Interntionl Journl of Soft Computing nd Engineering (IJSCE) ISSN: 2231-2307, Volume-4, Issue-1, Mrch 2014 Some EOQ Model for Weiull Deteriortion Items with Selling Price Dependent Demnd A. Lkshmn Ro, B. Neel Ro, R. Snthi Kumr Astrct - In this pper we develop nd nlyze n inventory model ssumption tht deteriortion rte follows Weiull two prmeter distriutions with selling price dependent demnd. Here it is ssumed tht demnd rte is function of selling price. With shortge nd without shortge oth cses hve een tken cre of in developing the inventory models. Shortges re fully cklogged whenever they re llowed. Through numericl exmples the results re illustrted. The sensitivity nlysis for the model hs een performed to study the effect chnges of the vlues of the prmeters ssocited with the model. Keywords: EOQ model; deteriorting items; shortge; selling price dependent demnd; Weiull distriution. I. INTRODUCTION The influence nd mintennce inventories for deteriorting items with shortges hve received much ttention of severl reserchers in the recent yers ecuse most of the physicl goods deteriorte over period of time. In rel life, mny of the items re either dmged or decyed or ffected y some other fctors nd is not in perfect condition to stisfy the demnd. Food items, drugs, phrmceuticls, rdioctive sustnces re exmples of such items where deteriortion cn tke plce during the norml storge period of the commodity nd consequently this loss must e tken into ccount when nlyzing the system. So decy or deteriortion of physicl goods in stock is very relistic feture nd reserchers felt the necessity to use this fctor into considertion in developing inventory models. Ghre nd Schrder (1963) who developed n economic order quntity model with constnt rte of decy. An orderlevel inventory model for system with constnt rte of deteriortion hve proposed y Shh nd Jiswl (1977), Aggrwl [1978], Dve nd Ptel [1981]. Inventory models with time dependent rte of deteriortion were developed y Covert nd Philip [1973], Mishr [1973] nd De nd Chudhuri [1986]. Some of the significnt recent work in this re hve een done y Chung nd Ting [1993], Fujiwr [1993], Hrig [1996], Hrig nd Benkherouf[1994], Wee [1995], Jln et l. [1999], Su, et l. [1996], Chkrorty nd Chudhuri [1997], Giri nd Chudhuri[1997], Chkrorty, et l. [1997] nd Jln nd Chudhuri, [1999],etc. At the eginning, demnd rte were ssumed to e constnt which is in generl likely to e time dependent nd stock dependent. Mnuscript received Mrch, 2014. Dr. A. Lkshmn Ro, Deprtment of Bsic Sciences & Humnities, Adity Institute of Technology nd Mngement, Tekkli, Andhr Prdesh Indi. B. Neel Ro, Deprtment of Bsic Sciences & Humnities, Adity Institute of Technology nd Mngement, Tekkli, Andhr Prdesh Indi. Dr. R. Snthi Kumr, Deprtment of Bsic Sciences & Humnities, Adity Institute of Technology nd Mngement, Tekkli, Andhr Prdesh Indi. Begum et l. [2010] hve developed economic lot size model for price-dependent demnd. Inventory model for meliorting items for price dependent demnd rte ws proposed y Mondl et.l [2003], with the motivtion of C. K. Tripthy., et l. [2010] nd Sushil Kumr., et l. [2013] we developed EOQ models for Weiull deteriorting items nd price dependent demnd. In this pper, we hve developed generlized EOQ model for deteriorting items where deteriortion rte follows twoprmeter Weiull nd demnd rte is considered to e function of selling price. For the model where shortges re llowed they re completely cklogged. Here we hve considered oth the cse of with shortge nd without shortge in developing the model. Using differentil equtions, the profit rte function re otined. By mximizing the profit rte function, the optiml production schedule nd optiml production quntity re derived. Through numericl illustrtion the sensitivity nlysis is crried. This model lso includes some of the erlier models s prticulr cses for prticulr or limiting vlues of the prmeters. II. ASSUMPTIONS AND NOTATIONS The following ssumptions re mde for developing the model: ) The demnd rre is function of selling price which is ) Shortges, whenever llowed re completely cklogged. c) The deteriortion rte is proportionl to time. d) Replenishment is instntneous nd led time is zero. e) T is the length of the cycle. f) Q: Ordering quntity in one cycle g) A: Ordering cost h) C: Cost per unit i) h: Inventory holding cost per unit per unit time Some EOQ Model for Weiull Deteriortion Items with Selling Price Dependent Demnd j) : Shortges cost per unit per unit time k) s: Selling price per unit nd l) The deteriortion of units follows the two prmeter Weiull distriution with proility density function, is scle prmeter nd is shpe prmeter nd. Therefore, the instntneous rte of replenishment is m) During time t 1, inventory is depleted due to deteriortion nd demnd of the item. At time t 1 the inventory ecomes zero nd shortges strt occurring. III. MATHEMATICAL FORMULATION OF THE MODEL Let I(t) e the inventory level t time t (. The differentil equtions governing the system in the cycle time [0, T] re 51 Pulished By: & Sciences Puliction Pvt. Ltd.
52 Pulished By: & Sciences Puliction Pvt. Ltd. Some Eoq Model for Weiull Deteriortion Items with Selling Price Dependent Demnd (1) With I(t) = 0 t t = t 1 Solving the equtions (1) nd (2) nd neglecting higher powers of, we get (2) (3) (4) Stock loss due to deteriortion in the cycle of length T is Let Hence we get the profit function (10) Ordering quntity Q in the cycle of length T is (5) (11) (6) Holding cost is otined y sustituting the equtions (3) nd (4), we get Our ojective is to mximize the profit function. The necessry conditions for mximizing the profit function re nd We get Neglecting higher powers of, we get nd Shortge cost during the cycle is Let e the profit rte function. Since the profit rte function is the totl revenue per unit minus totl cost per unit time, we hve Sustituting the vlues of equtions (6), (7) nd (8) in eqution (9), one cn get the profit rte function s (9) (13) Using the softwre Mt cd 15, we otin the optiml policies of the inventory system under study. To find the optiml vlues of T nd s, we otin the first order prtil derivtives of given in eqution (11) with respect to T nd s nd equte them to zero. The condition for mximiztion of is
Vritions in T Interntionl Journl of Soft Computing nd Engineering (IJSCE) ISSN: 2231-2307, Volume-4, Issue-1, Mrch 2014 IV. NUMERICAL EXAMPLE A. Cse I (with shortges) Let A = 500, C = 10, h = 2, = 0.5, = 10, = 0.5, = 0.4, = 100, = 2 Bsed on ove input dt nd Using the softwre Mtcd 6.0, we clculte the optiml vlue of = 1.1484, = 2.871, s = 34.976, Q = 62.627, = 310.964 B. Cse II (without shortges) Bsed on ove input dt nd Using the softwre Mtcd 6.0, we clculte the optiml vlue of = 0.8084, = 2.021, s = 18.39, Q = 120.43, = 166.199 V. SENSITIVITY ANALYSIS To study the effects of chnges of the prmeters on the optiml profit derived y proposed method, sensitivity nlysis is performed considering the numericl exmple given ove. Sensitivity nlysis is performed y chnging (incresing or decresing) the prmeters y 10%nd 20% nd tking one prmeter t time, keeping the remining prmeters t their originl vlues. The results re shown in Tle-1 nd Tle-2 for with shortge cse nd without shortge cse respectively. The reltionship etween the prmeters nd the optiml vlues re shown in Figure 1 nd 2. Tle 1 Sensitivity nlysis of the model (with shortges) Vrition Optiml Chnge in prmeters Prmeters Policies -20% -10% 0% 10% 20% Alf t 1 1.285 1.238 1.148 1.020 0.818 T 3.214 3.097 2.871 2.551 2.045 s 31.073 32.883 34.976 38.757 41.65 Q 35.161 52.005 62.627 78.614 138.202 143.527 174.373 310.964 406.978 472.826 Bet t 1 2.076 1.605 1.148 0.960 0.943 T 5.192 4.014 2.871 2.400 2.359 s 24.868 29.573 34.976 38.314 46.486 Q 99.519 88.286 62.627 40.418 54.076 428.809 411.187 310.964 264.305 172.377 Alf t 1 0.867 0.867 1.148 1.375 2.067 T 2.168 2.169 2.871 3.438 5.169 s 30.119 30.615 34.976 38.451 40.507 Q 67.11 65.225 62.627 51.286 50.349 350.938 345.726 310.964 215.199 171.88 Bet t 1 0.804 0.842 1.148 1.686 1.696 T 2.012 2.105 2.871 4.215 4.241 s 30.134 30.168 34.976 37.136 37.152 Q 66.023 65.719 62.627 56.506 50.485 343.059 343.344 310.964 291.625 282.324 Some EOQ Model for Weiull Deteriortion Items with Selling Price Dependent Demnd Vritions in t 1 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 5.5 5 4.5 4 3.5 3 2.5 2 Percentge chnge in prmeters Percentge chnge in prmeters 53 Pulished By: & Sciences Puliction Pvt. Ltd.
54 Pulished By: & Sciences Puliction Pvt. Ltd. Some Eoq Model for Weiull Deteriortion Items with Selling Price Dependent Demnd Vritions in s 47 44 41 38 35 32 29 26 23 Percentge chnge in prmeters Vritions in Q 140 130 120 110 100 90 80 70 60 50 40 30 Percentge chnge in prmeters Vritions in P 490 460 430 400 370 340 310 280 250 220 190 160 130 Percentge chnge in prmeters Fig 1: Reltionship etween prmeters nd optiml vlues with shortges We study from ove Tle-1 revels the following i) Increse in the vlues of either of the prmeters, will result in increse of T, s nd Q ut decrese t 1,. ii) Decrese in the vlues of either of the prmeters, will result in decrese of T, s nd Q ut increse t 1,. iii) Increse in the vlues of either of the prmeters, will result in increse of s ut decrese t 1, T, Q nd. iv) Decrese in the vlues of either of the prmeters, will result in decrese of t 1 nd s ut increse T, Q nd. v) Increse in the vlues of either of the prmeters, will result in increse of t 1 T nd s ut decrese Q nd. vi) Decrese in the vlues of either of the prmeters, will result in decrese of t 1 T nd s ut increse Q nd. vii) Increse in the vlues of either of the prmeters, will result in increse of t 1 T nd s ut decrese Q nd. viii) Decrese in the vlues of either of the prmeters, will result in decrese of t 1 T nd s ut increse Q nd. Tle 2 Sensitivity nlysis of the model (without shortges) Vrition Optiml Chnge in prmeters Prmeters Policies -20% -10% 0% 10% 20% t 1 0.770 0.829 1.501 1.504 1.542 T 1.541 1.658 3.002 3.008 3.084 s 17.675 17.814 18.107 18.815 19.411 Q 130.339 160.007 194.469 194.575 197.932 171.313 176.242 221.659 261.284 305.507 t 1 1.02 1.36 1.501 1.487 1.554 T 2.04 2.72 3.002 2.975 3.109 s 17.731 17.788 18.107 19.139 19.628 Q 130.015 164.106 194.469 197.072 198.043 182.2 184.828 221.659 192.267 182.685 t 1 1.538 1.530 1.501 1.411 1.326 T 3.076 3.061 3.002 2.823 2.653 s 18.039 18.083 18.107 18.113 18.149 Q 198.7 198.76 194.469 181.835 169.794 227.355 226.653 221.659 208.438 195.789 t 1 1.256 1.525 1.501 1.298 1.083 T 2.512 3.043 3.002 2.597 2.167 s 18.183 18.11 18.107 18.076 18.289 Q 161.089 167.699 194.469 165.891 134.723 289.195 223.586 221.659 194.391 164.166
Vritions in T 55 Pulished By: & Sciences Puliction Pvt. Ltd. Interntionl Journl of Soft Computing nd Engineering (IJSCE) ISSN: 2231-2307, Volume-4, Issue-1, Mrch 2014 Vritions in t 1 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 Percentge chnge in prmeters 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 Percentge chnge in prmeters Vritions in s 19.7 19.4 19.1 18.8 18.5 18.2 17.9 17.6 17.3 17 Percentge chnge in prmeters Vritions in Q 200 190 180 170 160 150 140 130 Percentge chnge in prmeters Vritions in P 320 300 280 260 240 220 200 180 160 Percentge chnge in prmeters Fig 2: Reltionship etween prmeters nd optiml vlues without shortges We study from ove Tle-2 revels the following i) Increse in the vlues of either of the prmeters, will result in increse of t 1, T, s, Q nd. ii) Decrese in the vlues of either of the prmeters, will result in increse of t 1, T, s, Q nd. iii) Increse in the vlues of either of the prmeters, will result in increse of t 1, T, s, Q nd. Some EOQ Model for Weiull Deteriortion Items with Selling Price Dependent Demnd iv) Decrese in the vlues of either of the prmeters, will result in increse of t 1, T, s, Q nd. v) Increse in the vlues of either of the prmeters, will result in decrese of t 1, T, s, Q nd. vi) Decrese in the vlues of either of the prmeters, will result in increse of t 1, T, s, Q nd. vii) Increse in the vlues of either of the prmeters, will result in decrese of t 1, T, s, Q nd. viii) Decrese in the vlues of either of the prmeters, will result in increse of t 1, T, s, Q nd. VI. CONCLUSION In this present pper we hve developed deterministic inventory model for deteriorting items for with shortge nd without shortge cses. The deterministic demnd rte is ssumed to e function of selling price. Whenever shortges re llowed nd they re completely cklogged. We cn mke good comprtive study etween the results of the with-shortge cse nd without-shortge cse. In the numericl exmples, it is found tht the optimum verge profit in with shortge cse is more thn tht of the without shortge cse. From the ove model one cn clculte the optimum verge profit mrgins for with shortge cse nd without shortge cse for the deterministic inventory model with vrying demnd rte. REFERENCES [1] Aggrwl, S.P., A note on n order-level inventory model for system with constnt rte of deteriortion, Opserch, 15(1978), pp: 84-187. [2] Burwell T.H., Dve D.S., Fitzptrick K.E., Roy M.R., Economic lot size model for price-dependent demnd under quntity nd freight discounts, Interntionl Journl of Production Economics, 48(2)(1997), pp: 141-155. [3] Chkrrti, et l, An EOQ model for items Weiull distriution deteriortion shortges nd trended demnd n extension of Philip s model. Computers nd Opertions Reserch, 25(1997), pp: 649-657. [4] Chkrorti, T., nd Chudhuri, K.S., An EOQ model for deteriorting items with liner trend in demnd nd shortges in ll cycles, Interntionl Journl of Production Economics, 49(1997), pp: 205-213.
Some Eoq Model for Weiull Deteriortion Items with Selling Price Dependent Demnd [5] Chung, K., nd Ting, P., A heuristic for replenishment of deteriorting items with liner trend in demnd, Journl of the Opertionl Reserch Society, 44(1993), pp: 1235-1241. [6] Covert, R.P., nd Philip, G.C., An EOQ model for items with Weiull distriution deteriortion.aiie Trnsctions, 5(1973), pp: 323-326. [7] Dve, U., nd Ptel. L.K., (T, S) policy inventory model for deteriorting items with time proportionl demnd, Journl of the Opertionl Reserch Society. 32(1981),pp: 137-142. [8] De, M., nd Chudhuri. K.S., An EOQ model for items with finite rte of production nd vrile rte ofdeteriortion, Opserch, 23(1986), pp:175-181. [9] Fujiwr, O., EOQ models for continuously deteriorting products using liner nd exponentil penlty costs, Europen Journl of Opertionl Reserch, 70(1993), pp:104-14. [10] Ghre, P.M., nd Schrder, G.F., An inventory model for exponentilly deteriorting items, Journl of Industril Engineering, 14(1963), pp:238-243. [11] Giri, B.C., nd Chudhuri, K.S., Heuristic models for deteriorting items with shortges nd time-vrying demnd nd costs, Interntionl Journl of Systems Science, 28(1997), pp:53-159. [12] Goh, M., EOQ models with generl demnd nd holding cost functions, Europen Journl of Opertionl Reserch, 73(1994), pp:50-54. [13] Hrig, M., Optiml EOQ models for deteriorting items with time vrying demnd, Journl of Opertionl Reserch Society, 47(1996),pp: 1228-1246. [14] Hrig, M.A., nd Benkherouf, L., Optiml nd heuristic inventory replenishment models for deteriorting items with exponentil timevrying demnd, Europen Journl of Opertionl Reserch. 79(1994), pp: 123-137. [15] Jln, A.K., nd Chudhuri, K.S., Structurl properties of n inventory system with deteriortion nd trended demnd, Interntionl Journl of systems Science, 30(1999), pp: 627-633. [16] Jln, A.K., Giri, R.R., nd Chudhuri, K.S., EOQ model for items with Weiull distriution deteriortion shortges nd trended demnd, Interntionl Journl of Systems Science. 27(1996),pp: 851-855. [17] Mishr, R.B., Optimum production lot-size model for system with deteriorting inventory, Interntionl Journl of Production Reserch, 3(1975), pp: 495-505. [18] Mondl, B., Bhuni, A.K., Miti, M., An inventory system of meliorting items for price dependent demnd rte, Computers nd Industril Engineering, 45(3)(2003), pp: 443-456. [19] Muhlemnn, A.P. nd Vltis-Spnopoulos, N.P., A vrile holding cost rte EOQ model, Europen Journl of Opertionl Reserch,4(1980), pp: 132-135. [20] Shh, Y.K., nd Jiswl, M.C., An order-level inventory model for system with constnt rte of deteriortion, Opserch, 14(1977), pp:174-184. [21] Su, C.T., et l, An inventory model under infltion for stockdependent consumption rte nd exponentil decy, Opserch, 33(1996), pp: 72-82. [22] Vn Der Veen, B., Introduction to the Theory of Opertionl Reserch, Philip Technicl Lirry, Springer-Verlg, New York, 1967. [23] Wee, H.M., A deterministic lot-size inventory model for deteriorting items with shortges nd declining mrket, Computers nd Opertions, 22(1995), pp: 345-356. [24] Weiss, H.J., Economic Order Quntity models with nonliner holding cost, Europen Journl of Opertionl Reserch, 9(1982), pp: 56-60. [25] You, S.P., Inventory policy for products with price nd timedependent demnds, Journl of the Opertionl Reserch Society, 56(2005), pp: 870-873. [26] C. K. Tripthy., et l, An Inventory Model for Weiull Deteriorting Items with Price Dependent Demnd nd Time- Vrying Holding Cost, Applied Mthemticl Sciences, 4(2010), pp: 2171 2179. [27] Sushil Kumr., et l, An EOQ Model for Weiull Deteriorting Items With Price Dependent Demnd, IOSR Journl of Mthemtics, 6(2013), pp: 63-68. Dr. A. Lkshmn Ro Fculty Deprtment of Bsic Sciences nd Humnities in Adity Institute of Technology nd Mngement, Tekkli, Andhr Prdesh (Indi). He received his Ph.D. degree from Andhr University, Viskhptnm nd he hs more thn Sixteen yers experience in cdemics nd reserch. He hs pulished two reserch ppers in reputed ntionl nd interntionl journls nd reviewer of Interntionl Journl of Opertions Reserch. His re of speciliztion is Opertions Reserch, Inventory Models. B. Neel Ro Fculty Deprtment of Bsic Sciences nd Humnities in Adity Institute of Technology nd Mngement, Tekkli, Andhr Prdesh (Indi). He received his M.Phil. degree from Andhr University, Viskhptnm nd he hs more thn Eighteen yers experience in cdemics nd reserch. His re of speciliztion is Fluid Dynmics, Mgneto Hydrodynmics. Dr. R. Snthi Kumr Fculty Deprtment of Bsic Sciences nd Humnities in Adity Institute of Technology nd Mngement, Tekkli, Andhr Prdesh (Indi). He received his Ph.D. degree from Achry Ngrjun University, Guntur nd he hs more thn Fifteen yers experience in cdemics nd reserch. He hs pulished Eight reserch ppers in reputed ntionl nd interntionl. His re of speciliztion is Reltivity nd Cosmology. 56 Pulished By: