Fuzzy EOQ Model for Time-Deteriorating Items Using Penalty Cost

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1 merican Journal of Operational Research 6 6(: -8 OI:.59/j.ajor.66. Fuzzy EOQ Moel for ime-eteriorating Items Using Penalty ost Nalini Prava Behera Praip Kumar ripathy epartment of Statistics Utkal University Bhubaneswar Inia bstract In this paper a fuzzy inventory moel for time-eteriorating items using penalty cost uner the conitions of infinite prouction rate is formulate an solve. Penalty cost is assume to be linear an exponential. Fuzziness is introuce in the cost component of holing cost an set up cost. eman rate is also assume to be fuzzy. In fuzzy environment all relate parameters are assume to be trapezoial. Representing these three costs trapezoial fuzzy numbers the optimum orer quantity is calculate using signe istance metho an grae mean integration metho for efuzzification. Numerical examples have been given in orer to show the applicability of the propose moel. Sensitivity analysis is also carrie out to etect the shift in the variables of interest of the system. Keywors Fuzzy Inventory Moel rapezoial Fuzzy Number efuzzification Penalty ost. Introuction Inventory control is very important for both real worl applications an research purpose. In conventional inventory moels the uncertainties are treate as ranomness an hanle using probability theory. he most wiely use inventory moel is the Economic orer quantity (EOQ moel. his moel was evelope arris [] Wilson []. Later aley [] analyze many inventory systems. But uncertainties ue to fuzziness primarily introuce Zaeh []. Zaeh et al [5] propose some strategies for ecision making in fuzzy environment. Kacpryzk et al [6] iscusse some long-term inventory policy making through fuzzy-ecision making moels. Proucts like fresh vegetables fruits bakery items etc. o not eteriorate at the beginning of the perio but they continuously eteriorate after some time. s a result the selling price of such prouct ecreases which can be consiere as a penalty cost. Srivastava an Gupta [] have propose an EOQ moel for time-eteriorating items using penalty cost. Fujiwara an Perera [8] have propose an EOQ moel for time continuously eteriorating items using linear an exponential penalty cost. Pevekar an Nagare [] evelope an inventory moel for timely eteriorating proucts consiering penalty cost an shortage cost. Park [7] an Vujosevic et al [9] evelope the inventory moel in fuzzy sense whereas orering cost an holing cost are orresponing author: nalinipbehera@gmail.com (Nalini Prava Behera Publishe online at opyright 6 Scientific & caemic Publishing. ll Rights Reserve represente fuzzy numbers. Margatham an Lakshmievi [] have propose a fuzzy inventory moel for eteriorating items with price epenent eman. Previously Park [7] has represente cost as trapezoial fuzzy numbers wherein Vujosevic et al [9] represente orering cost triangular fuzzy number an holing cost trapezoial fuzzy number. Jaggi et al [] applie the extension principle to obtain the fuzzy total cost an they efuzzifie the fuzzy total cost using grae mean integration metho an signe istance metho. In this article fuzzy EOQ moel for time eteriorating items using penalty cost is consiere where holing cost set up cost an eman rate are assume as trapezoial fuzzy numbers. For efuzzification of the total cost function signe istance metho an grae mean integration metho are use.. efinitions an Preliminaries efinition.. trapezoial fuzzy number ( a b c is represente with membership function as: x a L( x when a x b; b a whenb x c; ( x x R( x whenc x ; c otherwise a b c an efinition.. Suppose

2 Nalini Prava Behera et al.: Fuzzy EOQ Moel for ime-eteriorating Items Using Penalty ost B ( a b c are two trapezoial fuzzy number then arithmetic operations are efine as. B ( a b a b a b a b. B ( a b ab ab ab Θ B a b a b a b a b. a. a a a φ B b b b b 5. a ( a a a a ( a a a a < efinition.. Let ( a b c be a trapezoial fuzzy number then the signe istance metho of is efine as ( [ ] L R Where [ L R ] [ a ( b a ( c ] is a - cut of fuzzy set which is a close interval. ( a b c efinition.. Let ( a b c be a trapezoial fuzzy number then the grae mean integration representation of is efine as P [ ] a b c 6 L R. ssumptions an Notations he moel is evelope on the following assumptions an notations... ssumptions (i single prouct is consiere over a prescribe perio of unit of time. (ii he replenishment occurs instantaneously at an infinite rate. (iii No back orer is permitte. (iv elivery leas time zero... Notations Q Number of items receive at the beginning of the perio. eman rate. Inventory holing cost. Set-up cost per cycle. ime perio at which eterioration of prouct start. ( verage total variable cost per unit time. Length of replenishment cycle which will not excee prouct lifetime. Optimum value of. QQ Optimum value of Q. Fuzzy inventory holing cost. Fuzzy set up cost per cycle. Fuzzy eman rate ( verage total fuzzy variable cost per unit time. s efuzzifie value of ( applying signe istance metho. ( grae mean integration metho. efuzzifie value of ( applying. Mathematical Moel (for Infinite prouction rate.. risp Moel In this context we have consiere two types of penalty cost function of age (i Linear (ii Exponential penalty cost functions as a measurement of utility of the prouct. linear penalty cost function t t P (t π ; otherwise which gives the cost of keeping one unit of prouct in stock until age t where be the time perio at which eterioration of prouct starts an π is a constant. here will be no penalty cost incurre upon the proucts up to time perio (. n exponential penalty cost function is taken as ( t P (t e t otherwise Which also gives the cost of keeping one unit of prouct in stock until age t where be the time perio at which eterioration of prouct starts an an are constants. he total variable cost per cycle time consists of the inventory holing cost set up cost an penalty cost. Since the eman rate is unit per time. he total eman in one cycle of time-interval is. he number of items receive at the beginning of the perio is Q (.

3 merican Journal of Operational Research 6 6(: -8 Propose Inventory Moel in risp Sense ase I. When linear penalty cost function is use linear penalty cost function P ( π ( t t which gives the cost of keeping one unit of prouct in stock until age t where be the time perio at which eterioration of prouct start an ππ is constant. he cost ue to the eterioration of the prouct elivere uring the perio (t tt is given π ( t t. hus penalty cost ue to the eterioration of the prouct elivere is given uring the time interval ( π ( t t π Now inventory holing cost for the perio ( is given. Q ( Q herefore the average total variable cost per unit time ( is given π π ( π (. he optimal solution is obtaine ifferentiating ( with respect to an equating it to zero. hen the optimal cycle time is obtaine an expresse as π ( π he optimal economic orer quantity putting value of in eq.(. Q ( π ( π (. Q is obtaine (. From the above expressions (. an (. it is clear that if there is no perishability (i.e. π then these two expression become same as that of the non-perishable lot size moel. ase II. When exponential penalty cost function is use n exponential penalty cost function ( t P( t ( e t which gives the cost of keeping one unit of prouct in stock until age t where be the time perio at which eterioration of prouct starts an an are constants. he cost ue to the eterioration of the prouct elivere uring the perio ( t t t is given ( e t t. he penalty cost ue to the eterioration of the prouct elivere uring the time interval ( is given ( t ( ( e t e [( ( ] herefore total variable cost per unit time is given ( ( ( e ( By using secon orer approximation of the exponential ( term e in ( We get ( Τ he optimal solution is obtaine ifferentiating ( with respect to an equating it to zero. hen the optimal cycle time is obtaine an expresse as ( he optimal economic orer quantity putting value of in eq.(. Q ( ( (.5 Q is obtaine (.6 From the expression (.5 an (.6 it is clear that if an π then these two expressions are same as (. an (.... Fuzzy Moel ase I. When linear penalty cost function is use ue to uncertainty in the environment it is not easy to efine all the parameters precisely. ccoringly we assume some of these parameters an may change with some limit. Let ( ( an ( are trapezoial fuzzy numbers. he total variable cost per unit time in fuzzy sense is given π π ( π We efuzzify the fuzzy total cost ( using signe istance metho an grae mean integration metho. (i By Signe istance metho total cost is given s ( [ s ( s ( s ( s ( ]

4 Nalini Prava Behera et al.: Fuzzy EOQ Moel for ime-eteriorating Items Using Penalty ost Where π π ( π s π π ( π s π π ( π s π π ( π s ( ( ( ( ( s s s s s o minimize total cost function per unit time ( s the optimal value of can be obtaine solving the following equation: s ( (.7 Provie Equation (.7 is equivalent to ( ( c s > (.8 π π π π π π π π fter simplification we get the optimal cycle time obtaine an expresse as is ( π ( ( π ( he optimal economic orer quantity Q is Q (ii By Grae mean integration metho total cost is given 6 Where ( ( ( π π π π π π π π π π π ( π 6 o minimize total cost function per unit time ( the optimal value of can be obtaine solving the following equation. ( (.9 Provie ( ( c > (. Equation (.9 is equivalent to π π π π 6 π π π π fter simplification we get the optimal cycle time is obtaine an expresse as ( π ( ( π ( he optimal economic orer quantity Q is Q ase II. when exponential penalty cost function is use ue to uncertainty in the environment it is not easy to efine all the parameters precisely ccoringly we assume some of these parameters an may change

5 merican Journal of Operational Research 6 6(: -8 5 with some limit. Let an are as trapezoial fuzzy numbers. he total variable cost per unit time in fuzzy sense is given We efuzzify the fuzzy total cost signe istance an grae mean representation methos. (i By Signe istance metho total cost is given s s s s s Where s o minimize total cost function per unit time s the optimal value of can be obtaine solving the following equation: s (. Provie > c s (. Equation. is equivalent to fter simplification we get the optimal cycle time is obtaine an expresse as he optimal economic orer quantity Q is Q (ii By Grae mean integration metho total cost is given 6 Where 6 o minimize total cost function per unit time the optimal value of can be obtaine solving the following equation: (. Provie > c (. Equation (. is equivalent to 6 fter simplification we get the optimal cycle time is obtaine an expresse as he optimal economic orer quantity Q is Q 5. Numerical Illustration he ata for the stuy have been solicite after an

6 6 Nalini Prava Behera et al.: Fuzzy EOQ Moel for ime-eteriorating Items Using Penalty ost intensive search of the following research papers. ripathy P.K. an Prahan S. [] Srivastava M. an Gupta R. [] Jaggi. K. Pareek S. Sharma. an Nihi. [] 5.. risp Moel Let P 5 units per ay units per ay Rs.. per ay 5 ays.99. ase-. when linear penalty cost function is use then Optimum cycle time 5.9 ays Optimum orer quantity Q 55.7 units ase-. when exponential penalty cost function is use then Optimum cycle time. Optimum orer quantity Q. Sensitivity nalysis of risp Moel he sensitivity analysis is performe for checking the effectiveness of the EOQ moel for infinite prouction rate with respect to the parameter on optimum cycle time an optimum orer quantity Q. able an able epict the values of the optimum policies for ifferent values of the parameter in case of linear an exponential penalty cost. Percentage changes of these values are shown with respect to the parameter 5 in the ata set are taken. he able shows the effect of parameter on optimum policies in case of linear penalty cost. If the value of the parameter is increase 6% the value of optimum cycle time an optimum orer quantity are also increase 56%. Further if the parameter is ecrease % the optimum cycle time an optimum orer quantity are ecrease 7%. able. Effect of parameter on optimal policies in case of linear penalty cost hange (% in QQ -6%.6(-5% 7.8(-5% -%.(-7% 7.(-7% 5 5.9(% 55.7(% 6 % 6.5(8% 8.5(8% 8 6% 8.(56%.(56% he able shows the effect of parameter on optimum policies in case of exponential penalty cost. If the value of the parameter is increase 6% the value of optimum cycle time an optimum orer quantity are also increase 58%. Further if the parameter is ecrease % the optimum cycle time an optimum orer quantity are ecrease 9%. able. Effect of parameter on optimal policies in case of exponential penalty cost hange (% in QQ -6%.88(-57% 56.(-57% -%.55(-9% 6.5(-9% 5.(%.(% 6 % 5.7(9% 58.(9% 8 6% 7.(58%.(58% 5.. Fuzzy Moel Let P 5 units per ay 5 ays.99 ( (9955 (55 ase-. When linear penalty cost function is use then (By using signe istance metho Optimum cycle time 5.8 ays Optimum orer quantity Q ( units (By using grae mean integration metho Optimum cycle time 5.8 ays Optimum orer quantity Q ( ase-. When exponential penalty cost function is use then (By using signe istance metho Optimum cycle time 5.5 ays Optimum orer quantity Q ( units (By using grae mean integration metho Optimum cycle time 5.5 ays Optimum orer quantity Q ( units Sensitivity nalysis of Fuzzy Moel he sensitivity analysis is performe for checking the effectiveness of the EOQ moel for infinite prouction rate with respect to the parameter on optimum cycle time an optimum orer quantity Q in fuzzy sense using signe istance metho an grae mean integration metho. able- an able- epict the values of the optimum policies for ifferent values of the parameter in case of linear an exponential penalty cost. Percentage changes of these values are shown with respect to the parameter 5 in the ata set are taken.

7 merican Journal of Operational Research 6 6(: -8 7 able. Effect of parameter on optimal policies in case of exponential penalty cost hange (% in By using Signe istance metho By using Grae mean integration metho QQ QQ -6%.5 (-5%.5 (-5%.5 (-5% (-5% -%.5 (-7%.5 (-7%.5 (-7% (-7% (% 5.8 (% 5.8 (% (% 6 % 6. (8% 6. (8% 6. (8% (8% 8 6% 8. (56% 8. (56% 8. (56% (56% able. Effect of parameter on optimal policies in case of exponential penalty cost hange (% in By using Signe istance metho By using Grae mean integration metho QQ QQ -6%.6 (-57% (-57%.9 (-57% (-57% -%.7 (-9% (-9%.7 (-9% (-9% (% (% 5.8 (% (% 6 % 6. (9% (9% 6. (9% (9% 8 6% 8. (57% (57% 8. (59% (59% 6. onclusions In this paper a fuzzy EOQ moel for time-eteriorating items using penalty cost is stuie. he eman rate holing cost an set up cost are represente trapezoial fuzzy numbers. It has been also fuzzifie using signe istance metho an grae mean integration metho. From the table- an table- it shows that if it increase the parameter μμ in the crisp moel then the optimal cycle time an optimal orer quantity will also simultaneously increase. Finings from table- an table- shows that the change in parameter μμ will result in the change in an QQ. With the increase value of parameter μμ will also result in increase of optimal cycle time an optimal orer quantity. Similarly with the ecrease value of parameter μμ will also result in ecrease of optimal cycle time an optimal orer quantity. It is conclue that the value of the optimal cycle time an optimal orer quantity are not much sensitive to change in the value of the parameter μμ implying that fuzzy moel permits flexibility in the system inputs. he outcome of this research can be extene in future to the case of iscount moels. KNOWLEGEMENS he first author is grateful to Prof. R.S. Rey presient ORSI for giving opportunity to eliver this paper. Further the authors are grateful to the anonymous referees of merican Journal of Operational Research for their constructive comments. REFERENES [] arris F. (95 Operation an cost w shaw co. hicago. [] Wilson R. (9 scientific routine for stock control arvar Business Review. [] arly G. an Whitin.M. (96 nalysis of inventory systems Prentice-all Englewoo clipps NJ. [] Zaeh L.. (965 Fuzzy sets Information control [5] Zaeh L.. an Bellman R.E. (97 ecision making in a fuzzy environment Management sciences -6. [6] Kacpryzk J. an Stanicwski P. (98 Long-term inventory policy-making though fuzzy ecision making moels Fuzzy sets an systems [7] Park K.S. (987 Fuzzy sets theoretic interpretation of economic orer quantity IIIE ranactions on systems Man an ybernetics [8] Fujiwara O. an Perera U.L.J.S.R. (99 EOQ moel for continuously eteriorating proucts using linear an exponential penalty cost European journal of operation research 7 -. [9] Vujosevic M.et al. (996 EOQ formula when inventory cost is fuzzy International journal of prouction economics [] Srivastava M. an Gupta R. (9 EOQ moel for time-eteriorating items using penalty cost Journal of reliability an statistical stuies vol. issue [] Jaggi. K. Pareek S. Sharma. an Nihi. ( Fuzzy inventory moel for eteriorating items with time- varying eman an shortage merican journal of operational

8 8 Nalini Prava Behera et al.: Fuzzy EOQ Moel for ime-eteriorating Items Using Penalty ost research ( [] ripathy P.K. an Prahan S. ( n integrate partial backlogging inventory moel having weibull eman an variable eterioration rate with the effect of trae creit International journal of scientific & engineering research vol. issue.. [] Maragatham M. an Lakshmievi P.K. ( fuzzy inventory moel for eteriorating items with price epenent eman International journal of fuzzy mathematical stuies vol. 5 No [] Pevekar. an Nagare M.R. (5 Inventory moel for timely eteriorating proucts consiering penalty cost an shortage cost International journal of science technology & engineering vol. issue.. [5] Mishra S.S. Gupta S. Yaav S.K. an Rawat S. (5 Optimization of fuzzifie economic orer quantity moel allowing shortage an eterioration with full backlogging merican journal of operational research vol. 5(5 -. [6] Kumar S. an Rajput U.S. (5 Fuzzy inventory moel eteriorating items time epenent eman an partial backlogging pplie mathematics vol