OPTIMIZATION OF FUZZY INVENTORY MODEL WITHOUT SHORTAGE USING PENTAGONAL FUZZY NUMBER

Transcription

1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue, November 08, pp. 7, Article ID: IJMET_09 8 Available online at aeme.com/ijmet/issues.asp?jtypeijmet&vtype 9&IType ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed OPTIMIZATION OF FUZZY INVENTORY MODEL WITHOUT SHORTAGE USING PENTAGONAL FUZZY NUMBER R. Kalaiarasi Department of Mathematics, Cauvery College for women, Trichy-, Tamilnadu, India M. Sumathi Departmentt of Mathematics, Khadhir Mohideen College, Adhirampattinam, Tanjore District M. Sabina Begum Research scholar, Department of Mathematics, M.V. Muthiah Government Arts College For Women, Dindigul-, Tamilnadu. ABSTRACT This paper deals with an inventory model without shortages has been regarded in a fuzzy environment The intention of this study is to optimize the total cost and order quantity in fuzzy environment. In this model, the annual demand and setup cost are constituded by pentagonal fuzzy numbers. This paper develops an approach to determine the optimum order quantity and total annual integrated cost under the fuzzy new arithmetical operations using cut principle. Graded Mean Representation method is used to defuzzify the total cost function and the Lagrangian method is used to determine optimal order quantity. A numerical example is given to demonstrate the solution procedure. Keyword: Graded Mean Integration, Fuzzy Inventory System, Lagrangian Method. Cite this Article: R. Kalaiarasi, M. Sumathi and M. Sabina Begum, Optimization of Fuzzy Inventory Model without Shortage Using Pentagonal Fuzzy Number, International Journal of Mechanical Engineering and Technology, 9(), 08, pp IType editor@iaeme.com

2 R. Kalaiarasi, M. Sumathi and M. Sabina Begum. INTRODUCTION Inventory control is very important field of our, dispersal and retail infrastructure where demand plays an important role in inventory models. In earlier period the uncertainties of inventory models are treated as randomness and are handled by using probability theory. The first quantitative analysis of inventory was the simple EO model. This model was initiated by [], [] and they are explored in academics and industries. Later [] analyzed many inventory systems. In some situations uncertainties are due to fuzziness, primarily introduced by [] is applicable. Later on, so many researchers worked on these areas. Many applications of fuzzy set theory can be found in [6]. In EO model we have identified the order size that minimizes the sum of annual total cost of inventory. Thus EO model is a useful approximation to many real life problems. [] used trapezoidal fuzzy number to fuzzify the order cost in the total cost of inventory model with backorder. Then, they got fuzzy total cost. They obtained the estimate of the total fuzzy cost through centroid to defuzzify. Further, in a series of papers, Yao et al.[,,7], considered the fuzzified problems for the inventory with or without backorder models. Were also available in the literature which includes a new operation on trapezoidal fuzzy number. The ranking of fuzzy numbers has been a concern in fuzzy multiple criteria attribute decision making since its inception. More than fuzzy ranking methods have been proposed since 976.various techniques are applied in the literature that compares fuzzy numbers. R.Helen G.Uma, introduced a new operation and ranking on pentagonl fuzzy numbers In [],they applied the extension principle to obtain the fuzzy total cost, and then for defuzzification, the study shows that the Graded mean integration method used. In this article investigate the inventory model without shortage is considered. Two inventory models are discussed. The first model is fuzzy inventory model for crisp order quantity. The second model, fuzzy inventory model for fuzzy order quantity. To find the estimate of total cost in the fuzzy sense and then derive the corresponding optimal order quantity. An algorithm is developed to find the optimal order quantity and also for minimizing the total cost. Sensitivity analysis is carried out through the numerical examples.. MATHEMATICAL MODEL.. Notations and Assumptions. The following notations and assumptions are used throughout to develop the integrated inventory model.... Notations We define the following symbols: C: Holding cost per unit quantity per unit time S: Set up or ordering cost per order T: Iength of the plan : Order quantity per cycle D Total demand over the planning time period [0, T] TC Total cost for the period[0, T] TC % Fuzzy total cost for the period [0, T] F De-fuzzifyed total cost for [0, T] F() Mimimum de-fuzzifyed total cost for [0, T] : Optimal order quantity editor@iaeme.com

3 Optimization of Fuzzy Inventory Model without Shortage Using Pentagonal Fuzzy Number... Assumptions In this paper, the following assumptions are considered: (i) Total demand is considered as constant. (ii) Time of plan is constant. (iii) Shortages are not allowed. (iv) Only holding cost and setup cost are fuzzy in nature.. Proposed Inventory Model In this model, the economic lot size is obtained by the following equation: SD CT (..) The total cost for the period [0, T] is equal to carrying cost plus ordering cost. CT SD TC + (..) The optimum and can be obtained by equating the first partial derivatives of TC to zero. TC * 0 Optimal order quantity * SD (..) CT Minimum total cost TC * SCDT (..). METHODOLOGY.. DEFINITION A fuzzy set A % on the given universal set X is a set of order pairs % ( µ ) where µ ( x ) [0,] is called a membership function. A.. DEFINITION %. The cut of is defined by A { x : ( x) ; 0} µ A {, A : } A x x x X.. DEFINITION A pentagon fuzzy number A % p ( a, a, a, a, a ) ) where a, a, a, a,a are real numbers and its membership is given below. 6 editor@iaeme.com

4 R. Kalaiarasi, M. Sumathi and M. Sabina Begum 0, x < a x a, a x a a a y a, µ A% p x a y + a x a a a, a x a a a a x, a x a a a 0, x > a.. DEFINITION A % is called a Canonical pentagon fuzzy number if it is a closed and bounded pentagon fuzzy number and its membership function is strictly increasing on the interval [ a,a ] and strictly decreasing on the interval[a,a ]. REMARK: Pentagon fuzzy number A % p ( a, a, a, a, a ) is the ordered quadruple ( P (x), ( y ), ( y ), P (x) ) for x [0,0.] andy [0.,] where, x a P ( x) ( x) + ( x) a a a, a a, a a, P x a x y a a y a.. Alpha cut The classical cut set is the set of elements whose degree of membership in A % ( a, a, a, a, a ) is no less than it is defined as p { / } A% x X µ x A A A ( ), ( ) [ 0, 0.] ( ), ( ) [ 0.,] P P for for (..) ( ) + a a a, for a a + a a a + a a, for a a + a a [ 0,0.] [ 0.,] (..) 7 editor@iaeme.com

5 Optimization of Fuzzy Inventory Model without Shortage Using Pentagonal Fuzzy Number.6. FUZZY ARITHMETICAL OPERATIONS UNDER FUNCTION PRINCIPLE.6.. ADDITION OF TWO CANONICAL PENTAGON FUZZY NUMBER: A% p ( a, a, a, a, a ) B% p ( b, b, b, b, b ) Let and be two Canonical pentagon fuzzy [ 0,] numbers for all. Let us add cuts A% and B % of A % p and B % p using internal arithmetic. A% B% ( ) + a a a, for a a + a a a + a a, for a a + a a ( ) + [ 0,0.] b b b, for b b + b b b + b b, for b b + b b [ 0,0.] [ 0.,] [ 0.,] (.6..) ( a + b ) ( a + b ) + ( a + b ) ( a b ) ( a b ) ( a b ), for [ 0,0. ] A% + B% ( a + b ) ( a + b ) + ( a + b ) ( a + b ), for ( a + b ) ( a + b ) + ( a + b ) ( a + b ).7. Graded mean integration representation method [ 0., ] The graded Mean Integration Representation of P with grade w A, where P ( () ()) ( a b ) 6( a b ) ( a b ) ( a b ) ( a b ) (.6..) (.6..) (.7.). 8. Extension of the Lagrangian method. Taha [] discussed how to solve the optimum solution of nonlinear programming problem with equality constraints by using Lagrangian Method, and showed how the Lagrangian method may be extended to solve inequality constraints. The general idea of extending the Lagrangian procedure is that if the unconstrained optimum the problem does not satisfy all 8 editor@iaeme.com

6 R. Kalaiarasi, M. Sumathi and M. Sabina Begum constraints, the constrained optimum must occur at a boundary point of the solution space. Suppose that the problem is given by Minimize y f(x) Sub to g i (x) 0, i,,, m. The non negativity constraints x 0 if any are included in the m constraints. Then the procedure of the Extension of the Lagrangian method involves the following steps. Step : Solve the unconstrained problem Min y f(x) If the resulting optimum satisfies all the constraints, stop because all constraints are redundant. Otherwise set K and go to step. Step : Activate any K constraints (i,e., convert them into equality) and optimize f(x) subject to the K active constraints by the Lagrangian method. If the resulting solution is feasible with respect to the remaining constraints and repeat the step. If all sets of active constraints taken K at a time are considered without encountering a feasible solution, go to step. Step : If K m, stop; no feasible solution exists. Otherwise set K K + and go to step.. FUZZY INVENTORY MODELS.. Fuzzy inventory model for crisp order quantity We consider the model in fuzzy environment. Since the ordering and holding cost are fuzzy in nature,we represent them by pentagonal fuzzy numbers. Let : fuzzy carrying or holding cost per unit quantity per unit time " : fuz zy set up or ordering cost per order The total demand and time of plan are considered as constants. Total cost is CT SD TC + The fuzzy total cost is T C%, (..) C% T S% D + (..) To apply Graded Mean method to defuzzify the fuzzy total cost, and then obtain the optimal order quantity * by using new arithmetic operation on -cut technique. Suppose (C, C, C, C,C ), and % (S, S, S, S,S ) are fuzzy pentagonal numbers, in LR form, where 0 < S < C and S, S, S, S,S,C, C, C,C and C are known positive numbers.from (..), we have: 9 editor@iaeme.com

7 Optimization of Fuzzy Inventory Model without Shortage Using Pentagonal Fuzzy Number T D ( ( (, % ) C% S% T D ( C, C, C, C, C ) ( S, S, S, S, S ) T T T T T D D D D D C, C, C, C, C S, S, S, S, S T T T C C + C, T T T C C C + A T T T T C C + C C, T T T T C C + C C B D D D S S, D D D S S D D D D S S + S S, D D D D S S S S + TC% ( C%, S% ) [ 0.,] [ 0,0.] T D T D T D C C + C [ 0,0. ] T D T D T D C C + C T D T D T D T D C C + C C T D T D T D T D C C + C C [ 0., ] (..) Defuzzifying TC ( C, S ) % % % by Graded Mean Method,we have 0 editor@iaeme.com

8 R. Kalaiarasi, M. Sumathi and M. Sabina Begum T D T D F C + C T D T D + C + C T D + C Computation of * at which F() is minimum, when * ( S + 6S + S + S S ) ( ) D T C C C C C (..) df 0 d and where d F 0 d > (..).. Fuzzy integrated inventory model for fuzzy order quantity In this section, we introduce an integrated inventory model by changing the crisp order qu antity into fuzzy production quantity. Suppose fuzzy production quantity )* be a pentagonal fuzzy number % (,,,, ) with 0 <. Thus we can get the fuzzy total production inventory cost T D T D T D C C + C [ 0, 0. ] T D T D T D C C + C P( TC% ( C%, S% )) T D T D T D T D C C + C C T D T D T D T D C C + C C [ 0.,] (..) We can obtain the Graded Mean Integration Representation of P( TC % ( C %, S % )) by formula (.7.) as (, ) P( TC% C% S% T D T D C + 6 C T D T D ) C S C S T D C (..) editor@iaeme.com

9 Optimization of Fuzzy Inventory Model without Shortage Using Pentagonal Fuzzy Number with 0 <. It will not change the meaning of formula (..) if we replace inequality conditions 0 <. into the following 0 inequality, 0, 0, 0 0 In the following steps, extension of the Lagrangian method is used to find the solutions of,,,, to minimize P( TC % ( C %, S % )) in (..). Step : Solve the unconstraint problem. Consider min P( TC % ( C %, S % )).To find the min P( TC % ( C %, S % )) we have to find the derivative of P( TC % ( C %, S %)) with respect to,,,,. CT SD + δ δ δ δ δ 6C T S D C T S D C T 6S D C T S D + Let all the above results partial derivatives equal to zero and solve,,,,. δ Let 0 Then S D C T δ Let 0 Then 0SD 6C T δ Let 0 Then 6S D C T δ Let 0 Then SD C T editor@iaeme.com

10 R. Kalaiarasi, M. Sumathi and M. Sabina Begum δ Let 0 Then SD C T Because the above show that > > > >, it does not satisfy the constraint 0 <. Therefore set K and go to Step. Step : Convert the inequality constraint 0 into equality constraint 0 and optimize P( TC % ( C %, S % )) subject to 0 by the Lagrangian Method. We have Lagrangian function as L(,,,,, λ) P( TC% ( C%, S% ) λ ( ) Taking the partial derivatives of (,,,,, ) ). L λ with respect to,,,,,and λ. Let all the partial derivatives equal to zero and solve,,,, andλ. Then we get, ( S D S D) 6C T C T 6S D C T SD C T SD C T Because the above results show that > >, it does not satisfy the constraint 0 <.. Therefore it is not a local optimum. Similarly we can get the same result if we select any other one inequality constraint to be equality constraint, therefore set K and go to Step. Step : Convert the inequality constraints 0, 0, into equality constraints 0 and 0. We optimize P( TC % ( C %, S % )) Subject to 0 and 0 by the Lagrangian Method. Then the Lagrangian method is (,,,,, λ, λ ) ( (, )) λ ( ) λ ( ) L P TC % C % S %, we take the partial derivatives of (,,,,,, ) L λ λ with respect to,,,,, λ, λ and Let all the above results partial derivatives equal to zero and solve,,,,. λ, and λ. 6S D + 0S D S D C T + 6C T C T SD C T editor@iaeme.com

11 Optimization of Fuzzy Inventory Model without Shortage Using Pentagonal Fuzzy Number SD C T The above results >, does not satisfy the constraint 0 <. Therefore it is not a local optimum. Similarly we can get the same result if we select any other two inequality constraints to be equality constraint, therefore set K and go to Step. Step : Convert the inequality constraints 0, 0, 0,into equality constraints 0, 0 and 0. We optimize P( TC % ( C %, S % )) Subject to 0, 0 and 0 by the Lagrangean Method. The Lagrangean function is given by L,,,,, λ, λ, λ P( TC% C%, S% ) λ λ λ we take the partial derivatives of (,,,,,, ) L λ λ λ with respect to,,,,, +,,+.,+ / and let all the partial derivatives equal to zero and solve,,,,, +,,+., and + /. Then we get ) S D + 6S D + 0S D S D C T + C T + 6C T C T SD C T The above results >, does not satisfy the constraint 0 <. Therefore it is not a local optimum. Similarly we can get the same result if we select any other two inequality constraints to be equality constraint, therefore set K and go to Step. Step : Convert the inequality constraints 0, 0, 0,and 0 into equality constraints 0, 0, 0 and - 0.We optimize P( TC % ( C %, S % )) Subject to 0, 0, 0 and - 0. by the Lagrangean Method. The Lagrangean function is given by (,,,,, λ, λ, λ, λ ) ( % ( %, %)) λ ( ) λ ( ) λ ( ) L P TC C S λ ( ) we take the partial derivatives of (,,,,,,,, ) L λ λ λ λ with respect to,,,, +,,+.,+ /,+ and let all the partial derivatives equal to zero and solve,,,,, +,,+., + /,and+ Then we get ) S D + S D + 6S D + 0S D S D C T + C T + C T + 6C T C T Because the above solution )* (,,,, ) satisfies all inequality constraints, the procedure terminates with )* as a local optimum solution to the problem. Since the above local optimum solution is the only one feasible solution of formula (..). So it is an optimum solution of the inventory model with fuzzy order quantity according to extension of editor@iaeme.com

12 R. Kalaiarasi, M. Sumathi and M. Sabina Begum the Lagrangian Method. Let )( Then the optimal fuzzy order quantity is )( ( *, *, *,*, * ) S D + S D + 6S D + 0S D S D * C T + C T + C T + 6C T C T. ALGORITHM FOR FINDING FUZZY TOTAL COST AND FUZZY OPTIMAL ORDER UANTITY Step : Calculate total cost for the crisp model for the given crisp values of C, S, T and D. Step : Now, determine fuzzy total cost using fuzzy cut operations on fuzzy holding cost, and fuzzy ordering cost, taken as pentagonal fuzzy numbers. Step : Use Graded Mean Integration, then we find the fuzzy order quantity )* which can be obtained by putting the first derivative of F() is equal to zero zero and where the second derivative is positive. Step : Use extension of the Lagrangian Method, we find the fuzzy optimal order quantity )( ( *, *, *,*, * ) is the special method for pentagonal fuzzy number, which can be putting first derivative of P( TC % ( C %, S % )) is equal to zero. Let ) ( Then the optimal fuzzy order quantity is )( ( *, *, *,*, * ). Step : To check whether, the economic order quantity obtained by Graded Mean Integration is same as the crisp order quantity. 6. NUMERICAL EXAMPLE Example: Crisp Model: Let us consider an inventory system with the following data: S 0/- Per unit, D 00 unit/year, C Rs./- Per unit, 6 Days By minimizing the total cost we obtain the optimal value * 6.67 and TCRs.00/- Fuzzy Model: Let demand per unit time. D00. Let the fixed setup cost be a pentagonal number S % (7,8,,,), C % (9,0,,,6),T 6 Days. Then we find that * 6.67 and TC% ( C%, S % ) Rs.00.00/- editor@iaeme.com

13 Optimization of Fuzzy Inventory Model without Shortage Using Pentagonal Fuzzy Number Again, let S % (8,9,0,,6), C % (9,,,,),Then, we get the same value * 6.67 and TC% ( C%, S% ) Rs.00.00/- S.NO DEMAND S % (7,8,,,), C % (9,0,,,6) S % (8,9,0,,6), C % (9,,,,) * TC% ( C%, S% ) * TC% ( C%, S% ) From the above table we observed that: (i) The economic order quantity obtained by Graded Mean Integration is equal to crisp economic order quantity. (ii) Total cost obtained by Graded Mean Integration is equal to crisp total cost. (iii) For different values of ordering quantity, the economic order quantity remains fixed. Same is true for total cost. 7. CONCLUSION In the fuzzy environment to discuss the inventory model without shortage is discussed. In addition, the function principle is using the cut operation with pentagonal fuzzy number.we find that the optimal fuzzy order quantity )( ( *, *, *,*, * ) is the special type of pentagonal fuzzy number. The optimal solution of our proposed models can be specified to meet the classical inventory models REFERENCES [] Harris, F., Operations and cost, AW Shaw Co. Chicago, (9). [] Wilson, R., A scientific routine for stock control. Harvard Business Review,, 9, 6 8, [] Hadley, G., Whitin T.M., Analysis of inventory systems, Prentice-Hall, Englewood clipps, NJ, 96. [] Zadeh L.A., Fuzzy sets, Information Control, 8, 8-, 96. [] Zadeh, L.A., Bellman, R.E., Decision Making in a Fuzzy Environment, Management Science, 7, 970, 0-6. [6] Jain, R., Decision making in the presence of fuzzy variables, IIIE Transactions on systems, Man and Cybernetics, 7, 976, [7] Kacpryzk, J., Staniewski, P., Long-term inventory policy-making through fuzzy-decision making models, Fuzzy Sets and Systems, 8, 98, 7-. [8] Zimmerman, H.J., Using fuzzy sets in operational Research, European Journal of Operational Research, 98, [9] Urgeletti Tinarelli, G., Inventory control models and problems, European Journal of Operational Research,, 98, -. [0] Park, K.S., Fuzzy Set Theoretical Interpretation of economic order quantity, IEEE Trans. Systems Man. Cybernet SMC, 7, 987, [] Chan,Wang, Backorder fuzzy inventory model under function principle, Information Science, 9, 996, -, editor@iaeme.com

14 R. Kalaiarasi, M. Sumathi and M. Sabina Begum [] Vujosevic, M., Petrovic, D., Petrovic, R., EO Formula when Inventory Cost is Fuzzy, International Journal of Production Economics,, 996, 99-0 [] Yao, J.S., Lee, H.M., Fuzzy Inventory with or without backorder for fuzzy order quantity with trapezoidal fuzzy number, Fuzzy sets and systems, 0, 999, -7. [] Yao, J.S., Lee, H.M., Economic order quantity in fuzzy sense for inventory without backorder model, Fuzzy Sets and Systems, 0, 999, -. [] W. Ritha, R. Kalaiarasi, Young Bae Jun, Optimization of fuzzy integrated vendor-buyer inventory models, Annals of Fuzzy Mathematics and Informatics pp [6] Hsieh, C.H., Optimization of Fuzzy Production Inventory Models, Information Sciences, 6, 00, -, 9-0. [7] Yao, J.S., Chiang, J., Inventory without back order with fuzzy total cost and fuzzy storing cost defuzzified by centroid and signed distance, European Journal of Operational Research, 8, 00, [8] R. Kalaiarasi And W.Ritha, Optimization Of Fuzzy Integrated Two-Stage Vendor-Buyer Inventory System,International Journal of Mathematical Sciences and Applications,Vol. No. (May, 0) [9] Abbasbandy.s., Asady.B., Ranking of fuzzy numbers by sign distance, Information sciences,76,0-6,006. [0] Cheng.C.H, A new approach for ranking fuzzy numbers by distance method, Fuzzy sets andsystem,9,07-7,988. [] Chu.T.C & Tsao.C.T., Ranking fuzzy numbers with an area between the centroid point and original point, Computers and mathematics with applications,,,00. [] Dubois.D & Prade.H., Operations of fuzzy numbers,internet.j.systems sci 9(6),6-66,978. [] R.Helen, G.Uma,A new operation and ranking on pentagonfuzzy numbers, int jr. of mathematical sciences & applications 7 editor@iaeme.com