Optimal inventory model with single item under various demand conditions

Transcription

1 Optimal inventory model wit single item under various demand conditions S. Barik, S.K. Paikray, S. Misra 3, Boina nil Kumar 4,. K. Misra 5 Researc Scolar, Department of Matematics, DRIEMS, angi, Cuttack, Odisa (India, 4 Department of Matematics, VSS, Burla, Odisa (India 3 Department of Matematics, Govt. Science College, Malkangiri, Odisa (India 4 Department of Matematics, NIS, Berampur, Odisa (India BSRC e goal of tis model is to talk about te stock model for time fluctuating demand and constant demand wit time dependent olding expense and consistent olding cost for case and case separately. Scientific model as been created for deciding te optimal request amount, te optimal process duration and optimal aggregate stock expense for bot te cases. Numerical illustrations are given for bot cases to approve te proposed model. Sensitivity analysis as been done to analyze te effect of canges in te optimal solution wit respect to cange in various parameters. Keywords: olding cost, Inventory system, time dependent demand. I. INRODCION Inventory management aims to minimize te inventory carrying cost. Previously EO models are constructed assuming demand rate as constant. It is watced tat demand for a specific item can be affected by internal factors, for example, value, time and accessibility. Demand elasticity means alteration in demand to te inventory. us, wen te demand rate is constant, marketing decisions plays an important role in Inventory management. Different models ave been discussed for constant demand rate wit constant olding cost. n EO model was talked about by eng et al. [6] on ideal valuing and requesting strategy under allowable postponement in installments was viewed as tat te offering cost is unavoidably iger tan te buying cost. suitable model was given by tem in wic retailer discovered its ideal cost and parcel size, at te same time, wen te supplier offered an allowable postponement in installment. consistent rate EO model was given by Mulemann and Valtis-Spanopoulos [] wit variable olding cost were rate of te normal estimation of capital investigated in stock. n EO inventory model was developed by Vander Veen [8] were olding cost is taken as a nonlinear function of inventory. Weiss [9] establised an EO model were te olding cost is a nonlinear function of lengt of time an item was eld in stock. Go [6] found an EO model wit general request and olding cost capacity, demand rate for a ting was considered as an element of existing stock level and conveying cost per unit was allowed to cange. lfares [] displayed te stride arrangement of te olding cost in perspective of te stock approac were request rate relies upon stock level and olding cost relies on capacity time. e olding cost per unit ting per 577 P a g e

2 unit time is taken as expanding capacity of time spent in storage. In a portion of te stock model, it is expected tat te olding cost canges wit time. n EO model was given by Giri et al. [5] for crumbling (deteriorating tings wit deficiencies, were bot te interest rate and te olding expense are considered as continuous function of time. Datta and Pal [3] talked about deterministic stock model witout deficiency were time orizon is boundless and as a level-dependent demand rate up to a specific stock-level and a consistent interest for watever is left of te cycle. Pal et al. [] develop a deterministic stock model in wic te interest rate was stock-dependent and tat te tings crumble at a steady rate. Discounted cas flow approac in inventory was introduced by Dye et al. [4]. ey found te ideal stock and evaluating metodologies amplifying te net present estimation of te aggregate benefit over te infinite orizon. tilizing DCF approac Cung and Liao [] built up an ideal requesting arrangement of EO model under excange credit depending upon te ordering amount. ey examined te ideal request amount of te EO model wic is reliant on te stock strategy and in addition firm credit arrangement utilizing marked down income (DCF approac and excange credit contingent upon te amount requested. Researcers like Jaggi et al [9], Hsu [8], and Roy et al. [3] created stock lot size models under excange credit financing in wic te interest rate is consistent. n EO model wit excange credit financing was given by eng et al. [7] for non decreasing request and ideal arrangement. n EO model was built up by Sarkar [4] were delay in installments and time sifting weakening rate was considered and were te retailers were permitted an excange credit offer by suppliers to purcase more tings wit various rebate rates on te acquiring cost. Hung [7] found a stock model wit summed up interest, deterioration and backorder rates. Hung [7] augmented te model from incline sort request rate and Weibull deterioration rate to arbitrary demand rate and subjective weakening rate in considering partial backorder. Kanra et al. [0] set up an EO model for crumbling ting wit time dependent quadratic interest under allowable deferral in installment were an exertion as been made to analyze an EO model for deteriorating down ting. Sana [5] set up an EO model were time orizon is boundless, ideal offering cost and parcel size wit time canging decay and partial backlogging. In te present section, te interest rate is time canging and olding expense is steady for case ; and in case te condition is vice-versa. e target of tis part is to acquire least aggregate stock cost, request amount and relating request cycle for bot cases. n algoritm tat minimizes te aggregate stock expense is produced. e Model is tried wit Numerical examples. II. SSMPIONS ND NOIONS e model needs te accompanying suppositions: i. Inventory utilized are of one sort. ii. Lead time is zero. iii. Sortages are not permitted. iv. e request rate R(t is decreasing function of time wit increase of for case. v. e annual demand rate is constant for case. 578 P a g e

3 vi. e olding expense is time dependent and olding cost parameter is given by ( t. t for case. vii. e olding expense is steady for case. We utilize te accompanying notations: I (t = On and inventory level at any time t, were t 0. = e lengt of cycle time. = e requesting cost per unit time = e consistent yearly request rate. = e stock olding expense of te ting of case (t = e time dependent olding cost for case. R (t = e time canging interest rate given by ( t t were 0, 0 is te demand parameter. otal inventory cost per cycle Ordering quantity III. FORMLION Case: R. Here t time t 0 te inventory level on-and is I (t. e demand rate is always assumed to be positive. Due to effect of demand te amount of stock reduces in te interval [ 0, ], and at time te stock acieves zero. Consequently, te stock level at any moment of time is given as follows. e inventory on-and at time t t in0, will be: I( t t I( t d( t. t, t t 0, Dividing by t and ten taking limit t 0 we get di ( t (3. dt. t under te boundary condition (3. I ( 0. ; 0 t We ave, solution of te differential equation (3. is found to be: (3.3 I( t. t e ordering quantity of stock is given by (3.4 were 0. From (3.4, we obtain 579 P a g e

4 580 P a g e (3.5. ( Now te average total cost per cycle is given by (3.6 ( [Ordering Cost + Holding Cost] dt t I 0 ( 0 dt t sing (3.5, we obtain (3.7 (. ( e necessary condition for minimization of ( is, ( ( e sufficient condition for minimization of ( is ( ( (3.0. ( ( ( (3.. ( ( (3 Now te function ( will be maximum if (3. 0 ( wic is obvious from (3.. Now from (3.8 we ave ( ( (

5 Now on solving (3.3, and using in te above equation we gate optimal wic is given by (. and ence te optimal cost ( can be evaluated. If 0, ten te optimal and (. Case: t time t 0 te inventory level on-and be I (t. e demand rate is always assumed to be constant. e amount of stock reduces in te period [ 0, ] because of te impact of demand and te stock acieves zero at time. subsequently, te stock level at any moment of time is depicted as follows. e inventory on-and at time t t, will be: I( t t I( t d( t. t, t t 0, Dividing by t and taking limit t 0, we get di ( t (3.4 dt ; t [0, ] Wit te boundary condition (3.5 I ( 0. e solution of te differential equation (3.4 is given by, (3.6 I( t t e stock ordering quantity is given by (3.7 I( 0. From (6.3.7, we obtain (3.8. Now te average total cost per cycle is given by (3.9 ( [Ordering Cost + Holding Cost] 0 t. I( t dt 58 P a g e

6 .. t t 0 dt 6 sing (3.8, we obtain (3.0 (. 6 6 e necessary condition for minimization of ( is, ( (3. 0. e sufficient condition for minimization of ( is ( (3. 0. ( 3 (3.3. ( 3 3 (3.4. Now te function ( will be minimum if ( (3.5 0 Now from (3. we ave 3 ( wic is obvious from (3.. Now on solving (3.6, implies minimizing total cost to determine optimal given by 3 3 and ence te optimal cost ( can be evaluated. IV. COMPIONL LGORIHM Step-: Start. Step-: Initialize te value of te variables,,,. Step-3: Evaluate (. 58 P a g e

7 ( Step-4: Evaluate. ( Step-5: Solve te equation 0. Step-6: Coose te solution from Step-5. ( Step-7: Evaluate. Step-8: If te value of Step-7 is greater tan zero ten tis solution is Optimal (minimum and move to Step-0. Step-9: Oterwise move to Step-6. Step-0: End. V. NMERICL EXMPLE e estimations of te parameters in legitimate units are considered as below: 500, 400, 0., 40 For case- Optimal , , For case- Optimal , , VI. SENSIIVIY NLYSIS BLE : 0(CSE β BLE : CSE CSE P a g e

8 BLE : 3 CSE CSE Important points from te table e effect of optimality due to cange of values of different parameters associated in tis model is discussed below. For Case:. & increase as decreases for increase in value of te parameter.. increases wile & decrease wit increase in value of te parameter. 3., & increase wit increase in value of te parameter. For Case:. increases wile & decrease wit increase in value of te parameter.., & increase wit increase in value of te parameter. Variation of ime duration, Ordering uantity and otal cost w.r.t. different parameters. CSE: 584 P a g e

9 .Variation of ime duration.variation of Ordering uantity 3.Variation of otal cost 585 P a g e

10 CSE:.Variation of ime duration.variation of Ordering uantity 3.Variation of otal cost 586 P a g e

11 VII. CONCLSION deterministic stock model for time fluctuating interest and steady request; and time subordinate olding expense and consistent olding cost for case and case individually are inferred. Matematical model as been produced for deciding te ideal (optimal request amount, te ideal process duration and ideal aggregate stock expense for bot te cases. Numerical examples are given for bot cases to accept te proposed model along wit sensitivity investigation. e model can furter be enriced wit sortage state and for multiple items under similar conditions. is can also be extended for deterioration conditions and also for discounted cas flow approac. REFERENCES [.] H. K. lfares, Inventory model wit stock level dependent demand rate and variable olding cost. International Journal of Production Economics, 08, 007, [.] K. J. Cung, and J. J. Liao, e optimal ordering policy of te EO model under trade credit depending on te ordering quantity from te DCF approac. European Journal of Operational Researc, 96, 009, [3.].K Datta, and. K. Pal, note on inventory model wit inventory level demand rate. Journal of te Operations Researc Society, 40, 990, [4.] C.Y. Dye, L.Y. Ouyang and. P. Hsie, Inventory and pricing strategies for deteriorating items wit sortages: discounted cas flow approac. Computers & Industrial Engineering, 5, 007, [5.] B. C. Giri,. Goswami, and K. S. Cauduri. n EO model for deteriorating items wit timevarying demand and costs. Journal of te Operational Researc Society, 47, 996, [6.] M. Go, EO model wit general demand and olding cost function. European Journal of Operational Researc, 73, 994, [7.] K.C. Hung, n inventory model wit generalized type demand, deterioration and backorder rates. European Journal of Operational Researc, 08, 0, [8.] L. F. Hsu, note on n economic order quantity (EO for items wit imperfect quality and inspection errors. International Journal of Industrial Engineering Computations, 3(4, 0, [9.] C. K. Jaggi,S. K. Goel, and M. Mittal, Economic order quantity model for deteriorating items wit imperfect quality and permissible delay on payment.international Journal of Industrial Engineering Computations,, 0, [0.] S. Kanra, S. K. Gos, and K.S. Cauduri (0. n EO model for a deteriorating item wit time dependent quadratic demand under permissible delay in payment. pplied Matematics and Computation, 8, 0, - 9. [.]. P. Mulemann and N. P. Valtis-Spanopoulos, variable olding cost rate EO model. European Journal of Operational Researc, 4, 980, [.] S. Pal,. Goswami, and K. S. Cauduri, deterministic inventory model for deteriorating items wit stock dependent demand rate. International Journal of Production Economics, 3(5, 993, P a g e

12 [3.] M. Roy, R. K. Gupta, and. Dasgupta, tecnique for determining te optimum mix of logistics service providers of a make-to-order supply cain by formulating and solving a constrained nonlinear cost optimization problem. Decision Science Letters,, 0, [4.] B. Sarkar, n EO model wit delay in payments and time varying deterioration rate.matematical and Computer modeling, 55(3 4, 0, [5.] S. S. Sana, Optimal selling price and lot size wit time varying deterioration and partial backlogging. pplied Matematics and Computation, 7, 00, [6.] J.. eng, C.. Cang and S. K. Goyal, Optimal pricing and ordering policy under permissible delay in payments. International Journal of Production Economics, 97, 005, -9. [7.] J.. eng, J. Min, and. Pan, Economic order quantity model wit trade credit financing for nondecreasing demand. Omega, 40, 0, [8.] B. Van der Veen, Introduction to te teory of operational Researc. Pilips ecnical Library, Springer- Verlag, New York 967. [9.] H. J. Weiss, Economic order quantity models wit non linear olding cost. European Journal of Operational Researc, 9, 98, P a g e