Fuzzy Optimization Technique in EOQ Model Using Nearest Interval Approximation Method

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1 ISSN(Online): ISSN (Print): International Journal o Innovative esearch in Science, (An ISO 397: 007 Certiied Organization) Vol. 4, Issue 0, October 05 Fuzzy Optimization Technique in EOQ Model Using Nearest Interval Approimation Method A.Faritha Asma, E.C.Henry Amithara Assistant Proessor, Department o Mathematics, Government Arts College, Trichy, Tamil Nadu, India Associate Proessor, Department o Mathematics, Bishop Heber College, Trichy, Tamil Nadu, India ABSTACT: This paper discusses an Economic Order Quantity (EOQ) model with shortage, where the setup cost, the holding cost, the shortage cost are considered as uzzy numbers. The uzzy parameters are then transormed into corresponding interval numbers. Minimization o the interval obective unction (obtained by using interval parameters) has been transormed into a classical multi-obective EOQ problem. The order relation that represents the decision maer s preerence among the interval obective unction has been deined by the right limit, let limit,center and hal width o an interval. This concept is used to minimize the interval obective unction. The problem has been solved by uzzy programming technique. Finally, the proposed method is illustrated with a numerical eample KEYWODS: Inventory, Interval number, EOQ, Fuzzy sets, Fuzzy optimization technique, Multi-obective Programming. I. INTODUCTION In traditional mathematical problems, the parameters are always treated as deterministic in nature. However, in practical problem, uncertainty always eists. In order to deal with such uncertain situations uzzy model is used [],[5].in such cases, uzzy set theory, introduced by Zadeh [4] is acceptable. There are several studies on uzzy EOQ model. in et al. [7] have developed a uzzy model or production inventory problem. Katagiri and Ishii[5] have proposed an inventory problem with shortage cost as uzzy quantity This paper discusses a uzzy EOQ model with shortage. Demand, Holding cost, ordering cost, shortage cost are taen as triangular uzzy numbers, and epression or uzzy cost is established. For minimizing the cost unction we transormed the uzzy obective unction into interval obective unction. Now, this single obective unction is then converted to multi-obective problem by deining let limit, right limit and center o the obective unction. This multiobective is then solved by uzzy optimization technique. inear membership unction is considered here. This model is illustrated by a numerical eample The article is organized as ollows: In section preliminary deinitions o uzzy set, interval number, basic interval arithmetic optimization in interval situation and nearest interval approimation is briely described. Section contains model ormulation. The uzzy optimization technique is section 3. In section 4 the process is illustrated by a numerical eample and in the last section the entire wor is concluded. II. EATED WOK For several years, classical economic order quantity (EOQ) problems with dierent variations were solved by many researches and had been published since 95. The main area in which research articles have been published, may be classiied as crisp and uzzy inventory models, relating to Economic order quantity, economic production quantity, optimization, deuzzyication. The research papers relating to the optimization problems and deuzzyications are o special interest. The maor assumption in the classical EOQ model is that the demand rate is constant and deterministic. Copyright to IJISET DOI:0.5680/IJISET

2 ISSN(Online): ISSN (Print): International Journal o Innovative esearch in Science, (An ISO 397: 007 Certiied Organization) Vol. 4, Issue 0, October 05 The main obective o this paper is to construct uzzy EOQ model and to determine the optimum order quantity so as to minimize the average total cost. Preliminaries Deinition : A uzzy set is characterized by a membership unction mapping elements o a domain, space, or universe o discourse X to the unit interval [0,]. (i.e) A={(, µ A ()) ; ϵ X.}, here µ A :X [0,] is a mapping called the degree o membership unction o the uzzy set A and µ A is called the membership value o ϵ X in the uzzy set A. Deinition : et be the set o all real numbers. An interval, may be epressed as a a, a : a a, a, a () where a and a are called the lower and upper limits o the interval a, respectively. I a = a then a = [a, a ] is reduced to a real number a, where a= a = a. alternatively an interval a can be and wa a a are respectively the mid-point and hal-width o the interval a.the set o all interval numbers in is denoted by I(). Optimization in interval environment epressed in mean-width or center-radius orm as a =, m a w a,where ma a a Now we deine a general non-linear obective unction with coeicients o the decision variables as interval numbers as Minimize Z n r a, a i i i l n q b, b i i i () subect to >0, =,,,n and ϵ S where S is a easible region o, 0 a a, 0 b b and r i,q are positive numbers. Now we ehibit the i ormulation o the original problem () as a multi-obective non-linear problem. Now ( ) Z( ) Z ( ), Z ( ) Z can be written in the orm a i where i Z( ) l n b Z ( ) n n i i a i i l n b i i r q r q The center o the obective unction i ZC ( ) Z( ) Z( ) (5) Thus the problem () is transormed in to minimizez ( ), Z ( ); S (6) C i i (3) (4) Copyright to IJISET DOI:0.5680/IJISET

3 ISSN(Online): ISSN (Print): International Journal o Innovative esearch in Science, (An ISO 397: 007 Certiied Organization) Vol. 4, Issue 0, October 05 subect to the non-negativity constraints o the problem, where Z C,Z are deined by (4) and (5). III. NEAEST INTEVA APPOXIMATION METHOD According to Gregorzewsi [3] we determine the interval approimation o a uzzy number as: et A =(a,a,a 3 ) be an arbitrary triangular uzzy number with a α-cuts [A (α),a (α)] and with the ollowing membership unction a ; a a a a a a a3 0; otherwise (7) Then by nearest interval approimation method, the lower limit C and upper limit C o the interval are 3 ( ) ; A a3 a 0 0 C ( ) A d a a a d a a C ( ) A d a a a d a a 3 Thereore, the interval number considering A as triangular uzzy number is 3 (8) a a a a,. Model ormulation: In this model, an inventory with shortage is taen into account. The purpose o this EOQ model is to ind out the optimum order quantity o inventory item by minimizing the total average cost. We discuss the model using the ollowing notations and assumptions throughout the paper. Notations: C : Holding cost per unit time per unit quantity. C : Shortage cost per unit time per unit quantity. C 3 : Setup cost per period D: The total number o units produced per time period. Q : The amount which goes into inventory Q : The unilled demand Q: The lot size in each production run. Assumptions: (i) (ii) (iii) (iv) Demand is nown and uniorm. Production or supply o commodity is instantaneous. Shortages are allowed. ead time is zero. et the amount o stoc or the item be Q at time t=0 in the interval (0,t(=t +t )), the inventory level gradually decrease to meet the demands. By this process the inventory level reaches zero level at time t and then shortages are allowed to occur in the interval (t,t). The cycle repeats itsel.(fig. ) Copyright to IJISET DOI:0.5680/IJISET

4 ISSN(Online): ISSN (Print): International Journal o Innovative esearch in Science, (An ISO 397: 007 Certiied Organization) Vol. 4, Issue 0, October 05 Fig. Inventory level Q Q 0 t t Time Q t The order level Q>0 which minimizes the average total cost (Q) per unit time is given by Q Q D min C( Q) C C C3 Q Q Q (9) Up to this stage, we are assuming that the demand, ordering cost, holding cost etc. as real numbers i.e.o ied value. But in real lie business situations all these components are not always ied, rather these are dierent in dierent situations. To overcome these ambiguities we approach with uzzy variables, where demand and other cost components are considered as triangular uzzy numbers. et us assume the uzzy demand D ( D, D, D ) uzzy holding cost C ` ( C, C, C ),uzzy shortage cost, uzzy ordering cost C 3 ( C3, C3, C3 ) eplacing the real valued variables D,C,C & C 3 by the triangular uzzy variables D, C, C, C we get, 3 Q Q D C ( Q) C C C 3 Q Q Q (0) Now we represent the uzzy EOQ model to a deterministic orm so that it can be easily tacled. Following Grzegorzewsi [3], the uzzy numbers are transormed into interval numbers as D ( D, D, D ) =[D,D ] C ` ( C, C, C ) =[C,C ] C ` ( C, C, C ) = [C,C ] C 3 ( C3, C3, C3 ) =[C,C ]. 3 3 Using the above epression (0) becomes C ( Q), () Where, Copyright to IJISET DOI:0.5680/IJISET

5 ISSN(Online): ISSN (Print): International Journal o Innovative esearch in Science, (An ISO 397: 007 Certiied Organization) Vol. 4, Issue 0, October 05 Q Q D C C C3 Q Q Q Q Q D C C C (3) 3 Q Q Q The composition rules o intervals are used in these equations. Hence the proposed model can be stated as Minimize Q, Q, (4) Generally, the multi-obective optimization problem(4), in case o minimization problem, can be ormulated in a conservative sense rom (3) as Minimize Q, Q, (5) C Subect to Q 0.Where C. Here the interval valued problem (4) is represented as Q, Q, Q, Minimize C Subect to Q 0 (6) The epression (6) gives a better approimation than those obtained rom (4). Fuzzy programming technique or solution: To solve multi-obective minimization problem given by (6), we have used the ollowing uzzy programming technique. For each o the obective unctions ( Q), (Q), ( Q ),we irst ind the lower bounds,, (best C () C values) and the upper bounds U, UC, U (worst values), where,, C are the aspired level achievement and U, UC, U are the highest acceptable level achievement or the obectives ( Q), C (Q), ( Q) respectively and d U is the degradation allowance or obective ( ) Q, =,C,. Once the aspiration levels and degradation allowance or each o the obective unction has been speciied, we ormed a uzzy model and then transorm the uzzy model into a crisp model. The steps o uzzy programming technique is given below. Step : Solve the multi-obective cost unction as a single obective cost unction using one obective at a time and ignoring all others. Step : From the results o step, determine the corresponding values or every obective at each solution derived. Step 3: From step, we ind or each obective, the best and worst U value corresponding to the set o solutions. The initial uzzy model o (0) can then be stated as, in terms o the aspiration levels or each obective, as ollows: ind Q satisying,, C, subect to the non negativity conditions Step 4: Deine uzzy linear membership unction ;, C, or each obective unction is deined ; by (7) ; U d 0; U Step 5: Ater determining the linear membership unction deined in(7) or each obective unctions ollowing the problem (6) can be ormulated an equivalent crisp model Ma α, ( );, C,. Copyright to IJISET DOI:0.5680/IJISET

6 ISSN(Online): ISSN (Print): International Journal o Innovative esearch in Science, α 0,Q 0. (An ISO 397: 007 Certiied Organization) Vol. 4, Issue 0, October 05 IV. NUMEICA EXAMPE In this section, the above mentioned algorithm is illustrated by a numerical eample. Here the parameters demand, ordering cost, holding cost and shortage cost are considered as triangular uzzy numbers (TFN). Ater that, the uzzy numbers are transormed into interval numbers using nearest interval approimation ollowing [3]. et C =5, C =5, C 3 =00, D=5000 units. Taing these as triangular uzzy numbers we have, C =(3,5,7), C =(,5,3), C 3 =(85,03,09), D =(4000,5000,6000). The uzzy numbers D, C, C, C 3 are transormed into interval numbers as, uzzy numbers D, C, C, C 3 are transormed into interval numbers as, D =[D,D ]=[4500,5500] C =[C,C ]=[4,6] C = [C,C ]=[3,8] C 3 =[C 3,C 3 ]=[94,06] Individual minimum and maimum o obective unctions,, are given in Table. Obective unctions C Now we calculate min,, Optimize ' = ' C = ' = =69.69; U ma,, min,, =03.53UC ma C, C, C min,, =400.9U ma,, C C C C C Optimize C Optimize = '' = '' C = '' =40.66 = = Using the equation (8), we ormulate the ollowing problem as: Ma α Q Q 4500 (4) (3) 94 (8.09) Q Q Q Q Q 5000 (5) (5.5) 00 (4.3) Q Q Q ''' =69.69 ''' C =03.53 ''' =400.9 Copyright to IJISET DOI:0.5680/IJISET

7 ISSN(Online): ISSN (Print): International Journal o Innovative esearch in Science, (An ISO 397: 007 Certiied Organization) Vol. 4, Issue 0, October 05 Q Q 5500 (6) (8) 06 (7.57) Q Q Q (8) esults and Discussions: The solutions obtained rom (8) are given in table and 3. Using non- linear programming technique eq.(8) solved and the optimum value o α is ound. It is given in table. Table : optimum value o α Maimum α The optimum results or the total average cost, the economic order quantity and the level o inventory are ound and given in table 3. Table 3: The optimum results C Q Q The comparison is done between crisp and uzzy model and it is represented in table 4. Table 4:comparison table model C C C 3 Q Q C(Q) Α Crisp Crisp Crisp Fuzzy [4,6] [3,8] [94,06] [698.7,403.] V. CONCUSION In this paper, we have presented an inventory model with shortage, where carrying cost, shortage cost, ordering or setup cost and demand are assumed as uzzy numbers instead o crisp or probabilistic in nature to mae the inventory model more realistic. At irst, epression or the total cost is developed containing uzzy parameters. Then each uzzy quantity is approimated by interval number. Ater that the problem o minimizing the cost unction is transormed into a multi-obective inventory problem, where the obective unctions are let limit, right limit and the center o the interval unction. Fuzzy optimization technique is then used to ound out the optimal results. A numerical eample illustrates the proposed method. EFEENCES [] Bellman,.E., Zadeh,.A.; Decision-maing in a uzzy environment. Management science;(970);7(4),464. [] Chen, S.H., Wang, C.C., Chang, S.M.; Fuzzy economic production quantity model or items with imperect quality. International Journal o Innovative Computing, Inormation and Control ;(007);3(), [3] Grzegorzewsi,P.; Nearest interval approimation o a uzzy number Fuzzy Sets and Systems; (00); 30, [4] Ishibuchi, H.,Tanaa, H., Multi obective programming in optimization o the interval obective unction. European Journal o Operational esearch; (990);48,9-5 [5] Katagiri,H.,Ishii,H.; Some inventory problems with uzzy shortage cost. Fuzzy Sets and Systems; 000;(), [6] i,., Kabadi, S.N.,Nair, P.K.; Fuzzy models or single-period inventory problem. Fuzzy Sets and Systems;00; 3(3), [7] in, D.C., Yao, J.S.; Fuzzy economic production or production inventory. Fuzzy Sets and Systems; 000;, [8] Mahapatra, G. S., oy, T.K.; Fuzzy multi-obective mathematical programming on reliability optimization model. Applied Mathematics and Computation; 006;74(), Copyright to IJISET DOI:0.5680/IJISET

8 ISSN(Online): ISSN (Print): International Journal o Innovative esearch in Science, (An ISO 397: 007 Certiied Organization) Vol. 4, Issue 0, October 05 [9] Par, K.S.;Fuzzy set theoretic interpretation o economic order quantity. IEEE Transactions on systems, Man and Cybernetics, SMC;(987);7, [0] Susovan Charaborty, Madhumagal Pal, Prasun Kumar Naya, Solution o interval-valued manuacturing inventory models with shortages, International J. o Engineering and Physical Sciences, 4() (00) [] Susovan Charaborty, Madhumagal Pal, Prasun Kumar Naya, Multi section technique to solve interval-valued purchasing inventory models without shortages, Journal o Inormation and Computing Sciences, 5(3), (00) [] Tang, S.; Interval number and uzzy number linear programming. Fuzzy Sets and Systems; (994); 66, [3] Wole, M.,A.; Interval mathematics, algebraic equations and optimization; Journal o Computational and Applied Mathematics; (000); 4, [4] Zadeh,.A.; Fuzzy sets. Inormation and control;(965); 8, [5] Zimmermann, H.J.; Description and optimization o uzzy system. International Journal o general system; (976) ; (4),09-5. [6] Zimmermann, H.J.; Fuzzy linear programming with several obective unctions.. Fuzzy Sets and Systems; (978);, Copyright to IJISET DOI:0.5680/IJISET