AN EOQ MODEL FOR TWO-WAREHOUSE WITH DETERIORATING ITEMS, PERIODIC TIME DEPENDENT DEMAND AND SHORTAGES

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1 IJMS, Vol., No. 3-4, (July-December 0), pp Serials Publications ISSN: X AN EOQ MODEL FOR TWO-WAREHOUSE WITH DETERIORATING ITEMS, PERIODIC TIME DEPENDENT DEMAND AND SHORTAGES Karabi Dutta Choudhury, Mantu Das & Sumit Saha Abstract: A deterministic inventory model with periodic time dependent demand for deteriorating items with two warehouses is developed when the replenishment rate is finite, demand is at a uniform rate and shortages are allowed. It is assumed that the rates of deterioration of items in the two warehouses are different and time-proportional. To make the model more realistic backlogged are allowed. To illustrate the variation in the optimal inventory level and the optimal cost a numerical example with sensitive analysis is done. Keywords: Inventory, Production, Periodic time dependent demand, Deterioration, Shortages, Two-Warehouses.. INTRODUCTION The inventory model deals with different parameters viz. demand, replenishment time, shortages, warehouse, EOQ etc. Demand is one of the important parameter. A good number of inventory models are developed with different types of demand such as stock-dependent, price-dependent, time-dependent demand. Some works in this field are worth mentioning viz. Baker and Urban [, ], Datta and Pal [7], Gayen and Pal [5], Paul et al., [4], Dutta Choudhury and Dutta [9, 0], Datta and Paul [3], Chen and Min [], Gallego and Ryzin [4], Bhunia and Maiti [3], Bose et al., [5], Khanna et al., [0], Maihami et al., []. But in the present scenario of globalization, periodic demand is an important factor. Again the classical inventory models are mainly developed with the single storage facility. But, in many practical situations, there exists many factors like temporary price discounts making retailers to buy a capacity of goods exceeding their own warehouse (OW). In this case, retailers will either rent other warehouses or rebuild a new warehouse. However, from economical point of views, they usually choose to rent other warehouses. Hence, an additional storage space known as rented warehouse (RW) is often required due to limited capacity of showroom facility. Besides, the ideas of time-varying demand in the field of inventory management, when a purchase (or production) of large amount of units of an item that can t be stored in its existing storage viz. OW at the market place due to limited st International Conference on Mathematics and Mathematical Sciences (ICMMS), 7 July 0.

2 380 Karabi Dutta Choudhury, Mantu Das & Sumit Saha capacity is made, then excess units are stocked in a rented warehouse with sophisticated facilities (RW). The RW is of infinite capacity. The actual service to the customer is done at OW only. If the OW is empty it may creates negative impression amongst the customers. So, OW should remain full with goods as far as possible. For this reason, the stock at OW is replenished from the stock of RW. Further, units at RW are exhausted fully by satisfying the required demand first. A two-warehouse model was presented by Hadley [8], where it is assumed the holding cost in RW is greater than the OW, Hadley presented a basic two-warehouse model, in which the cost of transporting a unit from rented warehouse (RW) to own warehouse (OW) was not considered. Sarma [8, 9] extended Hadley s model by introducing the transportation cost with and without shortage. Goswami and Chaudhuri [, 7] showed the generalization of Sarma s model by considering a linear demand. Bhunia and Maiti [3] formulated and solved a deterministic inventory model with inventory level-dependent consumption rate for two warehouses. Later, Bhunia and Maiti [4] also extended their model by taking into consideration the inventory deterioration under the assumption of continuous release. On the other hand, Lee and Ma [] proposed an optimal policy for deteriorating items with two-warehouses and time-dependent demands. In addition, for certain types of commodities, such as medicine, volatile liquids, blood bank, foodstuffs deterioration is usually observed during their normal storage period. By assuming constant demand rate, Sarma [, 7] developed a deterministic inventory model for a single deteriorating item with shortages and two levels of storage. Pakkala and Achary [3] extended the two-warehouse inventory model for deteriorating items with finite replenishment rate and shortages. Karmakar and Dutta Choudhury [9] reviewed the inventory models for deteriorating items with shortages. In this paper, for the first time, periodic time dependent demand has been introduced in a two-warehouse inventory control system with different deterioration rate which is also time proportional. Here, the two-warehouse inventory models with backlogged shortages have been formulated. The retailer possesses two warehouses OW (of finite capacity) and RW (infinite capacity). OW is situated in the main marketing place and RW at a distance from OW. Firstly the OW is filled up with the item and then excess units are kept in RW. Numerical examples with sensitivity analyses on the inventory model due to the change in some parameters like rate of deterioration and rate of lost sale etc. calculated. We assume that the deterioration rate of the items stored might be different in the two warehouses due to the difference in the environmental conditions or preserving conditions. However, the model is applicable even when the deterioration rate remains the same in both the warehouses. Assuming the production rate is finite and uniform demand and shortages are allowed, we also make certain observations on the single storage version of this model discussed by Dutta Choudhury et al., [8].

3 An EOQ Model for Two-Warehouse with Deteriorating Items, Periodic Time Dependent In this paper, we discussed an inventory model with the following assumptions: (i) The demand is a periodic function of time, (ii) The production rate is finite and varies with the demand, (iii) The unit cost of production is inversely proportional with the demand rate, (iv) The deterioration rate in OW and RW are different and time proportional, (v) Shortages in inventory are allowed, and (vi) Holding cost per unit per unit time in two warehouses are different. Numerical examples have been included to illustrate the derived results, and also the results are compared with the single storage model. Lastly, sensitivity analysis of the optimal solution with respect to the parameters of the system is carried out to identify the most sensitive parameters.. NOTATIONS AND ASSUMPTIONS The two-warehouse inventory models proposed in this research article are based on the following notations and assumptions: (i) (ii) (iii) The demand rate D(t) = a + bt + c sint, where a 0, b 0 and c 0. Also assumed that b > a and c > a to ensure that the demand rate is non-negative. The production rate is given by k (t) = D (t), where >, a constant. A variable fraction (t) = t and (t) = t in OW and RW respectively are the on-hand inventory deterioration per unit time, 0 < < <<. (iv) The unit production cost is inversely proportional to the demand rate as = D, where > 0, > 0. (v) (vi) (vii) (viii) (ix) (x) (xi) C o3 and C r3 are the holding cost per unit per unit of time in OW and RW respectively with C o3 < C r3. C is the shortage cost per unit per unit of time. D o and D r are the deterioration cost per unit per unit of time in OW and RW respectively. Capacity of the OW is limited i.e. W-units. When the production stopped, the inventory level reaches the level S in the RW. Replenishment rate is finite. The lead time is zero.

4 38 Karabi Dutta Choudhury, Mantu Das & Sumit Saha (xii) (xiii) (xiv) Shortages are fully backlogged. The capacity of the RW is unlimited. Inventory items are stored in RW only after OW is fully occupied. 3. THE MATHEMATICAL MODEL In the development of this model, we assume that during the production stage demand are met directly and the inventory is stocked first in the OW till it filled to the capacity W. Then the inventory is carried in the RW till a level S is reached, at which stage the production is stopped. It is also assumed that the carrying cost to transfer the goods from RW to the OW is negligible. After stopping production the demands are met from the RW till it is vacant, and then the inventory in the OW is used to meet the demand, it is further assumed that the items deteriorated in inventory are not replaced by good ones. The amount of stock in OW and RW are both zero at time t = 0 at which all backlogged are cleared. Production begins at time t = 0 and continues up to time t = t and at time t = t the stock level is W in the OW, after which the demands are met from the production and excess of the items are transferred to the RW till time t = t. At time t = t the inventory level in the RW reaches S and the production is stopped at that point. After meeting the demand in [0, t ] accumulated inventory is used in [t, t 4 ]. During [t, t 3 ] in RW the inventory level gradually decreases mainly to meet the demands and partly because of deterioration and it is zero at time t = t 3. In the OW, the inventory decreases during [t, t 3 ] only due to deterioration, and during [t 3, t 4 ] the decrease in inventory is both due to demand and deterioration. Now the stock level reduces at zero at time t = t 4. Then the shortage begins to start and accumulate till t = t 5. Again the production starts at time t = t 5 and clears the backlogged quantities during [t 4, t 5 ] in the interval [t 5, t ], which is shown in the Figure-. The inventory level gradually falls to zero at time t = t. This cycles repeats after time t. Consider the inventory level Q o (t) and Q r (t) in 0 t t for OW and RW respectively. Therefore, the instantaneous state of Q o (t) and Q r (t) are described by the following differential equations for both OW and RW respectively. Figure

5 An EOQ Model for Two-Warehouse with Deteriorating Items, Periodic Time Dependent dqo ( t) o( t) Qo ( t) ( ) D( t) 0 t t Q (0) 0 and Q ( t ) W o r o dqr ( t) r ( t) Qr ( t) ( ) D( t) t t t Q ( t ) 0 and Q ( t ) S r r 3 dqr ( t) r ( t) Qr ( t) D( t) t t t Q ( t ) S and Q ( t ) 0 r 3 o 4 3 dqo ( t) o( t) Qo( t) D( t) t t t Q ( t ) W and Q ( t ) 0 o dqo ( t) D( t) t t t Q ( t ) 0 o 4 5 dqo ( t) ( ) D( t) t t t Q ( t ) 0 o 5 Integrating all the equations from () to () and using the initial conditions we will get the respective Q o (t) and Q r (t) for OW and RW respectively with the given boundary conditions. After solving the above equations we have the following results respectively, () () (3) (4) (5) () 3 bt ct at 0( ) ( ) ( ) Q t c at 3 4 bt c cos t c ( t sin t cos t) 0 t t 8 Qr ( t) ( ) ( t t ) ( a c sin t) ( t t ) ( b c cos t ) a ( t t ) c ( ) (cos t cos t) at t ( t t) b ( t t t t t t t ( / ) ( / )

6 384 Karabi Dutta Choudhury, Mantu Das & Sumit Saha b a Qr ( t) a( t3 t) ( t3 t ) (t 3 t t3 t3 ) b ( t3 t t t3 ) 8 c ( )(cos t cos t ) cos t ( t t ) t sin t tsin t) t t t b a Qo ( t) a( t t) ( t t ) (t 3 t t t ) b ( t t ) c ( )(cos t cos t ) cos t ( t t ) t sin t tsin t) t t t ` (3 / ) (4 / ) b Q ( t) a ( t4 t) ( t4 t ) c(cos t4 cos t) t4 t t5 (5 / ) b Q( t) a ( t t ) ( t t ) c(cos t cos t ) t5 t t ( / ) Total holding cost in the time interval [0, t 4 ] is given by t t3 t4 t t3 ( ) ( ) ( ) 0 ( ) ( ) t t t t C C Q t Q t Q t C Q t Q t hol o o 3 Solving, we have C c C t C t t t t t t cos t hol o r 3 Cr t3 ( t t3) cos t3 C0( t3 t4) cos t4 ( ) { C0( ) 3 C ( t t t )}sin t C ( ) sin t { C ( ( t t t ) r r r 3 3 ( )) C ( )} sin t C ( ( t t t ) ( )) sint C t (cos t cos t ) C t (cos t cos t ). (7) r o t 3 But, W D( t) D( t) 0 0 t bt W ( ) at c( cos t) Total shortage cost in the time interval [t 4, t ] is given by sho t5 t ( ) ( ) t 4 t5 C C Q t Q t (8)

7 An EOQ Model for Two-Warehouse with Deteriorating Items, Periodic Time Dependent a Csho C { t4 t4t5 t5 ( ) ( t t5t ) b {( ) (t 3 t5t ) (t4 3 t4t5 t5)} c{( t t )cos t ( ) {( t t )cos t (sin t sin t )}}. (9) Total production cost in the time interval [0, t ] is given by The unit production cost in [t, t + t] is k and is D( t) k D( t) D( t) ( bt c sin t) a a The production cost in [0, t ] is t t t P k k k 0 t t5 b P a a{ t t5 t} ( ) ( t t5 t ) c (cost cost5 cos t) (0) Total amount of deterioration occurred in OW and RW is t t4 t t3 ( ) D( t) ( ) 0 D t ( ) ( ) t D t D t t t And the deterioration cost is 0 3 t t4 t t3 ( ) ( ) ( ) 0 R ( ) ( ) t t t DC D D t D t D D t D t 3 b DC D0 a{( ) t t3 t4} {( ) t t3 t4} c{( )cos t cost3 cos t4} b RR a{( ) t tt3} {( ) t tt3} c{( )cos t cost cos t3}. () So, the cost per unit time is tcq { Chol Csho P DC } () t

8 38 Karabi Dutta Choudhury, Mantu Das & Sumit Saha By solving the equations (7)-() and putting the respective values we can find the value of tcq. t ( b c)( ) C tcq a { a( ) C a ( )( b c)} t t t 5 5 t ( b c) C ( ) ac t t ( ) a( C D D ) t {( a b c) C C C ac ( D D )( b c)} t ( b c) t a { C C ) t 3 5 o3 o r o3 3 t o 3 r r3 o r r3 o3 Cr3 Co3 5 Cr 3( b c) acr3 3 3 b t acr 3tt t t ( t t tt ) 5 40 bc cc ( b c) C C C t t t t ( b c ) t r3 r3 4 r3 3 o t 3 C 3 Cr 3 Cr 3 3 Co3 C t 5 t t 3 t t 3 t 3 t o3 3 C 4 t3t4 t4 t5 { cc a( D D )} t { a( C C ) ( b c) ( D D )} t t r3 o r 3 o3 r3 o r 3 C C cc 3 3 o3 r3 r3 3 4 ( b c) t3 a{ Cr3 Co3 } t3 C C { ( ) ( )} 5 40 r3 o3 5 b t3 adot 4 a C Co3 Do b c t4 ac bc ( t t t ) t {( b c)( ) a ac } t 5 o3 4 3 o ac bc cc ac t t ( t t t t ) ac t t t t 8 r3 3 r3 r3 4 r o aco3 3 bco3 cco3 4 t3t4 t3t4 act4t5 8 ( ad cc a ) t { ac ( b c) ( D a ( )} t t r r3 r3 r C ( b c) cc a b t ( c ) t t r3 r r3

9 An EOQ Model for Two-Warehouse with Deteriorating Items, Periodic Time Dependent We have to find a minimum value of tcq and corresponding values of t, t, t 3, t 4, t 5, t. The necessary conditions of tcq to be minimum are tcq 0, i,, 3, 4, 5,. (3) t t The optimal solutions t (= t * ), t (= t * ), t 3 (= t 3 * ), t 4 (= t 4 * ), t 5 (= t 5 * ) and t (= t * ) can be obtained from the equation (), provided the sufficient conditions H i > 0 (i =,, 3, 4, 5, ) holds for t = t *, t = t *, t 3 = t 3 *, t 4 = t 4 *, t 5 = t 5 * and t = t * where H i is the Hessian determinant of order i given by c c c i c c ci Hi ; c c c i i ii tcq Where Cij, ( i, j,, 3, 4, 5, ). tt tj Corresponding minimum average cost during [0, t ] is tcq * = tcq (t *, t *, t * 3, t * 4, t * 5, t * ). Solving the equation number (3), we have tcq 0 t tcq 0 t tcq 0 t 3 tcq 0 t 4 tcq 0 t 5 tcq 0 t 4. SOLUTION PROCEDURE (4.) (4.) (4.3) (4.4) (4.5) (4.) The total average cost tcq (t, t, t 3, t 4, t 5, t ) given above, is a function of the six variables t, t, t 3, t 4, t 5, t. The necessary condition for tcq to be minimum is given by the equations

10 388 Karabi Dutta Choudhury, Mantu Das & Sumit Saha (4.) to (4.) respectively. These equations are highly non-linear in nature, which can be easily solved by Newton Method of higher variables when the values of the parameters are prescribed. The optimal solutions of the equations (4.) to (4.) are t *, t *, t 3 *, t 4 *, t 5 * and t * provided these values of t i * (i =,, 3, 4, 5, ) satisfy the conditions H i > 0 (i =,, 3, 4, 5, ). Substituting these values in equation number (), the optimal average cost tcq (t *, t *, t 3 *, t 4 *, t 5 *, t * ) can be obtained. 4. Numerical Example Consider a = ; b = ; c = 3; C o3 =.; C r3 =.; C = ; D o =.4; D r =.7; = 0.0; = 0.; =.5; =.5; =.8. After solving we get t * = 0.09; t * = 0.; t 3 * = 0.; t 4 * = 0.08; t 5 * =.9; t * = 5.00, and tcq * = 94.89,units respectively. 4. Sensitivity Analysis In this section, we study the sensitivity analysis to examine the effect of changes in the input parameters on the optimal results obtained in the example. We first find the optimal values of variables t, t, t 3, t 4, t 5, t and tcq by changing (increasing or decreasing) one parameters by 5% and 50% and all other parameters remains unchanged. Then we calculate the percentage change of tcq with respect to the other values. % Change in % Change % Change % Change % Change % Change % Change % Change * * * * * * Parameters Parameters t t t 3 t 4 t 5 t in tcq * a b c C D (Table Contd...)

11 An EOQ Model for Two-Warehouse with Deteriorating Items, Periodic Time Dependent % Change in % Change % Change % Change % Change % Change % Change % Change * * * * * * Parameters Parameters t t t 3 t 4 t 5 t in tcq * D r C o C r The following are the conclusions made from the table given below: () We see that the percentage change in the t i *, (i = to ) and tcq * cost is same for the change in and. () tcq * is slightly sensitive to the +ve changes in the values of b,, D r and ve changes in the values of C o3,, c, D o. Tcq * is highly sensitive to the changes in the values of a, C, Cr 3,. (3) t *, t *, t 3 * are less sensitive to the change in the respective others values.

12 390 Karabi Dutta Choudhury, Mantu Das & Sumit Saha (4) t 4 *, is highly sensitive to the change in the values D o, C o3, and less sensitive to the rest of the values. (5) t 5 * is less sensitive to the change in the values c, D o and highly sensitive to the rest of the other values. () t * is less sensitive to the change in the values D o and for the rest of the variables the same is highly sensitive. (7) In the table above indicate that the values of the respective parameters are in negative. 5. DISCUSSION AND CONCLUSION In this paper, some realistic features are considered. These features are likely to be associated with an inventory of consumer goods. The assumption of a periodic time-dependent demand rate and production rate is very realistic in the market. The deterioration rate is also assumed to be time-dependent and unit production cost is inversely proportional with the demand rate. The occurrence of shortages in inventory is a very natural phenomenon. Shortages are allowed and backlogged. If we equate some variables related to the RW to zero, we get the same result in the article of Choudhury et al., (0) with the demand rate D(t) = a + bt + c sint, which was considered with single warehouse only. But two ware house problem is the realistic one which cannot be ignored. The realistic features of demand pattern may be applicable in industry and in other sector of Economy of a Country. The sensitivity analysis of the solution to changes in the values of different parameters has been discussed. REFERENCES [] Baker R. C., and Urban T. L., (988b), A Deterministic Inventory System with an Inventory Level-Dependent Demand Rate, Journal of Operational Research Society, 39: [] Baker R. C., and Urban T. L., (988a), Single Period Inventory Dependent Demand Model, Omega, : [3] Bhunia A. K., and Maiti M., (994), A Two Warehouse Inventory Model for a Linear Trend in Demand, Opsearch, 3: [4] Bhunia A. K., and Maiti M., (998), A Two Warehouse Inventory Model for Deteriorating Items with a Linear Trend Demand and Shortages, Journal of Operational Research Society, 49: [5] Bose S., Goswami A., and Chaudhuri K. S., (995), An EOQ Model for Deteriorating Items with Linear Time-Dependent Demand Rate and Shortages Under Unflation and Time Discounting, Journal of Operational Research Society, 4: [] Chen C. K., and Min K. J., (994), A Multi-Product EOQ Model with Pricing Consideration T.C.E. Cheng s Model Revisited, Computers Ind Engng., :

13 An EOQ Model for Two-Warehouse with Deteriorating Items, Periodic Time Dependent [7] Datta T. K., and Pal A. K., (990), Deterministic Inventory System for Deteriorating Items with Inventory-Level Dependent Demand Rate and Shortages, Opsearch, 7: 3 4. [8] Dutta Choudhury K., Das M., and Saha S., (0), An EQO Model for Deteriorating Items for Shortages with Periodic Time Dependent Demand, Deterioration and Unit Production Cost, Assam University Journal of Science and Technology, 9: [9] Dutta Choudhury K., and Dutta T. K., (00), An Inventory Model with Stock-Dependent Demand Rate and Dual Storage Facility, Assam University Journal of Science and Technology, 5: [0] Dutta Choudhury K., and Dutta T. K., (009), Ordering and Pricing Policy for Multi-Quality Product with Dual Storage Facility, Assam University Journal of Science and Technology, 4: 7. [] Dutta Choudhury K., Saha S., and Das M., A Note on Inventory Model, In the RAMSA-0 proceeding (In Press). [] Dutta Choudhury K., Saha S., and Das M., (0), An Inventory Model with Lot Size Dependent Carrying / Holding Cost, Assam University Journal of Science and Technology, 7: [3] Dutta T. K., and Paul K., (00), An Inventory System with Stock-Dependent, Price-Sensitive Demand Rate, Production Planning and Control, : 3 0. [4] Gallego G., and Ryzin G. V., (994), Optimal Dynamic Pricing of Inventories with Stochastic Demand Over Finite Horizon, Management Science, 40: [5] Gayen M., and Pal A. K., (009), A Two Ware House Inventory Model for Deteriorating Items with Stock Dependent Demand Rate and Holding Cost, Springer-Verlag. [] Goswami A., and Chaudhuri K. S., (99), An EOQ Model for Deteriorating Items with Shortage and Time-Varying Demand and Costs, Journal of the Operational Research Society, 4: [7] Goswami A., and Chaudhuri K. S., (99), Variation of Order-Level Inventory Models for Deteriorating Items, International Journal of Production Economics, 7: 7. [8] Hadley G., (94), A Comparison of Order Quantities Computed Using the Average Annual Cost and the Discounted Cost, Management Science, 0: [9] Karmakar B., and Dutta Choudhury K., (00), A Review on Inventory Model for Deteriorating Items with Shortage, Assam University Journal of Science and Technology, : [0] Khanna S., Ghosh S. K., and Chaudhuri K. S., (0), An EOQ Model for a Deteriorating Item with Time Dependent Quadratic Demand Under Permissible Delay in Payment, Applied Mathematics & Computation, 8: [] Lee C. C., and Ma C. Y., (000), Optimal Inventory Policy for Deteriorating Items with Two-Warehouse and Time Dependent Demands, Production Planning & Control, (7), [] Maihami R., and Kamalabadi I. N., (0), Joint Pricing and Inventory Control for Non-Instantaneous Deteriorating Items with Partial Backlogging and Time and Price Dependent Demand, International Journal of Production Economics, 3:.

14 39 Karabi Dutta Choudhury, Mantu Das & Sumit Saha [3] Pakkala T. P. M., and Achary K. K., (99), A Deterministic Inventory Model for Deteriorating Items with Two Warehouses and Finite Replenishment Rate, European Journal of Operational Research, 57: 7 7. [4] Paul K., Datta T. K., Chaudhuri K. S., and Pal A. K., (99), An Inventory Model with Two-Components Demand Rate and Shortages, Journal of the Operational Research Society, 47: [5] Saha S., Dutta Choudhury K., and Das. M., An Inventory Model for Non-Deteriorating Items with Non-Linear Time Dependent Demand Rate, In the RAMSA-0 Proceeding (In Press). [] Sarma K. V. S., (987), A Deterministic Order Level Inventory Model for Deteriorating Items with Two Storage Facilities, European Journal of Operational Research, 9: [7] Sarma K. V. S., (983), A Deterministic Inventory Model with Two Levels of Storage and an Optimum Release Rule, Opsearch, 0: [8] Sarma K. V. S., (990), A Note on the EOQ Model with Two Levels of Storage, Opsearch, 7: 9 7. [9] Sarma K. V. S., and Sastry M. P., (988), Optimum Inventory for Ssystems with Two Levels of Storage, Industrial Engg. Journal., 8: 9. Karabi Dutta Choudhury, Mantu Das & Sumit Saha Department of Mathematics, Assam University, Silchar, India karabidc@gmail.com