Optimal Policy for an Inventory Model without Shortages considering Fuzziness in Demand Holding ost and Ordering ost 1 2 : : Abstract In this paper an inventory model without shortages has been considered in a fuzzy environment In real situations the ordering cost holding cost and demand quantity have little disturbances every day That s why we consider the model in fuzzy environment We fuzzify them by using both the triangular fuzzy number and the trapezoidal fuzzy number For defuzzification purpose the signed distance method is used Then just after a little arrangement of terms we obtain the expression for economic order quantity EOQ and optimal total inventory cost To illustrate the results of the proposed model we give a numerical example with computational results Sensitivity analysis is also performed to determine the most sensitive input parameter of the inventory system Keywords : EOQ Fuzzy Numbers Fuzzy Inventory Model Defuzzification I INTRODUTION ℎ@ ℎ @ In today s fast growing world of industry and innovation the significant of inventory management has extremely increased In 1915 Harris [1] developed first inventory model Economic Order Quantity Later in 1934 Wilson [2] generalized this inventory model and he gave a formula to obtain economic order quantity In 1963 the model was further extended by changing the assumptions with the objective to make it more realistic by Hadley & Whitin [3] Inventory control is very important field for both real world applications and research purpose In conventional inventory models the uncertainties are treated as randomness and are handled by using probability theory In certain situations uncertainties are due to fuzziness primarily introduced by Zadeh [4 5] More applications of fuzzy set theory can be found in Zimmerman [6 7] In literature there are many papers on fuzzified problems of EOQ model Then they got fuzzy total cost They obtained the estimate of the total fuzzy cost through centroid to defuzzify In 1983 Urgeletti [8] treated EOQ model in fuzzy sense and used triangular fuzzy number In 1987 Park [9] considered fuzzy inventory cost in the economic order quantity model In 1996 Vujosevic et al [10] used trapezoidal fuzzy number to fuzzify the order cost in the total cost of inventory model with backorder hen & Wang [11] considered trapezoidal fuzzy number for fuzzification of the ordering cost inventory holding cost and backorder cost in the total cost of inventory model without backorder Then they found the estimate of the total cost in the fuzzy sense by functional principle In 2002 320
Kao & Hsu [14] introduced a single-period inventory model with fuzzy demand and a method for ranking fuzzy numbers is adopted to find the optimal order quantity From 1999 to 2003 Yao et al [12 13 15] considered the fuzzified problems for the inventory with or without backorder models In [12] they applied the extension principle to obtain the fuzzy total cost and then they defuzzified the fuzzy total cost by centroid In [13] they considered the fuzzified problems for the inventory with or without backorder models using trapezoidal fuzzy number In [15] they considered the fuzzified problems for the inventory without backorder models and they fuzzified the order quantity q as the triangular fuzzy number In 2004 Hsieh [16] introduced a fuzzy inventory model considering fuzzy average demand under safety stock In 1970 Zadeh & Bellman [5] defined the generalized mathematical operations for fuzzy numbers In 2007 Syed & Aziz [17] also used trapezoidal fuzzy number In basic EOQ model we identify the order size that minimizes the sum of annual costs of inventory holding and fixed setup to place orders Due to irregularities in fabrication or physical properties of a material we introduce Fuzzy set theoretic approach An inventory system is represented as a fuzzy system with the fuzzy inventory level as the output and fuzzy replenishment as an input De & Rawat [18] proposed an EOQ model without shortage cost by using triangular fuzzy number The total cost has been computed by using signed distance method In 2012 D Dutta and Pavan Kumar [19] proposed a fuzzy inventory model without shortage They consider the holding cost and the ordering cost as trapezoidal fuzzy number Recently in 2013 Mahata and Goswami [20] proposed two fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables They developed the inventory in both triangular fuzzy number and trapezoidal fuzzy number to get the optimal value of the profit function We extend our previous model D Dutta and Pavan Kumar [19] Here we consider holding cost ordering cost and demand quantity as trapezoidal fuzzy number The objective is to determine an optimal policy for total cost minimization and optimal order quantity The sensitivity analysis is also carried out to determine the most sensitive input parameter of the system II FUZZY ONEPTS Let R be a set of real numbers Then any convex normalized fuzzy subset of R with membership function :R [0 1] is called a fuzzy number Definition 1 A Trapezoidal Fuzzy Number TrFN denoted by A a b c d is defined by membership function as: μ x Lx 1 Rx 0 for a x b; for b x c; for c x d; otherwise If b c then trapezoidal fuzzy number TrFN becomes a triangular fuzzy number TFN A trapezoidal fuzzy number is shown in Fig1 and triangular fuzzy number is shown in Fig 2 Fig1: Trapezoidal Fuzzy Number TrFN Fig 2: Triangular Fuzzy Number TFN Definition 2 A fuzzy is called a LR-type trapezoidal fuzzy number if there are real numbers > 0 > 0 with for x m L 1 for m x m x μ for x m R Where m and m called the mean values of and & are called the left spread and right spread of Symbolically Definition 3 The mean of trapezoidal fuzzy number a b c d is given by 1 Definition 4 Suppose are two trapezoidal fuzzy numbers then arithmetical operations are defined as: 1 2 3 4 ø 5 6 α ø 0 <0 321
Total holding cost Average inventory * unit holding cost Definition 5 Let be a fuzzy set defined on R Then the signed distance of can be defined as: d 0 [ ] dα 2 where α [0 1] and [ ] [a b - a α d- d - c α] Total Ordering cost Number of orders * ordering cost 3 III NOTATIONS AND ASSUMPTIONS The proposed model is developed under the following assumptions: : Holding cost ie carrying cost per unit per unit time : Ordering cost per order Q : Order quantity per cycle ie quantity of material ordered Q : Optimal order quantity t : Length of a cycle in years T : Length of the plan time period could be as weekly monthly annually or some other period D : Total demand over the planning time period [0 T] : Total cost in crisp sense N : Average number of orders per cycle Fuzzy total cost for the period [0 T] Q De-fuzzified total cost We define the following notations: 1 Only one item is managed at a time 2 Plan of horizon is finite 3 Inventory replenishment is instantaneous 4 Holding cost and ordering cost both are variable Decision variable 5 Shortages Safety stock Deterioration and Quantity-discounts are not allowed Total Inventory cost Total holding cost Total ordering cost 5 can The optimum and be obtained by equating the first partial derivatives of to zero and solving the resulting equation: Optimal order quantity Minimum total cost 6 2DT 7 The behavior of inventory model with respect to time is shown in Fig3 Fig 3: Variation of In-hand Inventory with Time The economic order quantity or the quantity to be purchased to minimize the inventory related costs is shown in Fig 4 IV FORMULATION OF INVENTORY MODEL Since all demand meet and the complete order is delivered instantaneously there is no inventory shortage cost and no purchase quantity discounts In this model the economic lot size is obtained by the following model equation: Q where 4 Now for the plan period [0 T] we define the inventory parameters as follows: Maximum inventory Q and Minimum inventory 0 Average Inventory Fig4: ost urves in EOQ Model 322
[ Also V FUZZY SET THEORETIAL APPROAH This model is for independent demand inventory for examplepaper spare parts liquids oil fuel etc where demand is constant over time However in practical life the demand for order quantity and the concerned costs cannot be determined exactly; cannot be predicted correctly because every day there is a little disturbance over the demand quantity and over the costs So it is wise and better to consider each one of them as a fuzzy number The fuzzy total cost is given by: [ ] [ D ø Q] 8 where ø and are the fuzzy arithmetical operations by Function Principle Here is the average inventory that depends on the assumption of a constant sales rate producing a linear decline in the inventory position and D ø Q is the number of orders Without any loss of generality let > 0 We fuzzify the total cost function equation considering two cases: I triangular fuzzy number and II trapezoidal fuzzy number ase I Demand as Triangular Fuzzy Number TFN Let x Δ x x Δ and x μ x x μ are holding cost and ordering cost as triangular fuzzy numbers centered at x and x respectively with Δ > 0 μ > 0 Also let D x ρ x x ρ be the demand quantity as triangular fuzzy number with ρ > 0 Then the fuzzified total cost is given by: [ ] [ D ø Q] [x Δ x x Δ ] [x μ x x μ x ρ x x ρø Q] Δ [x Δ x x Δ ] [x μ x x μ ] x Δ x μ x μ x x say Let d be the value of defuzzification of distance Then d [ Q 2 Now we can get the optimal order quantity minimized Now by signed Q 11 Moreover the number of orders per cycle N is given by NDø D Then the average number of orders per cycle N that is the mean of N that is N 12 ase II Demand as Trapezoidal Fuzzy Number TrFN Suppose x x x x and y y y y are holding cost and ordering cost as trapezoidal fuzzy numbers in LR form Also y y y y x x x and x are known positive numbers Also let D d d d d be the fuzzy demand quantity as trapezoidal fuzzy number where d d d and d are positive numbers [ ] [ D ø Q] [x x x x ] [y y y y x y x ] x x x x y x a b c d say ab aα Now x x And y [x x y y y x 9 when Q is y d d c α [x x y d y d ] α In order to defuzzify the Fuzzy Total ost we use the signed distance method explained in 2 & 3 Hence the defuzzified total cost Q is given by [A α A α] dα Q d 0 x [x x y d y d ] [x x x y d y d y d y d ] x x x x ] x x x x y d y d at which Q is minimum when where [2y d y d y d y d y d Q x x x x omputation of 10 y d y d ] α y4d4 y2d2 y3d3 y1d1 y4d4] 14 Q 0 and hence > 0 This shows that Q is minimum at [2x x ] [ Q] 0 x Q ] y d y d 13 is minimum: [ Q] 0 and [ Q] > 0 323
[ Q] 0 gives the economic order quantity as: Now Also [ ] >0 and This shows that Q is minimum at & 11 we have Q x x x x y d y d 14 hence from 10 D y d y d 15 Moreover the number of orders per cycle N is given by NDø 16 Then the average number of orders per cycle N that is the mean of N that is N 2 2 VI 17 NUMERIAL ILLUSTRATION AND SENSITIVITY ANALYSIS Example: onsider an inventory situation with crisp parameters having the following values: Holding cost Rs12/- per unit Ordering cost Rs 20/per order Demand D 500 units/year Planning time period T 1 year Then determine the economic order quantity optimal number of orders per cycle and optimal total inventory cost risp Model: Here Rs12/- per unit unit D 500 unit Then Q and Rs 489898 Fuzzy Param eter 122474 Fuzzy Model: ase I Triangular Fuzzy Number T 1 year Let 9 12 15 15 20 25 and 200 500 800 Then Q units N and Q Rs For the sensitivity analysis we prepare the following table: Fuzzy Param eter Triangular Fuzzy Number 11 12 13 10 12 14 9 12 15 8 12 16 7 12 17 6 12 18 5 12 19 18 20 22 16 20 24 14 20 26 12 20 28 10 20 30 8 20 32 414327 420317 426224 432049 437798 443471 N 120678 118958 117309 115728 114208 112747 Q 497039 504381 511468 518459 525357 532165 449073 111340 538880 450 550 400 600 350 650 300 700 250 750 200 800 150 850 500 410792 121716 492950 500 413320 120972 495984 500 415832 120241 498999 500 418330 119523 501996 500 420813 118818 504975 500 118125 507937 500 425735 117444 510882 Table 1: TFN for Input Parameters Rs 20/- per 408248 units N 6 20 34 D Trapezoidal Fuzzy Number and Q N 9 11 13 15 7 11 13 17 6 11 13 18 5 11 13 19 4 11 13 20 3 11 13 21 2 11 13 22 16 19 21 23 409268 118097 491121 14 19 21 25 12 19 21 27 10 19 21 29 8 19 21 31 6 19 21 33 4 19 21 35 300 400 600 700 250 400 600 750 200 400 600 800 150 400 600 850 100 400 600 900 50 400 600 950 0 400 600 1000 416333 430116 436845 443471 450000 415832 116093 114187 112373 110642 108989 107407 120241 499600 507937 516140 524214 532165 54000 498990 418081 119594 501697 420317 118958 504381 422542 118331 507040 424755 117715 509706 426956 117708 512348 429146 116510 514976 Table 2: TrFN for Input Parameters and 324
Observations: From table 1 and 2 the following observations are made: i Table 1 and 2 show the variations in the optimum order quantity and in optimal total inventory cost due to fuzziness in the components of the model ii We notice that the optimal order quantity and the total inventory cost for fuzzy model with fuzzy decision variables are not the same as fuzzy model with crisp decision variables iii We are revealed that the optimum values of order quantity and total cost are fairly sensitive to the changes in the level of fuzziness of ordering cost and demand but not sensitive to the changes in the level of fuzziness of holding cost iv The decision variable ie order quantity and the total cost are highly sensitive due to fuzziness in the ordering cost v The results so obtained in trapezoidal fuzzy number case are much better & economic than the case of triangular fuzzy number VII ONLUSION In this paper we proposed a basic inventory model without shortage considering fuzziness in some input parameters ie demand quantity holding cost and ordering cost First the model is developed in crisp environment Then it is described in fuzzy environment The holding cost ordering cost and demand quantity are treated as both the fuzzy triangular number TFN and the fuzzy trapezoidal number TrFN In order to defuzzify the model the signed-distance method is applied The economic order quantity and optimal total cost have been calculated in both crisp and fuzzy environments The corresponding effect of changes in the input parameters of inventory model has been observed We observe that Economic Order Quantity increases with small amount which is under consideration and is within our level of expectation Finally we conclude that for an EOQ model with fuzzy demand quantity fuzzy holding cost and fuzzy ordering cost the results so obtained in fuzzy trapezoidal number case are much better & economic than the case of triangular fuzzy number [3] Hadley G Whitin TM 1963 Analysis of Inventory Systems Prentice-Hall Englewood clipps NJ [4] Zadeh LA 1965 Fuzzy sets Information ontrol 8 pp 338353 [5] Zadeh LA Bellman RE 1970 Decision Making in a Fuzzy Environment Management Science 17 pp140-164 [6] Zimmerman HJ 1983 Using Fuzzy Sets in operational Research European Journal of Operational Research 13 pp 201-206 [7] Zimmerman HJ 1983 Fuzzy Mathematical Programming omputers and Operations Research 10 pp 291-298 [8] Urgeletti Tinarelli G 1983 Inventory ontrol Models and Problems European Journal of Operational Research 14 pp1-12 [9] Park KS 1987 Fuzzy Set Theoretical Interpretation of economic order quantity IEEE Transactions on Systems Man and ybernetics SM 17 pp1082-1084 [10] Vujosevic M Petrovic D Petrovic R 1996 EOQ Formula when Inventory ost is Fuzzy International Journal of Production Economics 45 pp 499-504 [11] hanwang 1996 Backorder fuzzy Inventory Model under Function Principle Information Science 95 1-2 pp71-79 [12] Yao JS Lee HM 1999 Economic Order Quantity in Fuzzy Sense for Inventory without Backorder Model Fuzzy Sets and Systems 105 pp13-31 [13] Yao JS Lee HM 1999 Fuzzy Inventory with or without Backorder for Fuzzy Order Quantity with Trapezoidal Fuzzy Number Fuzzy sets and systems 105 pp 311-337 [14] Kao K Hsu WK 2002 A Single-Period Inventory Model with Fuzzy Demand omputers and Mathematics with Applications 43 pp 841-848 [15] Yao JS hiang J 2003 Inventory without Backorder with Fuzzy Total ost and Fuzzy Storing ost Defuzzified by entroid and Signed distance European Journal of Operational Research 148 pp 401-409 [16] Hsieh H 2004 Optimization of Fuzzy Inventory Models under Fuzzy Demand and Fuzzy Lead Time Tamsui Oxford Journal of Management Sciences 202 pp 21-36 [17] Syed JK Aziz LA 2007 Fuzzy Inventory Model without Shortages using Signed Distance Method Applied Mathematics & Information Sciences 12 pp 203-209 [18] De PK Rawat A 2011 A Fuzzy Inventory Model without Shortages using Triangular Fuzzy Number Fuzzy Information & Engineering 1 pp 59-68 [19] Dutta D Pavan Kumar 2012 Fuzzy Inventory Model without Shortage Using Trapezoidal Fuzzy Number with Sensitivity Analysis IOSR Journal of Mathematics Volume 4 Issue 3 Nov - Dec 2012 pp 32-37 [20] Mahata G Goswami A 2013 Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables omputers & Industrial Engineering 64 pp 190-199 AKNOWLEDGEMENT The authors express their heartfelt gratitude and boundless regards to the anonymous referees for timely reviewing of the paper Best efforts have been made by the authors to revise this paper REFERENES [1] Harris F 1915 Operations and ost AW Shaw o hicago [2] Wilson R 1934 A Scientific Routine for Stock ontrol Harvard Business Review 13 pp116 128 325