarwal, A text book of machine design (S K Kataria & Sons, Delhi-Ludhiana), 1991, Sharma K D, Fundamentals of machine design (Asia Publishing House), 1969, Shigley J E, Mechanical engineering design (Mc-Graw Hill Book Company), 1986, Saaty T L, The analytic hierarchy process (Mc-Graw Hill, New York), Buffa E S, Modern production/operations management (Wiley Eastern Ltd., New Delhi), Saaty T L, Euro J Operat Res, 48 (1990) 9. 8 Saaty T L, France J W & Valentine K R, Socio-Economic Plan Sci, 25 (1999) 155.