ISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 12, June 2014

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1 ISSN: ISO 900:008 Certifie International Journal of Engineering an Innovative echnology (IJEI) Volume, Issue, June 04 Manufacturing process with isruption uner Quaratic Deman for Deteriorating Inventory Nita H. Shah, Dushyant Kumar G. atel an Digesh Kumar B. Shah Department of Mathematics, Gujarat University, Ahmeaba-80009, Gujarat, Inia Department of Mathematics, Govt. oly. for Girls, Ahmeaba-8005, Gujarat, Inia Department of Mathematics, L. D. College of Engg., Ahmeaba-8005, Gujarat, Inia Abstract Each an every supply chain or logistics structure is subject to isruption. his isruption may be ue to uncertainty or unintene events like instrument break own, unavailability of raw materials, some crisis, natural calamities or strikes. his article first establishes the prouction time when there is no isruption an afterwars when system gets isrupte. After isruption in the prouction, we investigate whether it has been resulte in any amount of shortages or not. o maintain the goowill of the company, an aitional replenishment is suggeste if there are any shortages. For isrupte prouction system, isruption time is calculate. Also in the case of shortages, when to replenish an how much to replenish that is also shown in the article. Moreover, quaratic eman for proucts is analyze in the present article. his type of eman initially increases with time up to some extent an then it starts to ecrease. In aition, the units in the inventory system are subject to eterioration at a constant rate. Numerical example an relate graphical stuies are given to valiate the results. Effects of variations in inventory parameters on prouction time are shown for manufacturers to take avantageous ecisions. Keywors rouction ime, Disruption ime, Quaratic Deman, Deterioration, Shortages, Inventory I. INRODUCION Every prouction system is subject to isruption an the companies are always keen to concentrate on this issue. his isruption may be ue to unplanne events like instrument break own, unavailability of raw materials, unavailability of electricity, some crisis, strikes or natural calamities like earthquakes, floo etc. Lin an Kroll [5] iscusse the prouction system uner an imperfect prouction system subject to ranom breakowns. Lin an Gong [6] erive optimal prouction time uring normal an isrupte prouction perios for constant eman. Chung et al. [] evelope an economic prouction quantity moel for eteriorating items with stochastic machine unavailability time an shortages. Sheu an Chen [9] evelope an moel to etermine economic prouction quantity an level of preventive maintenance for an imperfect prouction process. Khelekar [4] iscusse a isruption prouction moel with exponential eman. Wee [0] efine eterioration as ecay, amage, spoilage, evaporation, obsolescence, pilferage, loss of utility or loss of marginal value of commoities that reuces utility. Foo grains, fruits an vegetables, beverages, meicines, raioactive chemicals, fertilizers an pesticies, volatile liquis like spray an perfumes, passenger seats are examples of eteriorating proucts, whose usefulness ecreases uring the regular storage. Raafat [7], Shah an Shah [8], Goyal an Giri [], Bakker et al. [] formulate a review of literature on eteriorating inventory moel. Most of the above cite articles consiere constant eman, which is not always seen in the market. his paper takes into consieration quaratic eman. his type of eman initially increases with time for some time an then ecreases. his article first fins the optimal prouction time when there is no isruption. Once any isruption comes into picture, prouction gets affecte. In such situation, if there are any shortages then to overcome the shortages new spot replenishment is one by the company to take care of the customers. For isrupte prouction system, isruption time an prouction time are calculate. Also in the case of shortages, time for placing new orer an quantity to be orere are also compute. Inventory moel eals with units which eteriorate at constant rate. Numerical example is stuie an graphical conclusions are exhibite. A. Notations II. NOAIONS AND ASSUMIONS H R t rouction Rate ime horizon Deman rate I t Inventory level at any instant of time t, 0 t Deterioration rate, 0 a Scale eman a 0 b c Qr p p r Linear rate of change of eman with respect to time0b Quaratic rate of change of eman 0c Orering quantity when shortages occur rouction stops at this time when there is no isruption rouction System isrupts at this time rouction stops at this time when there is isruption in the system ime of placing orer to overcome shortages 0

2 ISSN: ISO 900:008 Certifie International Journal of Engineering an Innovative echnology (IJEI) B. Assumptions. he inventory system has single item.. he items in inventory eteriorate at a constant rate.. We consier eman rate to be function of time as R t a bt ct () Volume, Issue, June 04 where a 0is scale eman, 0b enotes linear rate of change of eman with respect to time, 0c enotes quaratic rate of change of eman. 4. rouction rate is constant. p ac H ah b c / 4 a ch b c a b c b c ah ah b c a ac H ah b c a H p III. MAHEMAICAL MODEL A. Moel: I rouction without isruption In this moel, we optimize prouction system run, when prouction is smooth i.e. no isruption. Here, prouction continues till time p. hen prouction stops an inventory level reaches to zero at cycle time H ue to eman an eterioration as shown in fig.. B. Moel: II rouction with isruption In this case, we optimize prouction system run when prouction is isrupte ue to unplanne event. Due to this, prouction rate is change by. rouction begins isrupte at the time. rouction will be increase or ecrease as per positive or negative respectively. Fig. rouction system without isruption I t I t a bt ct ; 0 t p t with bounry conition I 0 0. () t I I t a bt ct ; p t H t with bounry conition I H 0 () hen solution of ifferential equation () an () is given by t a ab ac e I t a bt ct 4 a b ct ac bh ch c Ht e I t a b ch 5 bt ct b ct c Accoring to figure, Ip I p, which gives Fig. rouction system with isruption hen the governing ifferential equations are t I I t a bt ct ; 0 t t with bounry conition I 0 0. (6) I t t t I I t a bt ct ; t (7) p t p I t a bt ct ; t H ; I H 0 I t remains same as per equation (4). Solving Equation (7) using the bounary conition I I, we get 8 04

3 ISSN: ISO 900:008 Certifie International Journal of Engineering an Innovative echnology (IJEI) Volume, Issue, June 04 t IV. NUMERICAL EXAMLE AND SENSIIVIY e I t ANALYSIS Example: aking values H year, 50units, a bt ct t e 00 units, 0. 5year, a 00, b 05., c 0. 5, a b ct ac 5%, 9 we have a ab a c e p , p , Q r Solving equation (8) using bounary conition I H 0, an r we get Following figures show sensitivity of inventory parameters. I t t p ae H p Hb ch b ch c e 0 t p bt ct b ct c e bp c p 4cp b 4c Fig. 4 Vs. p If prouction system satisfies the eman of items after isruption then we fin optimal prouction time p such that at time H, entire stock will be sol-out an inventory level will be zero. I p I p, we can obtain Using Conition prouction time p. Fig. 5 a Vs. p Fig. rouction system after isruption an if prouction system after isruption oes not satisfy the eman of items then there will be shortages in the system. In this situation, we will fin the replenishment time r of placing the orer an respective orer quantity Q r. We can obtain replenishment time r using the conition I r 0. At obtaine r, we can fin orer quantity Q r using Qr I r. Fig. 6 b Vs. p 05

4 ISSN: ISO 900:008 Certifie International Journal of Engineering an Innovative echnology (IJEI) Fig. 7 Fig.8 c Vs. p Vs. p Fig. 9 Vs. r From fig.4 we can observe that Volume, Issue, June 04 p is irectly proportional to. It is evient from fig. 5, fig. 6 an fig. 7 that increase in scale eman a an linear rate of change of eman b ecrease p while increase in quaratic rate of change of eman c increases p. Fig 8 shows the obvious fact that larger eterioration rate increase prouction time when there is no isruption. From fig. 9, we can say that in case of shortages, instantaneous replenishment time gets extene accoring to increase in isruption time. V. CONCLUSIONS he prouction system experiences ifferent types of interruptions. his stuy etermines optimal prouction time initially when system runs smoothly without isruption an afterwars when system has some sort of isruption ue to any reason. It is also taken care that if shortages occur ue to isruption then how much new instantaneous replenishment shoul be mae. Quaratic eman for eteriorating inventory units is stuie. Graphical analysis is provie for a numerical example. Managers can take ecisions from the results e.g. what shoul be the optimal prouction time if eterioration rate changes or isruption starts or eman changes. Future research can be one by incorporating various eman structures like stock epenent eman, price sensitive eman or fuzzy eman. Different preservation technologies can also be implemente to reuce eterioration. REFERENCES [] Bakker, M., Riezebos, J. an eunter, R. H., "Review of inventory systems with eterioration since 00," European Journal of Operational Research., 5-8, 0. [] Chung, C. J., Wiyaana, G. A. an Wee, H. M. "Economic prouction quantity moel for eteriorating inventory with ranom machine unavailability an shortage," International Journal of rouction Research. 49: 88-90, 0. [] Goyal, S. K. an Giri, B. C., "Recent tren in moeling of eteriorating inventory," European Journal of the Operational Research Society. 4, -6, 00. [4] Khelekar, U. K., "A isruption prouction moel with exponential eman," International Journal of Inustrial Engineering Computations,, , 0. [5] Lin, G.C. an Kroll, D. E., "Economic lot sizing for an imperfect prouction system subject to ranom breakowns," Engineering Optimization. 8() 7-9, 006. [6] Lin, G.C an Gong D.C., "On a prouction-inventory moel for eteriorating items subject to an imperfect process, Journal of the Chinese Institute of Inustrial Engineers, 4(4), 9-6, 007. [7] Raafat, F., "Survey of literature on continuously eteriorating inventory moels". Journal of the Operational Research Society. 40, 7-7, 99. [8] Shah, Nita H. an Shah, Y. K., "Literature survey on inventory moels for eteriorating items," Economic Annals. 44, -7, 000. [9] Sheu, S. H. an Chen, J. A., "Optimal lot-sizing problem with imperfect maintenance an imperfect prouction," International Journal of Systems Science. 5(): 69-77, 004. [0] Wee, H. M., "Economic prouction lot size moel for eteriorating items with partial back orering," Computers & Inustrial Engineering. 4, , 99. AUHOR S ROFILE Dr. Nita H. Shah is a professor in the Department of Mathematics, Gujarat University, Ahmeaba, Inia. She receive her h.d. in inventory control management, operations research. Currently, she is engage in research in inventory control an management, supply chain management, forecasting an information technology an information systems, neural networks, sensors an image processing. She has more than 75 papers publishe in international an national journals. She is author of four books. She is serving as a member of the eitorial boar of Investigation Operational, Journal of Social Science an Management, International Journal of Inustrial Engineering an Computations an Mathematics oay. 06

5 ISSN: ISO 900:008 Certifie International Journal of Engineering an Innovative echnology (IJEI) Volume, Issue, June 04 Mr. Dushyantkumar G. atel is a lecturer in mathematics at Government olytechnic for Girls, Ahmeaba. He has publishe research papers in international journals viz. Mexican Journal of Operations Research, ISRN Operations Research, International Journal of Mathematics an Mathematical sciences (IJMMS) etc. His area of interest inclues inventory moels. Mr. Digeshkumar B. Shah is an assistant professor in the Department of Mathematics an Humanities, L.D. College of Engineering, Ahmeaba. He has publishe 5 research papers in international journals viz. International Journal of System Science (IJSS), OSEARCH, Journal of Mathematical Moelling an Algorithms in Operations Research (JMMA), British Journal of Applie Science & echnology, etc. His areas of interest inclue operations research an mostly inventory moels. 07

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