Optimum inventory control for imperfect quality item with maximum lifetime under Quadratic demand and Preservation Technology Investment

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1 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. Optimm inventory control for imperfect qality item with maximm lifetime nder Qadratic demand and Preservation echnology Investment Nita H. Shah,*, Urmila Chadhari and Mrdl Y. Jani 3 Department of Mathematics, Garat University, Ahmedabad 389, Garat, India. Government Polytechnic Dahod, Dahod-3895, Garat, India. 3 Department of Applied Sciences, Faclty of Engg & echnology, Parl University, Vadodara-3976, Garat, India. (*Corresponding Athor: Nita H. Shah) Abstract he model is applicable for dry-frits and food indstry. India is considered to be the contry of festivals and varieties of food. Dring the festival, the demand of dry-frits extremely increases for preparation of sweets. his article focses for dryfrits. o maintain the qality of dry frits, preservation technology investment is incorporated. he demand follows qadratic natre dring season. he obective is to minimize the total cost of the inventory system with respect to screening time cycle time and investment for preservation technology. Here, to collect defective items from the imperfect qality items, we se concept of learning crve process. he model is spported with nmerical examples and also established scenario of the model. Sensitivity analysis is done to assme decision-making insights. Keywords: Inventory control, imperfect items, learning process, maximm fixed life-time, preservation technology investment. INRODUCION he classical EOQ has been an extensively developed model for inventory control prposes de to its modest and instinctively pleasing mathematical formlation. Salameh and Jaber () established a mathematical model that permits some loose qality items or imperfect qality items reqirements. he researchers assmed that each lot is screened percent by learning process and that can be sold at lower price. Hang (4) established for flawed items in a (JI) manfactring environment, a model to determine an optimal layered vendor byer inventory strategy. Maddah and Jaber (8) stdied a new model that remedies a flaw in the one given by Salameh and Jaber () sing renewal theory. Jaber et al (8) extended it by assming that the percentage imperfect per lot diminishes according to a learning crve. hey inspected empirical data from the self-propelled indstry for several learning crve models and the S-shaped logistic learning crve (Carlson (973); Jordan (958)) was fond to fit well. Jaggi and Mittal () examined when the items are of imperfect qality, the effect of deterioration on a retailer s EOQ. In that paper, defective items is assmed to be kept in the same warehose ntil the end of the screening process. Jaggi et al () and Sana () presented inventory models for imperfect qality items nder the condition of credit limit in payments. Haidar et al (4) extended the work of Jaggi and Mittal () to allow for shortages. Moreover, Alamri et al (6) developed an inventory control model for imperfect qality items. In classical inventory problems, it is assmed that prodcts have an infinite shelf life, while the most of items lose their initial vales over time and for some of them this occrs faster than sal which is called deterioration. (Soni & Patel, 3). Ghare and Schrader (963) determined deteriorating item s inventory model. he criticise articles by Raafat (99), Shah and Shah (), Goyal and Giri (), Bakker et al. (), on deteriorating items for inventory system throw light on the part of deterioration. Chng and Cardenas-Barrón (3) established spply chain inventory modelling algorithm for stock-dependent demand which comprising of three players for deteriorating items. Frthermore, Shah and Barrón (5) developed when a distribtor offers order-linked credit period or cash discont, byer's decision for credit policies and ordering for deteriorating items. On the other hand to redce deterioration, se preservation technology, Hs et al. () determined a model nder preservation technology investment, an inventory model to minimize the deterioration rate of inventory for constant demand. Hsieh and Dye (3) evalated when demand is changing with time, a prodction inventory model inclding the effect of preservation technology investment. Recently, Shah, et al. (6a) established an integrated inventory model for time dependent deteriorating item nder time and price sensitive demand with preservation technology. Moreover, Shah et al. (6b) developed spply chain inventory model nder selling price and trade credit dependent qadratic demand for time dependent deteriorating item with preservation technology. In the earlier research, constant demand was considered in many research articles however demand rarely remains constant over infinite planning horizon. In or stdy, demand is depended on time and qadratic in natre which is more feasible for the stdy seasonal prodct for example food indstry, electronics items and fashion goods. Moreover, most of the prodcts lose their tility over time. So, we consider time dependent deterioration rate and to redce deterioration preservation technology investment is calclated. Frthermore, stdy of screening process is very interesting concept of inventory modelling. In or stdy we consider many aspects of bsiness to calclate cost fnction. herefore, this paper focs 475

2 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. an inventory control model for imperfect qality items to minimize total cost. he model is applicable for dry-frits. o maintain the qality of dry frits, preservation technology investment is incorporated. he demand follows qadratic natre dring season. In the model, each lot is sbect to a per cent screening where items that are not achieving to certain qality standards are stored in a different warehose. herefore, different holding costs for the perfect and imperfect items are considered in the mathematical model. Items deteriorate while they are in storage, with screening and deterioration rates being chance fnctions of time. he percentage of imperfect items per lot decreases according to a learning crve. After a per cent screening, defective qality items may be sold at a low price as a single batch at the end of the screening process. Under above assmptions, the obective is to minimize the cost of inventory system with respect to the screening time, cycle time, and investment of preservation technology. he rest of the paper is strctred as follow. Section is abot the notations and the assmptions that are sed. Section 3 is abot formlation of the mathematical model of the proposed inventory control problem. Section 4 validates the derived inventory model with nmerical example and its sensitivity analysis. his section also provides some managerial insights. Finally, Section 5 provides conclsion and ftre research directions. NOAION AND ASSUMPIONS he proposed inventory problem is based on the following notation and assmptions. Notation Inventory system s parameters: Cycle index a otal Scale demand of the prodct, a b c Rt A Linear rate of change of demand of the prodct, b Qadratic rate of change of demand of the prodct, c ime dependent qadratic demand rate; R t a bt ct, where a is scale demand, bc, are rates of change of demand, respectively. Ordering cost per order incrred by the inventory system ($/order) C prchase cost per nit item (in $) t First phase dration (i.e. screening time) () t Deterioration rate; t m Fixed life-time of the prodct (in years) xt P Screening fnction Percentage of defectives per lot redces according to a learning crve Preservation technology investment per nit time (in $)(decision variable) f Ig I g Id t t t ; proportion of redced deterioration item (in year), Good Inventory level of the inventory system for the item dring first phase t t (nits) Good Inventory level of the inventory system for the item dring second phase t t (nits) Defective Inventory level of the inventory system for item at t t (nits) Cycle time (in years) of the inventory system (decision variable) Q Order qantity at time t h g h d HC Holding cost rate for inventory system for good item per nit per annm Holding cost rate for inventory system for defective item per nit per annm, g d h h Holding cost of the inventory system for item ($/nit / nit time) C,, t otal cost of the inventory system for th cycle per nit time ($/nit / nit time) Relations between parameters: m t he problem is expressed as follows: min C t,, Sbect to constraints Assmptions: m. he inventory system involves single instantaneos deteriorating item.. he demand, screening and deterioration rates are arbitrary fnctions of time denoted by Rt, xt and t respectively. he percentage of defectives per lot redces according to a learning crve denoted by P, where is cycle index. Here, we consider screening fnction as xt t and learning 476

3 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. crve is defined as,, and,. P e. Where, 3. he demand rate, Rt a bt ct (say) is fnction of time, a is scale demand, b denotes the linear rate of change of demand with respect to time, c denotes the qadratic rate of change of demand. 4. ime horizon is infinite. 5. Shortages are not allowed. i.e. P x t R t, t. 6. Lead time is zero or negligible. 7. he instantaneos rate of deterioration is t,. mt t m for any finite vale of m, we have t. If m then t i.e. the item is non- deteriorating. 8. he proportion of redced deterioration rate, f is assmed to be a continos increasing and concave fnction of investment on preservation technology, i.e. f and f. WLOG, assme f. MAHEMAICAL MODEL In this model, two phases are analyzed. he phases are screening phase, t and non-screening phase t,. At the beginning of each cycle (,,3,...), Q nits are received in the inventory system, which flfil actal demand and deterioration dring the screening phase and the nonscreening phase. Each lot is sbected to a % screening process at a rate of xt that starts at the beginning of the cycle and terminates by time, by which point in time Q nits have been screened and y nits have been depleted, which is the combined with demand and deterioration. Dring this phase, items not conforming to certain qality standards are stored in a different warehose. Q (-P) Q - y Inventory Level P Q t ime Figre : Inventory variation of the model for one cycle he variation in the inventory level dring the screening and non-screening phases (see Fig. (as in Alamri 6)) and inventory level variation for the defective items (shaded area in Fig. ) is given by eqations respectively as follows. di g dt t t f I g t R t x t P, t t With bondary condition and di g dt t I g Q, where Q. x d t f I t R t, t t With the bondary condition g g I. In the different warehose, the inventory level of defective item is given by the following eqation did t xt P, t t with the bondary condition dt I. d Solving above differential eqations, we get inventory level at any instant of time in the different phase, I, I (t), (t) g I (t) g and I (t) g (see Appendix- to Appendix-5). Now, the total cost component per nit time of the inventory system is comprises of Ordering of the inventory cost per nit Prchase cost of the item per nit Screening cost of the item per nit Holding cost per nit: : OC A : : PC CQ SC dq g 477

4 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. t HC hg I g t dt I g t dt t t d Investment Preservation echnology d h I t dt for : PI he total cost of the inventory system for the item per nit time is C t,, OC PC SC HC PI Figre : Convexity behavior of the cost fnction for t.6 year NUMERICAL EXAMPLE AND SENSIIVIY ANALYSIS Nmerical Example Example: Dring Indian festivals (generally festival is going on dring whole year) there is demand of dry frits. So, at that time demand rates are a 5, kgs., b %, c %. he ordering cost is A $ / kg. Prchase cost of the dryfrits is C $ / kg Moreover, holding cost rates for good inventory of dry frits and defective dry frits are h $ 8. /kg/time nit, h $ 3. g d has maximal life-time is.5 /kg/time nit and m year. Rates for learning process are 7. 76,. 89, 89 and constants of screening rate for dry frits are 5, and screening cost rates is d $ 5.. Now, to redce deterioration rate of the dry-frits, rate of investment for preservation technology is 5. Here, we analyze only for one cycle conseqently. he vales of the decision variables are screening time is t. 6 year, cycle time of replenishment is. 49 year and investment for preservation technology is $ /kg. his reslts inventory system s minimm cost as $ Also, the convexity of the cost fnction obtained in Fig.-4. Figre 3: Convexity behavior of the cost fnction for.49 year Figre 4: Convexity behavior of the cost fnction for $

5 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. (No. of Cycle) P able : Learning process Q No. of Defective item PQ It is clear that from the ableand Fig 5-6 sing learning crve process we can separate the defective items from the inventory system dring the cycle. With increase in cycles percentage of defective items decreased which reslts in less nmber of defective items received in the inventory system. P No. of Cycle No. of Defective item (PQ) No. of Cycle Figre 5: Fraction of defective items in cycles Figre 6: Nmber of defective items dring cycles Sensitivity Analysis for the Inventory Parameters In able, the sensitivity analysis of example is carried ot by varying one variable at a time as-%, -%, % and %. Parameter Change % Vales able : Sensitivity analysis t (in year) (in year) a -% % (in $) otal Cost C (in $) % % %

6 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. Parameter Change Vales t otal Cost % (in $) (in year) (in year) C (in $) b -% % % % % c -% % % % % % % % % % % % % % % % % % % % % % % % % h -% d -% % % % h -% g -% % %

7 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. Parameter Change % Vales t (in year) (in year) (in $) otal Cost C (in $) % C -% % % % % A -% % % % % d -% % % % % % % % % % m -% % % % % % % % % % N.B.: he soltion is declared to be infeasible becase life time is less than the cycle time. In order to observe the sensitivity of the model parameters on the optimal soltion, we consider the data as given in nmerical Example. Optimal soltions for different vales of a, b, c,,,,, h, A, C, d,, m and are h g, d presented in able. he following observation cold be made from able.. In able., constants of Screening rate and prchase cost increases screening time rapidly whereas scale demand and holding cost rate for good item decreases screening time rapidly. However, linear rate of screening and screening cost increases screening time slowly wherever qadratic rate of change of demand of the prodct, ordering cost and rate of investment of preservation technology decreases screening time slowly. In addition, change in linear rate of change of demand, learning process rate,,, maximm life-time of the prodct and holding cost rate for defective item screening time remain constant.. In able., linear rate of screening, constants of Screening rate and prchase cost increases cycle time rapidly althogh scale demand and holding cost rate for good item decreases cycle 48

8 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. time rapidly. However, qadratic rate of change of demand of the prodct and screening cost increases screening time slowly wherever ordering cost and linear rate of change of demand decreases cycle time slowly. In addition, change in, rate of investment of preservation technology, learning process rate,,, maximm life-time of the prodct and holding cost rate for defective item cycle time remain constant. 3. In able., constants of screening rate and prchase cost increases investment of preservation technology rapidly whereas scale demand, rate of investment of preservation technology and holding cost rate for good item decreases investment of preservation technology rapidly. However, linear rate of screening and screening cost increases investment of preservation technology slowly wherever maximm life-time of the prodct decreases investment of preservation technology slowly. In addition, change in linear rate of change of demand, learning process rate,,, qadratic rate of change of demand of the prodct, ordering cost and holding cost rate for defective item investment of preservation technology remain constant. 4. In able., constants of screening rate and prchase cost increases total cost of the inventory system rapidly whereas scale demand decreases total cost of the inventory system rapidly. However, linear rate of screening, ordering cost and screening cost increases total cost of the inventory system slowly wherever qadratic rate of change of demand of the prodct and holding cost rate for good item decreases total cost of the inventory system slowly. In addition, change in linear rate of change of demand, learning process rate,,, maximm life-time of the prodct, holding cost rate for defective item and rate of investment of preservation technology total cost of the inventory system remain constant. CONCLUSION In this paper, we consider inventory control model for instantaneos deteriorating imperfect qality item nder screening time, replenishment time and preservation technology investment with qadratic demand. In this article, a general inventory control model for items with imperfect qality was presented. Each lot is sbected to a per cent screening and items not conforming to certain qality standards are stored in a separate facility with different holding costs of the perfect and imperfect items being considered. he obtained nmerical reslts reflect the learning process effects incorporated in the proposed model. De to time varying deteriorating item retailer invest money on preservation technology to redce deterioration. he total cost of the inventory system with respect to the screening time, replenishment time and investment of preservation technology is minimized. For nmerical examples, inventory control system reaches the minimm cost and carry-ot sensitivity analysis with respect to inventory parameters. Crrent research have nmeros possible extension like, model can be frther generalized by taken more items at a time. One can also analyzed mlti layered spply chain. Research can be extended for finite or infinite planning horizons. ACKNOWLEDGMENS he athors are thankfl to anonymos reviewers for their constrctive comments. he first athor is thankfl to DS- FIS file # MSI 97 for the financial assistance to carry ot this research. REFERENCES Alamri, A., A., Irina H. and Aris, A. S., (6). Efficient inventory control for imperfect qality items. Eropean Jornal of Operational Research, 54 (), 9-4. Bakker, M., Riezebos, J. and enter, R., (). Review of inventory system with deterioration since. Eropean Jornal of Operation Research, (), Carlson, J. G. (973). Cbic learning crve: Precession tool for labor estimating. Manfactring Engineering and Management, 7 (5), 5. Chng, K. J. and Cárdenas-Barrón, L.E., (3). he simplified soltion procedre for deteriorating items nder stock-dependent demand and two-level trade-credit in the spply chain management. Applied Mathematical Modelling, 37 (7), Ghare, P. M. and Scharender, G. H., (963). A model for exponentially decaying inventory system. Jornal of Indstrial Engineering, 4 (5), Goyal, S. K. and Giri, B. C., (). Recent trends in modeling of deteriorating inventory. Eropean Jornal of Operational Research, 34 (), -6. Hsieh,. P., and Dye, C. Y. (3). A prodction inventory model incorporating the effect of preservation technology investment when demand is flctating with time. Jornal of Comptational and Applied Mathematics, 39 (), 536. Hs, P. H., Wee, H. M. and eng, H. M. (). Preservation technology investment for deteriorating inventory.international Jornal of Prodction Economics 4 (), Hang, C. K. (4). An optimal policy for a single-vendor single-byer integrated prodction-inventory problem with process nreliability consideration. International Jornal of Prodction Economics, 9 (), Jaber, M., Goyal, S., and Imran, M. (8). Economic prodction qantity model for items with imperfect qality sbect to learning effects. International Jornal of Prodction Economics, 5 (), Jaggi, C. K., and Mittal, M. (). Economic order qantity model for deteriorating items with imperfect qality. Revista Investigación Operacional, 3 (), 7 3. Jaggi, C. K., Goel, S. K., and Mittal, M. (). Economic order qantity model for deteriorating items with imperfect qality and permissible delay on payment. International Jornal of Indstrial Engineering Comptations, (), Jordan, R. B. (958). Learning how to se the learning crve. NAA Blletin, 39 (5), Maddah, B., and Jaber, M. (8). Economic order qantity for items with imperfect qality. Revisited. International Jornal of Prodction Economics, (),

9 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. Mossawi-Haidar, L., Salameh, M., and Nasr, W., (4). Effect of deterioration on the instantaneos replenishment model with imperfect qality items. Applied Mathematical Modelling, 38 (4), Salameh, M. K., and Jaber, M. Y. (). Economic prodction qantity model for items with imperfect qality. International Jornal of Prodction Economics, 64 (), Sana, S. S. (). An economic order qantity model for nonconforming qality prodcts. Service Science, 4 (4), Shah Nita. H., Cárdenas-Barrón, L.E., (5). Retailer's decision for ordering and credit policies for deteriorating items when a spplier offers order-linked credit period or cash discont, Applied Mathematics and Comptation, 59 (), Shah, Nita H. and Shah, Y. K., (). Literatre srvey on inventory models for deteriorating items. Economic Annals, 44 (45), -37. Shah, Nita. H., Chadhari, U. B., Jani M.Y., (6a). An integrated prodction-inventory model with preservation technology investment for time-varying deteriorating item nder time and price sensitive demand. International Jornal of Inventory Research, 3(), Shah, Nita. H., Chadhari, U. B., Jani M.Y., (6b). Optimal policies for time-varying deteriorating item with preservation technology nder selling price and trade credit dependent qadratic demand in a spply chain. Int. J. Appl. Compt. Math, DOI.7/s Soni, H. N., & Patel, K. A. (). Optimal pricing and inventory policies for non-instantaneos deteriorating items with permissible delay in payment: fzzy expected vale model. International Jornal of Indstrial Engineering Comptations, 3(), 8 3. Appendix- I ( t) I ( t) I ( t) m t, t t. g g g Appendix- I t S g where, S 3 3 7ab 6ac 7 m 3e ab e ac 3ab ac m 8e ab 6e ac 8ab 6ac 8 m 3 3 7e ab 6e ac ac m ac m 6e acm 6ac m 7e abm e acm 7ab m ac m e acm 3e abm m4e acm 3ab m 4ac m 6e acm 6ac m 8e abm e acm 8ab m 9a e a 3a e a 6e a 6a e ab e ac ab ac m e acm acm e abm 4e acm abm 4ac m e a a Q 6 7 6e 7e e 6e 6 7 e ( m) 483

10 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. Appendix-3 7a 7 7e a 3t 3ab ac 3 m 3e ab e ac 4ac 4ace e ac 6a 6 3 ac m 6ae ace m 4acm 4e acm abe ab abm 5 e abm ac m 5ab 5 m 5e ab a e a t I g t S m t acm t e acm t acmt ace mt ac t 3ab t ac t e ac t 3abe t ace t 3ab m 4acm 4ac m e acm 3abme 4ace m 4ace m ac m 8ac m 8ace m ac t 5abe m ace t 5ab m act abt e act e abt Where, S e 3 Appendix-4 I t cm cm t c t 3bm 3b t 4cm c t 3b 4cm cmt c m t a ct 5bm bt 8cm ct 6 5b cm 4c bm 4cm 7 b c cm cm (6 7 ) c 3bm 3b 4cm c 3b m a 4cm cmt c c 5bm b 8cm c 6 5b cm 4c bm 4cm 7 b c t t g, 484

11 International Jornal of Applied Engineering Research ISSN Volme 3, Nmber 6 (8) pp Research India Pblications. Appendix-5 and t t Id t, t t. e 485