Research Article A Deterministic Inventory Model of Deteriorating Items with Two Rates of Production, Shortages, and Variable Production Cycle

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1 International Scholarly Research Network ISRN Applied Mathematics Volume 011, Article ID , 16 pages doi:10.540/011/ Research Article A Deterministic Inventory Model of Deteriorating Items with Two Rates of Production, Shortages, and Variable Production Cycle Jhuma Bhowmick 1 and G. P. Samanta 1 Department of Mathematics, Maharaja Manindra Chandra College, Kolkata , India Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah , India Correspondence should be addressed to G. P. Samanta, g p samanta@yahoo.co.uk Received 9 March 011; Accepted 19 April 011 Academic Editors: S. Cho and D. Spinello Copyright q 011 J. Bhowmick and G. P. Samanta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A Continuous production control inventory model is developed for a deteriorating item having shortages and variable production cycle. It is assumed that the production rate is changed to another at a time when the inventory level reaches a prefixed level and continued until the inventory level reaches the level S>. The demand rate is assumed to be constant, and the production cycle T is taken as variable. The production is started again at a time when the shortage level reaches a prefixed quantity Q. For this model, the total cost per unit time as a function of, Q, S, andt is derived. The optimal decision rules for, Q, S, andt are computed. The sensitivity of the optimal solution towards changes in the values of different system parameters is also studied. Results are illustrated by numerical examples. 1. Introduction EOQ inventory models have long been attracting considerable amount of research attention. For the last fifteen years, researchers in this area have extended investigation into various models with considerations of item shortage, item deterioration, demand patterns, item order cycles, and their combinations. The control and maintenance of production inventories of deteriorating items with shortages have received much attention of several researchers in the recent years because most physical goods deteriorate over time. In reality, some of the items are either damaged or decayed or vaporized or affected by some other factors and are not in a perfect condition to satisfy the demand. Food items, drugs, pharmaceuticals, and radioactive substances are examples of items in which sufficient deterioration can take place during the normal storage period of the units, and consequently this loss must be taken into

2 ISRN Applied Mathematics account when analyzing the system. Research in this direction began with the work of Whitin 1 who considered fashion goods deteriorating at the end of a prescribed storage period. Ghare and Schrader were the two earliest researchers to consider continuously decaying inventory for a constant demand. An order-level inventory model for items deteriorating at a constant rate was discussed by Shah and Jaiswal 3. Aggarwal 4 developed an order-level inventory model by correcting and modifying the error in Shah and Jaiswal s analysis 3 in calculating the average inventory holding cost. In all these models, the demand rate and the deterioration rate were constants, the replenishment rate was infinite, and no shortage in inventory was allowed. Researchers started to develop inventory systems allowing time variability in one or more than one parameters. Dave and Patel 5 discussed an inventory model for replenishment. This was followed by another model by Dave 6 with variable instantaneous demand, discrete opportunities for replenishment, and shortages. Bahari-Kashani 7 discussed a heuristic model with time-proportional demand. An economic order quantity EOQ model for deteriorating items with shortages and linear trend in demand was studied by Goswami and Chaudhuri 8. It is a common belief that a large pile of goods attracts more customers in the supermarket. This phenomenon is termed as stock-dependent demand rate. Baker and Urban 9 established an economic order quantity model for a power-form inventory-level-dependent demand pattern. Mandal and Phaujdar 10 then developed an economic production quantity model for deteriorating items with constant production rate and linearly stock-dependent demand. Later on, Datta and Pal 11 presented an EOQ model in which the demand rate is dependent on the instantaneous stocks displayed until a given level of inventory S 0 is reached, after which the demand rate becomes constant. Other papers related to this area are Urban 1, Giri et al. 13, Padmanabhan and Vrat 14, Pal et al. 15, Ray and Chaudhuri 16, Urban and Baker 17, Ray et al. 18, Giri and Chaudhuri 19, Datta and Paul 0, Ouyang et al. 1, Teng et al., Roy and Samanta 3, and others. Another class of inventory models has been developed with time-dependent deterioration rate. Covert and Philip 4 used a two-parameter Weibull distribution to represent the distribution of the time to deterioration. This model was further developed by Philip 5 by taking a three-parameter Weibull distribution for the time to deterioration. Mishra 6 analyzed an inventory model with a variable rate of deterioration, finite rate of replenishment, and no shortage, but only a special case of the model was solved under very restrictive assumptions. Deb and Chaudhuri 7 studied a model with a finite rate of production and a time-proportional deterioration rate, allowing backlogging. Goswami and Chaudhuri 8 assumed that the demand rate, production rate, and deterioration rate were all time dependent. A detailed information regarding inventory modeling for deteriorating items was given in the review papers of Nahmias 8 and Raafat 9. An order-level inventory model for deteriorating items without shortages has been developed by Jalan and Chaudhuri 30. Ouyang et al. 31 considered the continuous inventory system with partial backorders. A production inventory model with two rates of production and backorders is analyzed by Perumal and Arivarignan 3, Samanta and Roy 33. In the present paper, we have developed a continuous production control inventory model with variable production cycle for deteriorating items with shortages in which two different rates of production are available, and it is possible that production started at one rate and after some time it may be switched over to another rate. Such a situation is desirable in the sense that by starting at a low rate of production, a large quantum stock of manufactured items at the initial stage is avoided, leading to reduction in the holding cost.

3 ISRN Applied Mathematics 3. Notations and Modeling Assumptions The mathematical model in this paper is developed on the basis of the following notations and assumptions: i a is the constant demand rate; ii p 1 >a and p >p 1 are the constant production rates started at time t 0andat time t t 1 >o, respectively; iii C 1 is the holding cost per unit per unit time; iv C is the shortage cost per unit per unit time; v C 3 is the cost of a deteriorated unit. C 1, C,andC 3 are known constants; vi C 4 and C 5 are the constant unit production costs when the production rates are p 1 and p respectively C 4 >C 5 ; vii Qt is the inventory level at time t 0; viii A is the setup cost; ix replenishment is instantaneous and lead time is zero; x T is the variable duration of production cycle; xi shortages are allowed and backlogged; xii C is the average cost of the system; xiii the distribution of the time to deterioration of an item follows the exponential distribution gt, where e t, for t>0, gt 0, otherwise..1 is called the deterioration rate; a constant fraction 0 < 1 of the on-hand inventory deteriorates per unit time. It is assumed that no repair or replacement of the deteriorated items takes place during a given cycle. In this paper, we have considered a single commodity deterministic continuous production inventory model with a constant demand rate a. The production of the item is started initially at t 0ataratep 1 >a. Once the inventory level reaches, the rate of production is switched over to p >p 1, and the production is stopped when the level of inventory reaches S> and the inventory is depleted at a constant rate a. It is decided to backlog demands up to Q which occur during stock-out time. Thus, the inventory level reaches Q backorder level is Q, the production is started at a faster rate p so as to clear the backlog, and when the inventory level reaches 0 i.e., the backlog is cleared, the next production cycle starts at the lower rare p 1. We denote by 0, t 1, the duration of production at the rate p 1,byt 1, t, the duration of production at the rate p,byt, t 3, the duration when there is no production but only consumption by demand at a rate a, by t 3, t 4, the duration of shortage, and by t 4, T, the duration of time to backlog at the rate p. The cycle then repeats itself after time T. The duration of a production cycle T is taken as variable. This model is represented by Figure 1.

4 t 4 T 4 ISRN Applied Mathematics Inventory S O Time t 1 t t 3 Q Figure 1 3. Model Formulation and Solution Let Qt be the instantaneous state of the inventory level at any time t 0 t T,then the differential equations describing the instantaneous states of Qt in the interval 0, T are given by the following: dqt Qt p 1 a, dt 0 t t 1, dqt Qt p a, dt t 1 t t, dqt Qt a, dt t t t 3, dqt a, dt t 3 t t 4, dqt p a, dt t 4 t T. 3.1 The boundary conditions are Q0 0, Qt 1, Qt S, Qt 3 0, Qt 4 Q, QT The solutions of equations 3.1 are given by Qt 1 p1 a ) 1 e t), 0 t t p a ) [ e t t 1 1 p a )], t 1 t t 3.4 a S a ) e t t, t t t at t 3, t 3 t t Q p a ) t t 4, t 4 t T. 3.7

5 ISRN Applied Mathematics 5 From 3. and 3.3, we have 1 p1 a ) 1 e 1) t t 1 1 ] [1 log p1 a ) Q 1 p1 a ) neglecting higher powers of, 0 < 1 ) p1 a ) Q 1 p 1 a ) ) neglecting higher powers of, 0 < a 3.8b Again, from 3. and 3.4, we have 1 p a ) [ e t 1 t 1 p a )] S 3.9 [ ] 1 ] e t t 1 S 1 [1 p a ) p a ) 1 S p a ) S S p a ) t t 1 1 [ log 1 S Q ] 1 p a ) S S p a ) neglecting higher powers of ), 3.10a S Q S ) 1 p a ) p a ) ) neglecting higher powers of, t p1 a ) Q 1 p 1 a ) S Q S ) 1 p a ) p a ) 3.10b [ by 3.8b ] t 3 From 3.5, we have 1 a S 1 )e a t t 3 0, since Qt a t 3 t 1 log 1 S ) a S 3.1b a S ) neglecting higher powers of a S ) p1 a ) Q 1 p 1 a ) S Q 1 p a ) p a ) S a S a [ by 3.11 ]. 3.13

6 6 ISRN Applied Mathematics Again, from 3.6, we have at 3 t 4 Q, since Qt 4 Q, 3.14 t 4 t 3 Q a From 3.7 and QT0, we have Q p a ) T t 4 0 Q p a ) T t 3 Q ) [by ] , a Q a p a ) T p p1 a ) Q 1 p 1 a ) S p a ) [ by 3.13 ], S ) p a ) S a S a 3.17 D Total number of deteriorated items in 0,T [ p 1 a ) t 1 p a ) t t 1 S ] S at 3 t p 1 a )[ ] p1 a ) Q 1 p 1 a ) p a ) S Q S ) 1 p a ) p a ) a ) S a S [using ] 3.8b, 3.10b, and 3.1b a S ) p1 a ) S ) p a a, S 1 Total Shortage over the period 0,T t4 T [ at t 3 dt Q p a ) t t 4 ] dt t 3 t 4 p Q a p a ) [ ] by using 3.6 and 3.7, I T Total inventory carried over the period 0, T t1 0 Qtdt t t 1 Qtdt t3 t Qtdt. [ by 3.6 and 3.7 ]

7 ISRN Applied Mathematics 7 t1 0 Now, Qtdt 1 p1 a ) t 1 1 e t) dt 0 1 p1 a )[ t 1 1 ) ] e t 1 1 [ by 3.3 ] p 1 a )[ ] t 1 [by p 1 a ) ] 3.8a p 1 a )[ { 1 log 1 } p1 a ) Q 1 p1 a ) 3 Q 3 1 p1 a ) 3 [ by 3.8a and neglecting higher powers of ] ] p 1 a ) p 1 a )[ ] p1 a ) Q 3 1 p1 a ) Q p 1 a ) p1 a ) Q p 1 a ) 3 neglecting higher powers of ) t t 1 Qtdt Q 1 p 1 a ) Q p 1 a ), 3.1 t t 1 [ 1 p a ) { 1 p a )} ] e t t 1 dt [ by 3.4 ] 1 p a ) t t 1 1 { 1 p a )} { } e t t p a ) [ log 1 S Q ] 1 p a ) S S p a ) 3 S S p a ) 3 S [ ] by 3.9 and 3.10a p a )[ S p a ) S S p a ) S S p a ) S 3 p a ) 3. S S p a ) S Q ] p a ) 3 ) neglecting higher powers of S Q1) p a ) S 3 Q1) 3 3 p a ), S

8 8 ISRN Applied Mathematics t3 t3 Qtdt [ a S a ) ] e t t [ ] dt by 3.5 t t a t 3 t 1 S a ) { } e t 3 t 1 a log 1 S ) S [ ] by 3.1a a a [ S a S a S3 3 S 3a 3 ) neglecting higher powers of ] 3.3 S a S3 3a. Therefore, from 3.0, total inventory carried over the cycle 0,T Q 1 p 1 a ) Q p 1 a ) S Q1) p a ) S 3 Q1) 3 3 p a ) S a S3 3a, P Production cost over the period 0,T { C 4 p 1 t 1 C 5 p t t 1 p T t 4 } C ] 4p 1 [1 p1 a ) p 1 a ) C [ 5p p a ) S Q S Q )] 1 p a ) [ by 3.8b, 3.10b, and 3.16 ]. Average cost of the system C,S,T 1 T C 1I T C S 1 C 3 D P A 1 [ A C { } 5p T p a )S C1 C 3 p a p a ) C 5 p p a ) S C 1 ) a p p 3a p a ) S 3 C { 5p p a )Q C p a p a )Q C4 p 1 p1 a ) C 5p }Q p a ) 1 { f C 1 C 3 )} C4 p 1 p1 a ) C 5 p p a ) Q 1 C ] 1df 3k Q [ using 3.18, 3.19, 3.4, and 3.5 ],

9 ISRN Applied Mathematics 9 where d p 1 p a, k p 1 a ) p a ), ) p p 1 f, k 3.7 and from 3.17, Q a p a ) T S a ) p p 1 p p1 a ) ) a p a adf p a )S p p1 a )Q 1. p 3.8 The necessary conditions for C, S, T to be minimum are C 0, C C 0, S T 0, 3.9 that is, )} C 1 df Q 1 {C k C4 p 1 1 C 3 f p1 a ) C 5 p p a ) Q k C 5a C Q ) C 4 p 1 p p 1 df p1 a ) C 5p p a ) 0, C 1 p a p ) S ap { C1 C 3 p a ) C 5 a } S C 5 a p p a ) p ac 5 C Q { a p a ) a p ) S } 0, 3.31 TC 5 a C Q A { C4 p 1 p1 a ) C 5p p a ) } [ p p a ) C 5 Q S C { a Q C1 C 3 a S C ) ] 1 a p 3a p a ) S3 { C4 p 1 C 1 C 3 f p1 a ) C 5 p p a ) } C 5 p a ) )} Q 1 C 1df 3k Q

10 10 ISRN Applied Mathematics Solving these and using 3.8, we get the optimal values, Q, S,andT of, Q, S, and T, respectively, which minimize C, S, T provided they satisfy the following sufficient condition: H The Hessian Matrix of C C C C Q Q 1 1 S T C C C S S S T C C C T T S T 3.33 is positive definite. If the solutions obtained from equations 3.30, 3.31, and3.3 do not satisfy the sufficient condition 3.33, we may conclude that no feasible solution will be optimal for the set of parameter values taken to solve equations 3.30, 3.31,and3.3. Such a situation will imply that the parameter values are inconsistent and there is some error in their estimation. 4. Numerical Examples Example 4.1. Let A 00, C 1 1.5, C, C 3 18, C 4 15, C 5 13, a 3, p 1 4, p 6, and 0.00 in appropriate units. Based on these input data, the computer outputs are 6.681, S , Q , T , C These results satisfy the sufficient condition. Example 4.. The parameters are similar to those in Example 4.1, except that p is changed to 8units. Based on these input data, the computer outputs are 8.13, S , Q , T , C These results satisfy the sufficient condition. Example 4.3. The parameters are similar to those in Example 4.1, except that p is changed to 10 units. Based on these input data, the computer outputs are 8.95, S , Q 11.6, T 0.348, C These results satisfy the sufficient condition. These examples reveal that a higher value of p causes higher values of, S, Q, and C, but lower value of T. 5. Sensitivity Analysis Sensitivity analysis depicts the extent to which the optimal solution of the model is affected by changes or errors in its input parameter values. In this section, we study the sensitivity of

11 ISRN Applied Mathematics 11 the optimal inventory levels, S, backorder lever Q, cycle length T, and average system cost C with respect to the changes in the values of the parameters C 1, C, C 3, C 4, C 5,, A, a, p 1,andp. The results are shown in Tables 1 3. Careful study of Table 1 reveals the following facts. i It is seen that Q 1 is insensitive to changes in the values of parameters C 3 and, moderately sensitive to changes in the values of parameters C 1, C, A, anda, and highly sensitive to changes in the values of parameters C 4, C 5, p 1,andp. ii It is observed that S is insensitive to changes in the values of parameters C, C 3, and and moderately sensitive to changes in the values of parameters C 1, C 4, C 5, A, a, p 1,andp. iii It is seen that Q is insensitive to changes in the values of parameters C 3, and moderately sensitive to changes in the values of parameters C 1, C, C 4, C 5, A, a, p 1 and p. iv Table 1 reveals that T is insensitive to changes in the values of parameters C, C 3, and p, moderately sensitive to changes in the values of parameters C 1, C 5, A, and a, and highly sensitive to changes in the values of parameters C 4 and p 1. v It can be seen that the optimum total cost C is insensitive to changes in the values of parameters C 1, C, C 3,, A, andp and moderately sensitive to changes in the values of parameters C 4, C 5, a,andp 1. Careful study of Table reveals the following facts. i It is seen that Q 1 is insensitive to changes in the values of parameters C 3 and, moderately sensitive to changes in the values of parameters C 1, C, A, a, p 1,andp, and highly sensitive to changes in the values of parameters C 4 and C 5. ii It is observed that S is insensitive to changes in the values of parameters C, C 3,,and p, moderately sensitive to changes in the values of parameters C 1, C 5, A, a, and p 1, and highly sensitive to changes in the values of parameters C 4. iii It is seen that Q is insensitive to changes in the values of parameters C 3,, and p moderately sensitive to changes in the values of parameters C 1, C, C 5, A, a, and p 1, and highly sensitive to changes in the value of parameter C 4. iv Table reveals that T is insensitive to changes in the values of parameters C,C 3,, and p, moderately sensitive to changes in the values of parameters C 1, C 4, C 5, A,anda, and highly sensitive to changes in the value of parameter p 1. v It can be seen that the optimum total cost C is insensitive to changes in the values of parameters C 1, C, C 3,, A, andp and moderately sensitive to changes in the values of parameters C 4, C 5, a, andp 1. Careful study of Table 3 reveals the following facts. i It is seen that Q 1 is insensitive to changes in the values of parameters C 3 and, moderately sensitive to changes in the values of parameters C 1, C, A, a, p 1,andp, and highly sensitive to changes in the values of parameters C 4 and C 5. ii It is observed that S is insensitive to changes in the values of parameters C, C 3, and, p, moderately sensitive to changes in the values of parameters C 1, A, a,and p 1, and highly sensitive to changes in the values of parameters C 4 and C 5.

12 1 ISRN Applied Mathematics Changing parameter Change % Table 1: Sensitivity Analysis of Example 4.1. Change % in Q 1 S Q C C C C C A a p p T C

13 ISRN Applied Mathematics 13 Parameter Change % Table : Sensitivity Analysis of Example 4.. S Change % in C C C C C A a p p Q T C

14 14 ISRN Applied Mathematics Parameter Change % Table 3: Sensitivity Analysis of Example 4.3. S Change % in C C C C C A a p p Q T C

15 ISRN Applied Mathematics 15 iii It is seen that Q is insensitive to changes in the values of parameters C 3 and, moderately sensitive to changes in the values of parameters C 1, C, C 5, A, a, p 1, and p, and highly sensitive to changes in the value of parameter C 4. iv Table 3 reveals that T is insensitive to changes in the values of parameters C, C 3, and, p, moderately sensitive to changes in the values of parameters C 1, C 4, C 5, A,anda, and highly sensitive to changes in the value of parameter p 1. v It can be seen that the optimum total cost C is insensitive to changes in the values of parameters C 1, C, C 3,, A, andp and moderately sensitive to changes in the values of parameters C 4, C 5, a, andp Concluding Remarks In the present paper, we have dealt with a continuous production inventory model for deteriorating items with shortages in which two different rates of production are available, and it is possible that production started at one rate and after some time it may be switched over to another rate. It is assumed that the demand and production rates are constant and the distribution of the time to deterioration of an item follows the exponential distribution. Such a situation is desirable in the sense that by starting at a low rate of production, a large quantum stock of manufactured item, at the initial stage is avoided, leading to reduction in the holding cost. The variation in production rate provides a way resulting consumer satisfaction and earning potential profit. For this model, we have derived the average system cost and the optimal decision rules for, Q, S,andT when the deterioration rate is very small. Results are illustrated by numerical examples. However, success depends on the correctness of the estimation of the input parameters. In reality, however, management is most likely to be uncertain of the true values of these parameters. Moreover, their values may be changed over time due to their complex structures. Therefore, it is more reasonable to assume that these parameters are known only within some given ranges. References 1 T. M. Whitin, Theory of Inventory Management, Princeton University Press, Princeton, NJ, USA, P. M. Ghare and G. P. Schrader, A model for exponentially decaying inventories, Journal of Industrial Engineering, vol. 14, pp , Y. K. Shah and M. C. Jaiswal, An order-level inventory model for a system with constant rate of deterioration, Opsearch, vol. 14, no. 3, pp , S. P. Aggarwal, A note on an order level inventory model for a system with constant rate of deterioration, Opsearch, vol. 15, no. 4, pp , U. Dave and L. K. Patel, Policy inventory model for deteriorating items with time proportional demand, Journal of the Operational Research Society, vol. 3, no., pp , U. Dave, An order level inventory model for deteriorating items with variable instantaneous demand and discrete opportunities for replenishment, Opsearch, vol. 3, no. 4, pp , H. Bahari-Kashani, Replenishment schedule for deteriorating items with time-proportional demand, Journal of the Operational Research Society, vol. 40, pp , A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand, Journal of the Operational Research Society, vol. 4, no. 1, pp , R. C. Baker and T. L. Urban, A deterministic inventory system with an inventory level dependent demand rate, Journal of the Operational Research Society, vol. 39, pp , B. N. Mandal and S. Phaujdar, An inventory model for deteriorating items and stock-dependent consumption rate, Journal of the Operational Research Society, vol. 40, pp , 1989.

16 16 ISRN Applied Mathematics 11 T. K. Datta and A. K. Pal, A note on an inventory model with inventory-level-dependent demand rate, Journal of the Operational Research Society, vol. 41, no. 10, pp , T. L. Urban, An inventory model with an inventory-level-dependent demand rate and related terminal conditions, Journal of the Operational Research Society, vol. 43, no. 7, pp , B. C. Giri, S. Pal, A. Goswami, and K. S. Chaudhuri, An inventory model for deteriorating items with stock-dependent demand rate, European Journal of Operational Research, vol. 95, no. 3, pp , G. Padmanabhan and P. Vrat, EOQ models for perishable items under stock dependent selling rate, European Journal of Operational Research, vol. 86, no., pp. 81 9, S. Pal, A. Goswami, and K. S. Chaudhuri, A deterministic inventory model for deteriorating items with stock-dependent demand rate, International Journal of Production Economics, vol.3,no.3,pp , J. Ray and K. S. Chaudhuri, An EOQ model with stock-dependent demand, shortage, inflation and time discounting, International Journal of Production Economics, vol. 53, no., pp , T. L. Urban and R. C. Baker, Optimal ordering and pricing policies in a single-period environment with multivariate demand and markdowns, European Journal of Operational Research, vol. 103, no. 3, pp , J. Ray, A. Goswami, and K. S. Chaudhuri, On an inventory model with two levels of storage and stock-dependent demand rate, International Journal of Systems Science, vol. 9, no. 3, pp , B. C. Giri and K. S. Chaudhuri, Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost, European Journal of Operational Research, vol. 105, no. 3, pp , T. K. Datta and K. Paul, An inventory system with stock-dependent, price-sensitive demand rate, Production Planning and Control, vol. 1, no. 1, pp. 13 0, L. Y. Ouyang, T. P. Hsieh, C. Y. Dye, and H. C. Chang, An inventory model for deteriorating items with stock-dependent demand under the conditions of inflation and time-value of money, The Engineering Economist, vol. 48, no. 1, pp. 5 68, 003. J. T. Teng, L. Y. Ouyang, and M. C. Cheng, An EOQ model for deteriorating items with power-form stock-dependent demand, International Journal of Information and Management Sciences, vol.16,no.1, pp. 1 16, A. Roy and G. P. Samanta, An EOQ model for deteriorating items with stock-dependent time varying demand, Tamsui Oxford Journal of Mathematical Sciences, vol. 0, no., pp , R. P. Covert and G. C. Philip, An EOQ model for items with Weibull distribution deterioration, American Institute of Industrial Engineers Transactions, vol. 5, no. 4, pp , G. C. Philip, A generalized EOQ model for items with Weibull distribution deterioration, American Institute of Industrial Engineers Transactions, vol. 6, no., pp , R. B. Misra, Optimum production lot-size model for a system with deteriorating inventory, International Journal of Production Research, vol. 13, no. 5, pp , M. Deb and K. S. Chaudhuri, An EOQ model for items with finite rate of production and variable rate of deterioration, Opsearch, vol. 3, pp , S. Nahmias, Perishable inventory theory : a review, Operations Research, vol. 30, no. 4, pp , F. Raafat, Survey of literature on continuously deteriorating inventory model, Journal of the Operational Research Society, vol. 4, no. 1, pp. 7 37, A. K. Jalan and K. S. Chaudhuri, Structural properties of an inventory system with deterioration and trended demand, International Journal of Systems Science, vol. 30, no. 6, pp , L. Y. Ouyang, C. K. Chen, and H. C. Chang, Lead time and ordering cost reductions in continuous review inventory systems with partial backorders, Journal of the Operational Research Society, vol. 50, no. 1, pp , V. Perumal and G. Arivarignan, A production inventory model with two rates of production and backorders, International Journal of Management and Systems, vol. 18, pp , G. P. Samanta and A. Roy, A deterministic inventory model of deteriorating items with two rates of production and shortages, Tamsui Oxford Journal of Mathematical Sciences, vol. 0, no., pp , 004.

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