On Optimal Total Cost and Optimal Order Quantity for Fuzzy Inventory Model without Shortage

Transcription

1 International Journal of Fuzzy Mathemat and Systems. ISSN Volume 4, Numer (014, pp Researh India Puliations On Optimal Total Cost and Optimal Order Quantity for Fuzzy Inventory Model without Shortage 1 D. Stephen Dinagar and J. Rajesh kannan 1 PG. and Researh Department of Mathemat, T.B.M.L College, Porayar, INDIA. Department of Basi Engineering, S.S.P College, Puthur, INDIA 1 dsdina@rediffmail.om raja.npr16@gmail.om Astrat In th paper, we develop fuzzy optimal total ost and fuzzy optimal order quantity for the proposed inventory model. Holding ost, Ordering ost and Total demands are taken as hexagonal fuzzy numers. An inventory model without shortage has een onsidered in a fuzzy environment. Dtint fuzzy arithmeti operations on hexagonal fuzzy numers are proposed. Numerial examples are presented to illustrate the proess of otaining the fuzzy optimal order quantity and the fuzzy minimal optimal total inventory ost. Keywords: Hexagonal fuzzy numers, Fuzzy inventory model, Fuzzy optimal total ost, Fuzzy optimal order quantity, Introdution In 1915, the first inventory model was developed y F. Harr[3] later in 1965, first time the onept of fuzzy sets was introdued y Lofti A. Zadeh [11]. In 1970 L. A. Zadeh and R. E. Bellman proposed a mathematial model deion making in a fuzzy environment. The eonomi lot size models have een studied extensively. Sine Harr [3] & Wilson [9] presented the EOQ model serves useful approximation to many real life prolems. Urgeletti [8] treated, EOQ model in fuzzy sense and used triangular fuzzy numers. Chen and Wang [1] used trapezoidal fuzzy numers to fuzzify the order ost, inventory ost and ak order ost in the total ost of inventory model without ak order. Yao et al. [10] onsidered the fuzzified prolems for the inventory with or with or without akorder models. Jain [4] worked on the deion making in the presene of fuzzy variales. Kapryzk et al. [5] dussed some longterm inventory poliy making through the fuzzy-deion making models. Rajarajesvari et.al [6] proposed the hexagonal fuzzy numer without any restritions of parameters. Stephen and Rajesh [7] have developed the fuzzy inventory model with

2 194 D. Stephen Dinagar and J. Rajesh kannan allowale shortage using hexagonal fuzzy numers. We have modified the definition of the numer y inluding onditions for the onvexity of the numer and few more results are also inluded in the work. In th paper, the optimal total ost and optimal order quantity for the inventory model in fuzzy environment have een studied. An inventory model onsidering holding ost, ordering ost and total demand are all in terms of hexagonal fuzzy numers. The new arithmeti operations are defined and applied the fuzzy total ost and optimal order quantity. We apply fuzzy set theoreti approah. An algorithm developed to find the fuzzy optimal order quantity and also minimizing the fuzzy total ost. Sensitivity analys arried out through the numerial examples. In th artile, in setion, some asi definitions and new arithmeti operations on hexagonal fuzzy numers are presented. In setion 3, we desrie in rief notions and assumption used in the developed model. In setion 4, the fuzzy mathematial models and algorithm. In setion 5, numerial examples are given to illustrate the model and sensitivity analys has een mode for different hanges in the parameter values. In setion 6, the onluding remarks are given.. Definitions and preliminaries Definition.1: Fuzzy set A fuzzy set A in a universe of dourse x defined as the following set of pairs A {( x, ( x : x X}. Here : X [0,1] a mapping alled the memership A value of x X in a fuzzy set A. A Definition.: onvex fuzzy set A fuzzy set A {( x, ( x} X A alled onvex fuzzy set if all A are onvex sets x i.e. for every element x1 A and x A for every [0,1] x1 (1 x A [0,1]. Otherwe the fuzzy set alled non onvex fuzzy set. Definition.3: Hexagonal Fuzzy Numer A fuzzy numer on A h a Hexagonal fuzzy numer denoted y A h ( a1, a, a3, a4, a5, a6 where ( a a a a a a are real numers Satfying a a1 a3 a and a5 a4 a6 a5 and its memership funtion ( x given as; Ah

3 On Optimal Total Cost and Optimal Order Quantity 195 x a1 1 x a 1 a 1 x a a a1 1 1 x a a x a3 a3 a ( x 1, a A h 3 x a4 1 x a 4 1 a 4 x a5 a5 a 4 1 a6 x a x a a6 a5 0, x> a6 5 6 Remark: The Hexagonal fuzzy numers A h eomes trapezoidal fuzzy numers if a a1 a3 a and a5 a4 a6 a5. The Hexagonal fuzzy numers A h eomes non-onvex fuzzy numers if a a1 a3 a and a5 a4 a6 a5. Fig.1.Hexagonal fuzzy numer Definition.4: Equality of two Hexagonal fuzzy numers Two Hexagonal fuzzy numers A ( a1, a, a3, a4, a5, a6 and B (,,,,, are said to e equal i.e. A B if and only if a, a, a, a, a, a Definition.5: Symmetri Hexagonal fuzzy numer A fuzzy numer A ( a1, a, a3, a4, a5, a6 said to e a symmetri Hexagonal fuzzy numer, if a3 a1 a6 a4. Otherwe the fuzzy numer alled non symmetri fuzzy numer.

4 196 D. Stephen Dinagar and J. Rajesh kannan Fig. Symmetri Hexagonal fuzzy numer Definition.6: We define a ranking funtion R : f ( R R whih maps eah fuzzy numers to the real line f ( R represents the set of all hexagonal fuzzy numers. If R e any linear a1 a a3 a4 a5 a6 ranking funtion, then. R( A 6 Definition.7: Equivalent hexagonal fuzzy numers A fuzzy numer A said to e equivalent to a fuzzy numer B if its value of the Ranking funtion are the same. i.e. A B if R ( A R ( B Definition.8: (New Arithmeti Operations The new arithmeti operations etween hexagonal fuzzy numers are proposed given elow. Let us onsider A 1 ( a1, a, a3, a4, a5, a6 and A ( 1,, 3, 4, 5, 6 e two hexagonal fuzzy numers. Then, The addition of A 1and A A ( A ( a, a, a, a, a, a The sutration of A 1and A A ( A ( a, a, a, a, a, a The multipliation of A 1and A a1 a a3 a4 a5 a6 A 1( A,,,,, Where ( The divion of A 1and A 6a1 6a 6a3 6a4 6a5 6a6 A 1( A,,,,, if 0 Where (

5 On Optimal Total Cost and Optimal Order Quantity 197 If k 0 a salar ka defined as ( ka1, ka, ka3, ka4, ka5, ka6, if k > 0 ka ( ka6, ka5, ka4, ka3, ka, ka1, if k < 0 A a, a, a, a, a, a a, a, a, a, a, a Where a1, a, a3, a4, a5, a6 are non zero positive real numers. Case II Multipliation of A 1and A A ( A a, a, a, a, a, a Divion of A 1and A A A a a a a a a i (,,,,, where i 0, 1,,3, 4,5, Notations and Assumptions 3.1 Notations We define the following symols C : Fuzzy holding ost per unit quantity per unit time S : Fuzzy Setup ost (or ordering ost per order T: Length of the plan D : Fuzzy Total demand over the planning time period [0, 1] Q : Fuzzy Order quantity per yle T : Fuzzy total ost for the period [0, 1] F(Q : Minimum Fuzzy optimal total ost for [0, 1] Q : Fuzzy Optimal order quantity 3. Assumptions In th paper the following assumptions are onsidered as Total demand fuzzy nature Time plan onstant Holding ost, Ordering ost are fuzzy in nature Shortages are not allowed. 4. Fuzzy mathemati model I In th model, the new arithmeti operation as in ase - I have een utilized also fuzzy demand over the planning time period [0,1], fuzzy holding ost permit quantity per unit time and fuzzy set up ost or ordering ost are taken in terms of hexagonal fuzzy numers. Now we fuzzifying total ost given y []

6 198 D. Stephen Dinagar and J. Rajesh kannan CTQ SD T Q Our aim to apply the hexagonal fuzzy numer fuzzy total ost and otain the fuzzy optimal order quantity y using the simple alulus tehnique. Suppose C ( C1, C, C3, C4, C5, C6 S ( S1, S, S3, S4, S5, S6 D ( D1, D, D3, D4, D5, D6 are hexagonal fuzzy numers CTQ SD T Q By using new arithmeti operations and simplifying we get, TQ D1 TQ D TQ D3 TQ D4 TQ D5 TQ D 6 C1 a, C a, C3 a, C4 a, C5 a, C6 a T 6Q 6Q 6Q 6Q 6Q 6Q T F Q (4.1 The fuzzy optimal order quantity Q whih minimize the total inventory ost T F( Q otained as the solution of the first order fuzzy differential equation d T ( 0 and it found as dq Q ( Q1, Q, Q3, Q4, Q5, Q6,using defintion.8 (vi Where ( C C C C C C a ( S S S S S S ( Q1 Q Q3 Q4 Q5 Q6 d F( Q Also Q Q we have 0 dq and from (4.1 th show that F ( Q minimum at Q Q TQ 1 D1 a TQ D a TQ 3 D3 a TQ 4 a D4 a TQ 5 D5 a TQ 6 D6 a FQ,,,,, Fuzzy mathematial model II Suppose C ( C1, C, C3, C4, C5, C6 S ( S, S, S, S, S, S D ( D, D, D, D, D, D are hexagonal fuzzy numers

7 On Optimal Total Cost and Optimal Order Quantity 199 CTQ SD T Q By using arithmeti operations and simplifying we get, CTQ 1 S1D 1 CTQ SD CTQ 3 S3D 3 C4TQ S4D4 CTQ 5 S5D5 CTQ 6 S6D6 T,,,,, Q Q Q Q Q Q T F Q (4.1.1 The fuzzy optimal order quantity Q whih minimize the total inventory ost T F( Q otained as the solution of the first order fuzzy differential equation d T ( 0 and it found as dq S1D1 SD S3D3 S4 D4 S5D5 S6D6 Q,,,,, TC6 TC5 TC4 TC3 TC TC1 Q ( Q1, Q, Q3, Q4, Q5, Q6,using defintion.8 (vi d F( Q Also Q Q we have 0 dq th show that F( Q minimum at Q Q and from (4.1.1 TCQ 1 1 SD 1 1 TCQ SD ,, TCQ S D TCQ S D F Q,, TCQ S D , TCQ S D Q6 Q5 Q4 Q3 Q Q1 4.3 Algorithm for finding fuzzy total ost and fuzzy optimal order quantity. Step 1 Calulate the model fuzzy total ost for the fuzzy values of C, S, D and T Step Now determine fuzzy total ost using new arithmeti operations fuzzy holding ost, fuzzy ordering ost, fuzzy shortage ost and fuzzy demand taken in terms of hexagonal fuzzy numers. Step 3 Find the fuzzy optimal order quantity whih an e otain y putting the first derivative of F( Q equal to zero and seond derivate positive at Q Q 5. Numerial examples: Fuzzy model I Let C (6, 7,11,13,17,18, S (14,15,19, 1, 5, 6, D (50,350,500,550,650,700, T = 6days then Q (11.78,13.94,16.66,17.48,19.00,19.7 F( Q

8 00 D. Stephen Dinagar and J. Rajesh kannan =(78.40,97.88,108.40,198.78,1475.3, Sensitivity analys S. Demand( No. D S (14,15,19,1,5,6 C (6,7,11,13,17,19 Q F( Q S (14,15,18,, 5, 6 C (6,7,10,14,17,18 Q F( Q 1 (00,300,450, (10.54,1.90,15.81, (637.06,850.83, ,1 (10.54,1.90,15.81, (637.06,850.83, ,1 500,600, ,18.5, ,149.86, ,18.5, ,149.86,151.6 (5,35,475, (11.18,13.43,16.4, (683.96,890.06, ,1 (11.18,13.43,16.4, (683.96,890.06, ,1 55,65, ,18.64, ,145.94, ,18.64, ,145.94, (50,350,500, (11.78,13.94,16.66, (78.40,97.88,108.40,1 (11.78,13.94,16.66, (78.40,97.88,108.40,1 550,650, ,19.00, ,1475.3, ,19.00, ,1475.3, (75,375,55, (1.36,14.43,17.07, (771.0,964.36,137.35,13 (1.36,14.43,17.07, (771.0,964.36,137.35,13 575,675, ,19.36, , , ,19.36, , , (300,400,550, (1.90,14.90,17.48, (811.68,999.45,165.97,13 (1.90,14.90,17.48, (811.68,999.45,165.97,13 600,700, ,19.7, ,150.6, ,19.7, ,150.6, Fuzzy model II Let C (7,9,11,13,15,17, S (15,17,19, 1, 3,5, D (50,350, 450,550,650,750, T = 6days then Q (8.05,10.14,15.60,18.70, 7.81, F( Q (54.99, 401.7,10.8, ,300.87, Sensitivity analys S. Demand( No. D S (15,17,19, 1, 3,5 C (7,9,11,13,15,17 Q F( Q S (15,17,18,, 3, 5 C (7, 9,10,14,15,17 Q F( Q 1 (00,300,450, (7.0,9.39,14.80,17 (0.98,365.6,967.9,1404 (7.0,9.39,14.80,17 (0.98,365.6,967.9, ,600,650.83,6.7, ,94.16, ,6.7, ,94.16, (5,35,475, (7.63,9.77,15.1,18 (38.3,383.93,995.9,1437 (7.63,9.77,15.1,18 (38.3,383.93,995.9, ,65,675.7,7.7,31..38,990.05, ,7.7,31..38,990.05, (50,350,500, (8.05,10.14,15.60,1 (54.99,401.7,10.8,14 (8.05,10.14,15.60,1 (54.99,401.7,10.8,14 550,650, ,7.81, ,300.87, ,7.81, ,300.87, (75,375,55, (8.44,10.50,15.99,1 (70.89,418.98, ,15 (8.44,10.50,15.99,1 (70.89,418.98, ,15 575,675,75 9.1,8.34, ,305.48, ,8.34, ,305.48, (300,400,550, (8.81,10.84,16.36,1 (86.,435.54, ,153 (8.81,10.84,16.36,1 (86.,435.54, , ,700, ,8.86,3.91.3,3086.5, ,8.86,3.91.3,3086.5, From tale I, it oserved that, The fuzzy optimal order quantity loser to rp optimal order quantity The fuzzy total ost loser to rp total ost For different values of S and C, hanging only middle two spreads the fuzzy optimal order quantity remains fixed. The same true for fuzzy total ost. From tale II, it oserved that,

9 On Optimal Total Cost and Optimal Order Quantity 01 The fuzzy optimal order quantity inreases and th fuzzy total ost inrease linear. For the different values of S and C hanging only middle two spreads, the fuzzy optimal order quantity remains fixed, ut the fuzzy total ost not fixed on the middle two spreads are hanged. Conlusion In th paper we have studied the optimal order quantity and the optimal total ost under the fuzzy environment with the aid of hexagonal fuzzy numers. The new fuzzy arithmeti operations in fuzzy model-1 are proposed and applied in the proposed notions. By using hexagonal fuzzy numer ranking method and the orresponding hange have een oserved. From the tale, it oserved that the fuzzy optimal order quantity and fuzzy optimal total ost are very lose to the lassial model than the fuzzy model -. Thus it an e onluded that the proposed method more effetive and effiient for the omputational purpose. Referenes [1] Chan,Wang, Bakorder fuzzy inventory model under funtion priniple, Information Siene, 95, 1996, 1-, [] Dutta,D., Pavan Kumar, Fuzzy inventory model without shortage using trapezoidal fuzzy numer with sensitivity analys, IOSR Journal of Mathemat, Vol. 4, Issue 3, 01, [3] Harr, F., Operations and ost, AW Shaw Co. Chiago, (1915. [4] Jain, R., Deion making in the presene of fuzzy variales, IIIE Transations on systems, Man and Cyernet, 17, 1976, [5] Kapryzk, J., Staniewski, P., Long-term inventory poliy-making through fuzzy-deion making models, Fuzzy Sets and Systems, 8, 198, [6] Rajarajesvari,P.,Sahaya Sudha,A., A New operation on hexagonal fuzzy numer, Fuzzy logi systems,3, 013, [7] Stephen Dinagar,D., Rajesh Kannan,J., On fuzzy inventory model with allowale shortage, International Eletroni Journal of Pure and Applied Mathemat, Communiated. [8] Urgeletti Tinarelli, G., Inventory ontrol models and prolems, European Journal of Operational Researh, 14, 1983, 1-1. [9] Wilson, R., A sientifi routine for stok ontrol, Harvard Business Review, 13, 1934, [10] Yao, J.S., Chiang, J., Inventory without ak order with fuzzy total ost and fuzzy storing ost defuzzified y entroid and signed dtane, European Journal of Operational Researh, 148, 003, [11] Zadeh, L.A., & Bellman, R.E., Deion Making in a Fuzzy Environment, Management Siene, 17, 1970,

10 0 D. Stephen Dinagar and J. Rajesh kannan