Annals of Pure and Applied Mathematis Vol.,.,, 3-43 ISSN: 79-87X (P),79-888(online) Published on 5 September www.researhmathsi.org Annals of An EOQ Model with Paraboli Demand Rate and ime Varying Selling Prie Monika Mandal and Sujit Kumar De Department of Applied Mathematis with Oeanology and Computer Programming, Vidyasagar University, Medinipur-7, India Department of Mathematis, Midnapore College, Midnapur-7, India, E-mail: skdemamo8.om@gmail.om Reeived 9 August ; aepted 8 August Abstrat. his paper deals with the perishable items of seasonal produt where the demand rate follows a paraboli path. his path is symmetri about the time axis in whih the selling prie takes time varying linear dereasing funtion in one part and the inreasing funtion for the other part. However, we have assumed the gradient of the selling prie line is known for the first part and for the other the slope is unknown. Considering onstant deterioration rate the average profit funtion is developed. Solutions are made through analytial method. Graphial interpretations, numerial examples with sensitivity analysis have also been made to illustrate the model. AMS Mathematis Subjet Classifiation (): 9B5 Keywords: Inventory, paraboli demand funtion, time varying selling prie, deterioration. Introdution he lassial EOQ model was developed by assuming the demand rate as onstant. But in pratie, it does not onvey the reality. he atual fat is, for the ase of deaying items like fruits, vegetables et. the demand rate is growing and it reahes to a pik time then it began to derease within a stipulated period. he traditional onept on demand rate was first modified by Silver and Meal [8]. hen Resh et al. [4] and Donaldson [6] studied with linear trend demand in their models. Ghare and Shrader [7] initially proposed the model with exponentially deaying inventory. Subsequently, Covert and Phillip [5] and adikamalla [] developed an EOQ model with Weibull and Gamma distribution respetively. After that, a numerous researh papers were developed by several researhers onsidering the demand rate as time varying ontinuous funtion. Lot-size model with shortages and 3
An EOQ Model with Paraboli Demand Rate and ime Varying Selling Prie flutuating demand under inflation was developed by Yang et al []. Reently, Bose et al [], Giri et al. [9], Jamal et al. [], Ghosh and Chaudhuri [8] developed inventory models with time dependent deterioration rate. Moreover, Aggoun et al. [] was analyzed a stohasti proess with random harateristi over varying deterioration. Chang and Dye [4], Mehta and Saha [3], Roy [7] and Boukhel et al. [3] are some others researhers who onsidered the inventory-level dependent or prie dependent demand as a whole. Very reent, an eonomi prodution lot size (EPLS) model with random prie sensitive demand was developed by Roy et al. [5] and that with stok-prie sensitive demand and deterioration was explained by Roy and Chaudhuri [6]. Researhers like Goswami and Choudhuri [], Giri et al. [] have developed an inventory model onsidering time varying demand, shortages, and deterioration. Skouri et al. [9] have analyzed a model with weibull deterioration rate, partial baklogging and ramp type demand rate. In our paper, we have assumed the deterioration rate as onstant when the demand rate follows a paraboli path within a speifi season. However in a super market we observe that the sellers are usually unwilling to give a rebate / prie disount over an item for their ustomers to have more and more profit. Observation tells that, up to optimum demand the selling prie behaves like dereasing funtion and after that it beomes inreasing funtion in time with unknown gradient. Sine, till date no papers were published along this diretion, so onsidering above assumptions we have developed a model as well. Solution is made analytially, graphial interpretations; numerial examples with sensitivity analysis are done to illustrate the model. able- Major harateristis of inventory models on seleted areas Author(s) and publiation year Bose et. al. (995) Giri et al. (996) Jamal et al. (997) Chang & Dye (999) Aggoun et.al. () Mehta & Shah (3) Ghosh &Chaudhuri (4) Boukhel et.al. (5) Deision Criteria Deterioration Varying Demand Varying /Prie Baklogged Allowed Exponential imevaryin g Linear ime varying Exponential Constant Constant ime Varying Stohasti Stohasti Constant Exponential Weibull Constant ime Quadrati Inventory Level Constant 33
Monika Mandal and Sujit Kumar De Roy (8) Skouri et al. (9) Roy et.al. () Roy & Chaudhuri () Present Paper () Profit Profit Profit Profit Constant Selling Prie dependent Weibull Ramp type Partial Constant Constant Stohasti Prie Dependent Stok- Prie Sensitive ime- Paraboli Dereasing funtion Constant ime -Linear Assumptions and tations he following notations and assumptions are used to develop the model. Assumptions. Replenishments are instantaneous. Lead time is zero 3. Shortages are not allowed 4. Demand rate is a time dependent paraboli funtion D() t = d+ bt at for < t< t where a, b >, t ( = b + 4 ad / a) the axis of symmetry of the demand funtion and t = t + b/ a, a> is the time of zero demand. 5. he selling prie funtions below and above the axis t= t of symmetri demand are given by p() t = s λt for < t t and p() t = s+ µ t for t t t respetively. Also for ontinuity we have p( t) = p( t). 6. Deterioration is allowed and it is onstant tations i) q : he instantaneous inventory at time t. ii) D(t) : Instantaneous demand rate iii) Q : he order quantity per yle iv) d : the demand rate at time zero v) s : Unit selling prie ( $) vi) h : Inventory holding ost per unit quantity per week ($) vii) d : he deterioration ost viii) p : Purhasing prie of unit item ($) 34
An EOQ Model with Paraboli Demand Rate and ime Varying Selling Prie ix) : ordering ost per order ($) x) t : Growth period of demand xi) θ : he deterioration rate xii) : Cyle time in weeks xiii) W : Average Profit of the inventory ($). Model formulation he inventory starts at time t = with maximum order level Q and depletes with deterioration and paraboli demand rate D(t) for the period [, t ]. After time t the demand rate began to derease up to time. Also, the selling prie is a known time linear dereasing funtion within [, t ] and for the period [ t,] it began to inrease with unknown slope of linear time prie line. he governing differential equation of the inventory funtion is given by dq + θ q = D() t for < t dt () Subjet to the onditions q() = Q for < t t () and q ( ) = fo r t t (3) w, solving () using () and (3) we get θ t 3 qt ( ) = Qe at /3 bt / dt+ θ t ( d/ + bt/ 6 at /) < t b a θ and Q = d + + 6d + 4b 3a 3 (Assuming θ << ) w, using (5), the inventory holding ost per yle is given by HC = h q() t d t θ t t θ t = h Qe ( at + 3bt+ 6d) + ( 6d + bt at ) dt 6 3 θ = h d + b a + d + b a ( 6 4 5 ) ( 5 ) (4) (5) (6) 35
Monika Mandal and Sujit Kumar De he deterioration ost per yle an be found as DC = d Q D() t d t = d Q { d + bt at } dt (7) θ d = ( 6d + 4b 3a ) (using(5)) Purhasing ost of the ordered quantity per yle is given by PC = pq (8) and the set up ost per order = OC = (9) w the total selling prie per yle is given by t SP= p() t Dt () dt+ p() t Dt () dt t ( λ )( ) ( µ )( ) ( ) t = s t d+ bt at dt+ s + t d+ bt at dt t s s t s µ = + + + + + 6 ( 6d bt at ) ( 6d 3b a ) ( 6d 4b 3a ) where to simplify we have used p ( t ) = p ( t ) whih gives ( ) λ + µ = s s / t () Again if the selling prie is independent of time then total fixed selling prie is given θ by FSP s Q ( 6d 4b 3a ) s d b a = + = + 3 () herefore, total average profit per yle is given by W = [Selling prie-purhasing ost-holding ost-deterioration ost-set up ost]/cyle ime Model-I: he selling prie is a time dependent linear funtion We have the total average profit W = [ SP-PC-HC-DC-OC]/. So our inventory problem is () 36
An EOQ Model with Paraboli Demand Rate and ime Varying Selling Prie ( ) s s t s µ Maximize W = d + bt at + d + b a + d + b a 6 ( 6 ) ( 6 3 ) ( 6 4 3 ) h hθ pq/ 6d + 4b 5a d + 5b a d θ ( 6 d + 4 b 3 a ) / ( ) ( ) ( ) s s t s p ( µ θp θd) = + + + + + 6 ( 6d bt at ) ( 6d 3b a ) ( 6d 4b 3a ) h hθ ( 6d+ 4b 5a ) ( d+ 5b a ) / (3) Model-II: he selling prie is fixed throughout the season We have the total average profit W = [ FSP-PC-HC-DC-OC]/. So in this ase our inventory problem is s p θ ( p+ d ) Maximize W = d + b a + d + b a 6 h hθ ( 6d + 4b 5a ) ( d + 5b a ) / ( 6 3 ) ( 6 4 3 ) Speial Cases Case-I: In Model-I, if we put s = s = s and λ = = µ then we an easily get the Model-II. Case-II : If a and b then Model-II redues to the lassial profit model with deterioration, d θ θd W = ( s p) d θ ( d p) h and Q d + + + = + 3. Case-III : If a, b and θ then Model-II redues to the lassial profit model without deterioration where hd W = ( s p) d and Q = d w to optimize (3) we take, dw d = whih gives (4) 37
( ) ( ) Monika Mandal and Sujit Kumar De s s t 6d+ bt at s p ( µ θp θd ) + b a + 6 d + b a h hθ ( 6d+ 8b 5a ) ( 8d+ 45b 48a ) = ( 3 4 ) ( 6 8 9 ) ( ) ( ) dw s s t 6d+ bt at 6 ah θ h( 3 ) 3 3( ) a bθ = + a p d d 6 5 µ θ θ + h ( b+ θd) + θ b( p+ d) + a( s p) bµ < forany > of (5) (6) 3 (5) (6) w to obtain a solution of (5), we may use any searh tehnique or programme. Dt () = d+ bt at s p () t = s λ t p () t = s + µ t d t t Fig.- Paraboli Demand t Fig.- Selling Prie Funtion Inventory Level Q t ime t Fig.-3 EOQ model 38
An EOQ Model with Paraboli Demand Rate and ime Varying Selling Prie 3.. Numerial example If we put a =, b = 4, d = in the demand urve and taking setup ost = 5 $, holding ost h =3.5 $, deterioration ost d =.5 $, initial selling prie s = $, unit selling prie s =$, s = 3$, unit purhasing prie p = 3.5 $,initial selling prie line gradientλ =.4, deterioration rate θ =.3 and in the Model I and Model II we get the optimum result as follows able-: Results for varying and fixed selling prie t * * * µ Q t Model- 4.899 6.899 6.7536.797 4.6548 8.897 Model-II 4.899 6.899 6.7536 -- 4.6548 6.44 3.. Sensitivity analysis If we put a =, b = 4, d = in the demand urve and taking setup ost = 5 $, holding ost h =3.5 $, deterioration ost d =.5 $, initial selling prie s = $, s = 3$, unit purhasing prie p = 3.5 $,initial selling prie line gradient λ =.4, deterioration rate θ =.3 and onsidering the range of variations of the parameters {( a, b,, d, s, p) and ( λ, s, h, d, θ )} from -5% to + 5% we get the following optimal able-3 and 4. able-3. Sensitivity analysis for high sensitive parameters Parameter % hange * * W W µ * Q * W * * % W* +5.75 5.65 88.384 46.669-4.6 +3.96 5.73 99.46 58.8868-7.5 a -3.4449 8.574 73.378.59 36.74-5.67.835 4.995 39.4599 7.54 +5.63 7.8553 8.5545 33.4556 65. +3.7 7.4 56.89.3877 37.8 b -3.88 6.454 99.3 53.98-34.9-5.94 5.764 85. 36.88-55. +5.797 6.7536 4.6548 43.86-45.8 +3.797 6.7536 4.6548 58.694-77.48-3.797 6.7536 4.6548 3.399 7.48-5.797 6.7536 4.6548 7.8468 45.8 39 * W +5 ----- ----- ---- ---- ----
Monika Mandal and Sujit Kumar De +3.63 7.43 68.36.649 5.5 d -3.83 6. 84.648 36.48-54.96-5.365 5.46 6.997 3.75-95.95 +5 3. 6.7536 4.6548 9.74 38.45 +3.53 6.7536 4.6548 47.977 83.7 s -3.63 6.7536 4.6548 3.683-83.7-5 ----- ----- ----- ----- ------ +5.797 6.7536 4.6548 48.59-39.96 +3.797 6.7536 4.6548 6.4494-3.98 p -3.797 6.7536 4.6548. 3.98-5.797 6.7536 4.6548 3.3 39.96 able-4. Sensitivity analysis for low sensitive parameters Parameter % hange * * W W µ * Q * W * * % W* +5.77 6.7536 4.6548 79.89 -.7 +3.785 6.7536 4.6548 8.36 -.76 λ -3.89 6.7536 4.6548 8.4458.76-5.87 6.7536 4.6548 8.8565.7 +5.49 6.7536 4.6548 8.367 -.64 +3.634 6.7536 4.6548 8.5759 -.3 s -3.988 6.7536 4.6548 8.435.38-5.33 6.7536 4.6548 8.357.64 +5.797 6.7536 4.6548 8.35 -.64 +3.797 6.7536 4.6548 8.59 -.38 h -3.797 6.7536 4.6548 8.384.38-5.797 6.7536 4.6548 8.3443.64 +5.797 6.7536 4.6548 8.74 -.4 +3.797 6.7536 4.6548 8.764 -.8 d -3.797 6.7536 4.6548 8.8989.8-5.797 6.7536 4.6548 8.945.4 +5.797 6.7536 5.749 79.93 -. +3.797 6.7536 4.9668 8.899 -.67 θ -3.797 6.7536 4.347 8.3695.67-5.797 6.7536 4.346 8.793. 4
An EOQ Model with Paraboli Demand Rate and ime Varying Selling Prie 4. Comments on sensitivity analysis From the above able-3, we see that, the profit funtion is highly sensitive with the hange from -5% to +5% of the parameters of demand urve (a, b,d ) and the starting selling prie s relative to the other parameters. he objetive funtion is moderately sensitive whenever the setup ost and purhasing ost p are being hanged from -5% to +5%. However, from able-4, the holding ost h, deterioration rate θ and its orresponding ost d, selling prie line gradient λ and initial pik time selling prie s are negligible sensitive parameters over the profit funtion itself. hroughout the whole able-3 we see when the initial selling prie s is inreased to 5%, then the optimum profit is $ 9.74 with optimum yle time 6.75 weeks and the optimum order quantity is 4.6548 units. he other findings is that, after the pik time demand, the inrease of selling prie line gradient auses the inrease of the average profit funtion. From able-3 we see the highest gradient is 3. orresponds maximum average profit. 5. Conlusion and sope of future work We have developed our model with paraboli/ time quadrati demand rate under the time varying linear selling prie funtion and onstant deterioration. From past experiene we usually know the gradient of the inreasing selling prie funtion and after a pik time we usually ould not predit the gradient of the selling prie line. his happens due to the lak of prior knowledge and supply-onsumption of ommodities as well. Numerial examples shows before pik selling prie line, the slow derement of selling prie line gradient auses the high average profit in the model whih is a very ommon phenomenon in reality. For every seasonal produt, the market selling prie remains high at the beginning and at the end of the season, but, the purhasing ost remains onstant in many ases as the seller purhase huge goods from supplier for that partiular season. In our study we have shown the time dependent selling prie model is more profitable than onstant selling prie model. Finally, using ubi or higher powers of time in demand rate we may develop various inventory models. In future we shall develop several models inorporating various onsiderations like fuzzy, fuzzy stohasti, stohasti variable in the demand urve inluding deterioration and/or shortage in the model. REFERENCES. Aggoun, L., Benkherouf, L. and adj, L., On a stohasti inventory model with deteriorating items, IJMMS, 5(3) () 97-3.. Bose, S., Goswami, A. and Chaudhuri, K.S., An EOQ model for deteriorating items with linear time-dependent demand rate and shortages under inflation and time disounting, Journal of Operational Researh Soiety, 46 (995) 77-78. 4
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