Fuzzy Inventory Control Problem With Weibull Deterioration Rate and Logarithmic Demand Rate

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1 Volume 7 No. 07, 5-44 ISSN: (printed version); ISSN: 4-95 (on-line version) url: ijpam.eu Fuzzy Inventory Control Problem With Weibull Deterioration Rate and Logarithmic Demand Rate S. Balarama Murthy,S. Karthigeyan and J. Pragathi Bharathiar University, Coimbatore, India. balaramac@gmail.com Department of Mathematics, Dr. Ambedkar Govt. Arts College, Chennai, India karthigeyanshan@gmail.com WCC College, Chennai. pragathimaths95@gmail.com Abstract Inventory models in which the demand rate depends on the inventory level are based on the common real-life observation that greater product availability tends to simulate more sales. It involves the inventory holding cost, shortage cost, demand rate and deterioration rate. his paper proposes an inventory control model with Weibull deterioration rate and Logarithmic demand rate for the optimal stock of commodities to meet the future demand which may either arise at a constant rate or may vary with time. he proposed model is developed in both the crisp and fuzzy environments. Graded mean representation integration method is used to defuzzfy the fuzzified values. AMS Subject Classification: 90B05, E8, P0 Key Words and Phrases:Fuzzy numbers, Defuzzfication, Purchase inventory, shortage Quantity, Weibull deterioration, Logarithmic demand Introduction Inventory is an essential resource needed for day-to-day operations and it deals with how much to keep on hand and how frequently reorder. he term inventory refers to the goods or materials used by a business organization for the purpose of production and sale. Raw materials, work-in-progress goods and finished goods are three basic types of inventory. Raw materials are the items purchased by firms for use in production of finished product. Work-in-progress consists currently in the progress of production. hese are actually partly manufactured products. he 5

2 S. Balarama Murthy et al. main purpose of inventory is to maintain trade-off between the minimization of the total cost and maximization of the customer satisfaction. he inventory model assume certain or uncertain demand and supply. In reality both demand and supply are uncertain due to unpredictable events, change of orders or random capacity of supplies. Since some of the uncertainties within inventory systems cannot be considered appropriately using concepts of probability theory, fuzzy set theory has been used in model of inventory system. In the inventory process, the effect of deterioration is very important. While developing an optimal inventory policy for products such as fruits, vegetables, chemicals etc., (deteriorate has shortage period) the loss of inventory due to deterioration cannot be ignored. But as the inventory levels go up to provide these functions, the cost of storing, holding inventory and stock out also increases. Since the uncertainties with in inventory system cannot be considered appropriately using concepts of probability theory. Fuzzy set theory has been used in modeling of inventory systems. Fuzzy set theory, originally introduced by Zadeh, provides a framework for considering parameters that are vaguely or unclearly defined or whose values are imprecise or determined based on subjective beliefs of individuals. [8] Manish Pande and Gautam (0) had developed an inventory control model for fixed deterioration and logarithmic demand rates. hey used cost minimization technique to obtain the optimal value of stock, time and total cost. hey considered deterministic cases of demand by allowing shortage. hey obtained an approximate expression for initial inventory, total number of deteriorated units and total minimum average cost. [] Mishra etal (0) had developed an inventory model for deteriorating items with time-dependent demand and time varying holding cost under partial backlogging. hey considered a deterministic inventory model with time dependent demand and time varying holding cost where deterioration was time proportional. Shortages were allowed and the demand was partially backlogged. Using numerical examples, the model was solved analytically by minimizing the inventory cost. [0] Venkateswarlu and Mohan (04) had developed an inventory model with quadratic demand, constant deterioration and salvage value. hey considered two cases (i. Retarded Decline Model and ii. Accelerated Decline Model)to calculate the optimum total cost and total order quantity in this model. hey found that the retarded decline and accelerated decline have shown good results which will be useful to describe a realistic situation for any product. hey also formulated inventory models for constant deterioration rate together with salvage value. hey found the existence of retarded decline and accelerated decline models in this case. [] Kumar etal(05) had developed an inventory model with periodic demand, constant deterioration and shortages. hey used cost minimization technique to obtain the optimal value of stock, time and total cost. hey considered deterministic cases of demand (a sint )by allowing shortage. hey obtained the approximate expression for initial inventory, total number of deteriorated units and total minimum average cost. Fuzzy Inventory Control Models [5] Dutta and Kumar (0) developed fuzzy inventory model without shortage using trapezoidal fuzzy numbers. hey considered an inventory model without

3 Fuzzy Inventory Control Problem With... shortage in a fuzzy environment. o determine the optimal total cost and optimal order quantity trapezoidal fuzzy numbers were used. hey used signed distance method for defuzzification. hey concluded that fuzzy trapezoidal numbers were better and economic than fuzzy triangular numbers. [9] Ranganathan and hirunavukarasu (04) had developed an inventory control model for constant deterioration in fuzzy environment. hey considered an inventory model for fixed deterioration and demand rate. Shortages were allowed and fully backlogged. All inventory parameters were assumed to be trapezoidal fuzzy number. hey defuzzified the fuzzy model using grated mean integration representation method. After analyzing the result, they obtained the fuzzy optimal solution to minimize the total cost of the inventory system. [7] Karthigeyan etal (05) had developed fuzzy optimized production EOQ model with constant rate and allowing the shortages. hey analysed a continuous production inventory model for deteriorating items with shortages under fuzzy environment to estimate various fuzzy optimal quantities along with the respective defuzzified results by assuming the demand and production rates along with the holding cost, shortage cost and deteriorating cost as trapezoidal fuzzy numbers and the time of deterioration is exponential. hey obtained the estimate of accuracy by the fuzzy inventory control for constant deterioration and fuzzy shortages using trapezoidal fuzzification method and graded mean integration representation method. hey obtained the minimum total cost and optimal inventory quantity and optimal shortage quantity by a numerical example. WEIBULL DISRIBUION f(x) =β α( x γ α )β e x γ ( α ) β Where f(x) 0, x 0, β > 0, α > 0, < γ < α-scale parameter, β-shape parameter, γ-location parameter. (Frequently γ is not used and set to 0. hat means Weibull distribution reduced to two parameter i.e., γ = 0). he probability density function of a Weibull distribution is f(x) = β α ( x α )β e ( ( x α ))β, 0 < α <, β > 0 or f(x) = αβx β e αxβ At failure rate, θ(x) = f(x). f(x) F (x) = αβxβ, Where F(x)=is distribution function of Notations and Assumptions A. Notations Replenishment size is constant and replenishment rate is infinite. Lead time is zero is the fixed length of each production cycle log(+t) is the demand rate at time t. C h is the inventory holding cost per unit time. C s is the shortage cost per unit time. C d is the cost of each deteriorated unit. D is the total amount of deteriorated unit. q(t) is the on hand inventory at anytime t. Q is the total amount of inventory produced at the beginning of each period. S(> 0) is the initial inventory after fulfilling back order. 7

4 4 S. Balarama Murthy et al. ( avg) is the total average cost. B. Assumptions Q is the total amount of inventory at the beginning of each period. S is the initial inventory after fulfilling backorders. Inventory level gradually decreases during time(0, t ), (t, ) due to the reasons of market demand and deterioration. Inventory level falls to zero at timet=t. Shortages occur during time period (t,) which are fully backlogged. Fuzzy Concepts Fuzzy Number If a fuzzy set is convex and normalized and its membership function is defined in R and piecewise continuous is called as fuzzy number. So fuzzy number represents a real number interval whose boundary is fuzzy. rapezoidal fuzzy number A fuzzy set à = (a, a, a, a 4 ) where a < a < a < a 4 and defined on R, is called the trapezoidal fuzzy number, if the membership function of A is given by 0, if x a x a a µã = a, a x a, a x a a 4 x a 4 a, a x a 4 0, x a 4 (or) µã= max(min( x a a a,, a 4 x a 4 a ), 0) Fuzzy arithmetical operations : If Ã= (a, a, a, a 4 ) and B = (b, b, b, b 4 ) are two trapezoidal fuzzy real numbers, hen à + B = (a + b, a + b, a + b, a 4 + b 4 ) à B = (c, c, c, c 4 ) where = (a b, a b 4, a 4 b, a 4 b 4 ), = (a b, a b, a b, a b ), c = min, c = min, c = max andc 4 = max If a, a, a, a 4, b, b, b andb 4 are all non-zero positive real numbers, hen à B = (a b, a b, a b, a 4 b 4 ) à = a, a, a, a 4 = ( a, a, a, a 4 ) B = ( b 4, b, b, b ) and à B = (a b 4, a b, a b, a 4 b ) B = B ( ) = (/b 4, /b, /b, /b ) Ã/ B = (a /b 4, a /b, a /b, a 4 /b ) Graded mean Representation integration he method of defuzzification of a generalized trapezoidal fuzzy number Ã=(a, a, a, a 4 ) by its graded mean integration representation was proposed by Chen and Hsieh and is defined by GMRI (Ã) = ( h/[(a ) + h(a a a 4 + a )])/ hdh) = (a 0 + a + a + a 4 )/ 8

5 Fuzzy Inventory Control Problem With he Mathematical Model and Its Analysis Fuzzy inventory control problem with Weibull deterioration rate and logarithmic demand rate is developed. he formulae for total average cost, optimal amount of deteriorated unit and optimal amount of initial inventory after fulfilling back order under crisp and fuzzy environment are derived. Let q(t) denote on-hand inventory at time (0,), then the linear first order differential equation which the on hand inventory q(t) satisfies in two different parts of the cycle time are given by: d q(t) + θ(t)q(t) = Log( + t),0 t t dt (4.) d q(t) = Log( + t),t dt t (4.) with boundary conditions q(0) = S, q(t ) = 0 (4.) where θ(t) = αβt β Consider the equation (4.) d q(t) + (t)q(t) = Log( + t), 0 <= t <= t dt he first order linear equation is dy + P y = Q dx Its solution is ye P dx = Qe P dx dx + c Solving equation (4.), we obtain q(t)e αtβ = [ αtβ+ + t t + t4 ] + C (4.4) β+ put t=0, q(0)=s in 4.4, then S=c q(t) = [ αtβ+ + t t + t4 β+ ]e αtβ + Se αtβ, 0 t t (4.5) Solving equation (4.), we obtain q(t)= -[(+t) Log(+t)-(+t)]+c (4.) Put t = t, q(t ) = 0 in (4.), then c=( + t )log( + t ) ( + t ) q(t)=( + t )log( + t ) ( + t)log( + t) + t t,t t (4.7) Put t = t, q(t ) = 0in(4.5), then S= αtβ+ + t β+ t q(t)=e αtβ [( αtβ+ + t β+ t ) ( αtβ+ + t β+ t + t4 )], 0 t t (4.8) he deterioration units is D = S t αtβ+ Log( + t)dt = + t 0 β+ t ( + t )Log( + t ) + t Deterioration Cost = C d ( + t )Log( + t ) + t ] (4.9) Shortage Cost = Cs ( t q(t)dt = t q(t)dt (No. of Deterioration units) = C d [ αtβ+ t [( + t )Log( + t ) ( + t)log( + t) + t t ]dt =( + t )Log( + t )( t + +t (+t ) 4 + ( t ) β+ + t t (+ ) + ( Log( + )) (+t ) + ( t ) ) 4 Shortage Cost = Cs [( + t )Log( + t )( t + +t (+ ) ) + ( Log( + )) ] (4.0) Holding Cost = C h t q(t)dt 0 t 0 q(t)dt = t 0 ( αtβ )[ t t + αtβ+ β+ t + t t4 αtβ+ ]dt β+ 9

6 S. Balarama Murthy et al. = t t4 8 + t5 + αβtβ+ + 5 (β+)(β+) Holding Cost = C h [ t t4 8 + t5 αtβ+4 (β+)(β+4) αtβ+5 (β+)(β+5) + αβtβ+ + αtβ+4 5 (β+)(β+) (β+)(β+4) αtβ+5 ] (4.) (β+)(β+5) otal Inventory Cost (IC) = ( avg) = DeteriorationCost + ShortageCost + HoldingCost ( avg)= C d t )( + t [ αtβ+ + t β+ t ( + t )Log( + t ) + t ] + Cs [( + t )Log( + )+ (+ ) ( Log(+ )) (+t ) 4 + ( t ) ]+ C h [ t t4 8 + t5 αt β+4 αtβ+5 ] (β+)(β+4) (β+)(β+5) + αβtβ+ + 5 (β+)(β+) Optimality condition for minimization: d dt ( avg) = C d [t t + t + αt β+ Log( + t )] + Cs [Log( + t )( + t ) (+t )Log(+t ) ] + C h [t t + αβtβ+ + αtβ+ αtβ+4 ] β+ (β+) (β+) d dt ( avg) = C d [ t + t + α(β + )t β +t ] + Cs [ +t + t (+t ) Log( + t ) ] + C h [t t + 4t d dt ( avg) > 0 FUZZY MODEL + αβ(β+)tβ+ + α(β+)tβ+ β+ (β+) α(β+4)tβ+ ] (β+) We consider a model in fuzzy environment. Due to fuzziness, it is not easy to dene all the parameters precisely. We use the following variables:- α : fuzzy scale parameter in Weibull distribution, (C d) : fuzzy deterioration cost,(c h) : fuzzy holding cost,(c s) : fuzzy shortage cost. Suppose C d = ( C d, C d, C d, C d4 ), C h = ( C h, C h, C h, C h4 ), C s = ( C S, C S, C S, C S4 ), α = ( α, α, α, α 4 ) are non-negative trapezoidal fuzzy numbers. he otal Average Cost per unit time is given by ( avg) = C d (( α P ) L) ( CS M) CH (( α q) N) where, P = tβ+ (β+) L = [ t t ( + t )Log( + t ) + t ] M = + +, [ = (+ ) ( ( Log( + ))), = (( + t )Log( + t )( + t )), = ( ( t ) (+t ) )] 4 Q = [ βt β+ + t β+4 t β+5 ] (β+)(β+) (β+)(β+4) (β+)(β+5) N = [ t t4 8 + t5 ] 5 ( avg) = (P ( α Cd )) (L Cd ) ( C s M) (Q ( α Ch )) (N Ch ) ( avg(t )) = [( ( avg (t )), ( avg (t )), ( avg (t )), ( avg 4 (t ))] ( avg (t )) = P α C d + LC d + MC s + Q α C h + NC h ( avg (t )) = P α C d + LC d + MC s + Q α C h + NC h ( avg (t )) = P α C d + LC d + MC s + Q α C h + NC h ( avg 4 (t )) = P α 4 C d4 + LC d4 + MC s4 + Q α 4 C h4 + NC h4 Defuzzifying the total average cost ( avg(t )) by GMIR method, ( avg(t )) = P ( α C d + α C d + α C d + α 4 C d4 )+ L( C d +C d +C d + C d4 )+ M ( C s + C s + C s + C s4 ) + Q ( α C h + α C h + α C h + α 4 C h4 ) + N ( C h + C h + C h + C h4 ) 40

7 Fuzzy Inventory Control Problem With... 7 o minimize the total average cost per unit time, the optimal value of t can be obtained by solving the following equation, d dt ( avg(t )) = P ( α C d + α C d + α C d + α 4 C d4 ) + L ( C d + C d + C d + C d4 ) + M ( C s + C s + C s + C s4 ) + Q ( α C h + α C h + α C h + α 4 C h4 ) where, P = (tβ+ ) L = (t t + t Log( + t )) M = ( t +t Log( + t )) Q = tβ+) (β + tβ+ tβ+4 ) β+ (β+) (β+) N = t t Optimal amount of the initial inventory after fulfilling backorder S denoted by S by GMIR method is S S = + S + S + S 4 where, S = α t β+ + t β+ t, S = α t β+ + t β+ t S = α t β+ + t β+ t, S 4 = α 4t β+ + t β+ t Optimal amount of the unit deteriorated D denoted by D by GMIR method is D D = + D + D + D 4 where D = [ α t β+ β+ + t t )Log( + t ) + t ] D = [ α t β+ + t β+ t ( + t )Log( + t ) + t ] D = [ α t β+ + t β+ t ( + t )Log( + t ) + t ] D 4 = [ α 4t β+ + t β+ t ( + t )Log( + t ) + t ] hus, minimum value of the total cost ( avg(t )) denoted by ( avg(t )) by GMIR method is ( avg(t )) = (P α C d + LC d + MC s + Q α C h + NC h ) + (P α C d +LC d +MC s +Q α C h +NC h )+ (P α C d +LC d +MC s +Q α C h + NC h ) + (P α 4C d4 + LC d4 + MC s4 + Q α 4 C h4 + NC h4 ) 5 Numerical Examples In this paper, all the cost related to inventory (holding cost, shortage cost and deterioration cost) and scale and shape parameter in the Weibull distribution are assumed in crisp model and all the cost(holding cost, shortage cost and deterioration cost)and scale parameter in the Weibull distribution in fuzzy model as a trapezoidal fuzzy numbers using Graded Mean Integration Representation (GMIR) method. Better estimate for the minimum total cost is obtained from the result. ILLUSRAION he given values are C d = 5, C s = 4 and C h = 75 and we assume t = 0.5, =, α = 0. and β =. SOLUION: FOR CRISP MODEL:- d ( dt ( avg)) = C d = > 0 S = S = 0, D = D = 0459, ( avg) = 9.8 FOR FUZZY MODEL:- 4

8 8 S. Balarama Murthy et al. α = (0.05, 0., 0., 0.5), C d = (4, 5, 5, ), C s = (40, 4, 4, 4), C h = (70, 75, 75, 80), t = 0.5, =, =. P= 0.00,L= ,M= 0.905,N= 0.00 and Q= 0.00 ( avg(t )) = 5.59 Conclusion In this study, aking the deterioration cost, holding cost, shortage cost and scale and shape parameter in the Weibull distribution are considered as crisp and for a given cycle, deteriorated unit, initial inventory after fulfilling back order and minimum total cost are determined. aking the scale parameter in the Weibull distribution, deterioration cost, holding cost and shortage cost are considered as trapezoidal fuzzy number by Graded Mean Integration Representation method and for a given cycle, optimal amount of deteriorated unit, optimal amount of initial inventory after fulfilling back order and minimum total cost are determined. In this study, a purchase inventory model with Weibull deterioration rate and logarithmic demand rate is analyzed under fuzzy environment to minimize the total cost. All the related costs (deteriorating cost, holding cost and shortage cost) and scale parameter in Weibull distribution are considered as trapezoidal fuzzy numbers. o defuzzify the GMIR method were used. From the numerical illustration, the total average cost in the Crisp model is more than the Fuzzy model. o enhance the cost minimization, Fuzzy model is suitable. References [] Chandra K., Pareek J.S., Sharma A., and Nidhi, Fuzzy Inventory Model for Deteriorating Items with ime-varying Demand and Shortages, American Journal of Operation Research,(0), Vol., No., pp.8-9. [] Chang S.C.,, Fuzzy Production Inventory for Fuzzy Product Quantity with riangular Fuzzy Number, Fuzzy Sets and Systems, (999),Vol.07, pp [] Chen S. H., and Hsieh C. H., Optimization of Fuzzy Simple Inventory Models, 999 IEEE International Fuzzy System Conference Proceedings, (999), Vol., pp.40 44, Seoul, Korea. [4] Chen S.H., Operations on Fuzzy Numbers with Function Principle, amkang Journal of Management Sciences, (985), Vol., pp.. [5] Dutta D., and Kumar P., Fuzzy Inventory Model without Shortage using rapezoidal Fuzzy Numbers with Sensitivity Analysis, ISOR Journal of Mathematics, (0), Vol.4, No., pp.-7. [] Dutta D., N. P. Katyar and Kumar P., An Inventory Model with Periodic Demand, Constant Deterioration and Shortages, Industrial Engineering Letters, (05), Vol.5, No., pp.-0. 4

9 Fuzzy Inventory Control Problem With... 9 [7] Karthigeyan S., Balarama Murthy S., and Saranya A., Fuzzy Optimized Production EOQ model with Constant Rate and allowing the Shortages, International Journal of Applied Engineering Research, (05), Vol.0, No.80, pp.- 7. [8] Manish Pande, Gautam S. S., An Inventory Control Model for Fixed Deterioration and Logarithmic Demand Rates, Mathematical heory and Modelling, (0), Vol., No.7, pp.0-7. [9] Ranganathan V., and hirunavukarasu P., An Inventory Control Model for Constant Deterioration in Fuzzy environments, International Journal of Fuzzy Mathematics and Systems, (04), Vol.4, No., pp.7-. [0] Venkateswarlu R., and Mohan.R, Inventory Model with Quadratic Demand, Constant Deterioration and Salvage Value, Research Journal of Mathematical and Statistical Sciences, (04), Vol. (), pp.-5. [] Vinod Kumar Mishra, Lal Sahab Singh, Rakesh Kumar, Inventory Model for Deteriorating Items with ime-dependent Demand and ime Varying Holding Cost under Partial Backlogging, Journal of Industrial Engineering International, (0),Vol.9, Article id.4. 4

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