An Inventory Model with Time-Dependent Demand and Limited Storage Facility under Inflation

The main objective of this paper is to develop a two-warehouse inventory model with partial backordering and Weibull distribution deterioration. We consider inflation and apply the discounted cash flow in problem analysis. The discounted cash flow DCF and optimization framework are presented to derive the optimal replenishment policy that minimizes the total present value cost per unit time. When only rented or own warehouse model is considered, the present value of the total relevant cost is higher than the case when two-warehouse is considered. The results have been validated with the help of a numerical example. Sensitivity analysis with respect to various parameters is also performed. From the sensitivity analysis, we show that the total cost of the system is influenced by the deterioration rate, the inflation rate, and the backordering ratio.


Introduction
Deterioration is the change, damage, decay, spoilage, evaporation, obsolescence, pilferage, and loss of utility or loss of marginal value of a commodity that results in decreasing usefulness from the original one.Most products such as medicine, blood, fish, alcohol, gasoline, vegetables, and radioactive chemicals have finite shelf life and start to deteriorate once they are replenished.In addition, for certain types of commodities, deterioration is usually observed during their normal storage period.In most of the inventory models it is unrealistically assumed that during stockout either all demand is backlogged or all is lost.In reality often some customers are willing to wait until replenishment, especially if the wait will be short, while others are more impatient and go elsewhere.The backlogging rate depends Singh et al.21 presented a deterministic two-warehouse inventory model for deteriorating items with partial backlogging.Yang 22 extended Yang 16 model to incorporate threeparameter Weibull deterioration distribution with constant demand.
It has been observed that most researchers on inventory models do not consider time-varying rate of deterioration, inflation, and partial backordering simultaneously.Since this phenomenon is not uncommon in real life, the researchers should also incorporate them in their problem development.In this paper, the researcher has considered a twowarehouse inventory system for deteriorating items.Here, shortages are allowed and partially backlogged.The holding cost at RW is higher as compared to OW.The rate of deterioration in both warehouses is different and follows a two-parameter Weibull distribution.Further, this study takes inflation and applies the discounted cash flow DCF approach for problem analysis.In this study, the demand rate is exponentially increasing with time and shortages are partially backlogged with exponential backlogging rate.The discounted cash flow and optimization framework are presented to derive the optimal replenishment policy that minimizes the total present value cost.A numerical example and sensitivity analysis are presented to illustrate all the models.When only rented or own warehouse is considered, the present value of the total relevant cost is higher than the case when two-warehouse model is considered.From the sensitivity analysis, the total cost of the system is influenced by the deterioration rate, the inflation rate, and the backordering ratio.

Assumptions and Notations
In developing the mathematical models of the inventory system for this study, the following common assumptions were used.
1 Demand rate is known and is equal to ae bt , where a > 1 and 0 < b < 1 a > b are constants.
2 Shortages are allowed and backlogging rate is e −λt , when inventory is in shortage.
3 Deterioration of the item follows a two-parameter Weibull distribution.
4 Deterioration occurs as soon as items are received into inventory.
5 There is no replacement or repair of deteriorating items during the period under consideration.
6 Product transactions are followed by instantaneous cash flow.
7 The holding costs in RW are higher than those in OW.
8 The OW has a fixed capacity of W units and the RW has unlimited capacity.

2.1
The instantaneous rate of deterioration Z t xyt y−1 is used in the following model development.
When y > 1, deteriorating rate increases with time.When y < 1, deteriorating rate decreases with time.And when y 1, deteriorating rate is constant.The two-parameter Weibull distribution reduces to the exponential distribution.

Formulation and Solution of the Model
The OW inventory system in Figure 1 can be divided into three phases depicted by T 1 to T 3 .For each replenishment, a portion of the replenished quantity is used to backorder shortage, while the rest enters the system.W units of items are stored in the OW and the rest is dispatched to the RW.The RW is therefore utilized only after OW is full, but stocks in RW are dispatched first.Stock in the RW depletes due to demand and deterioration until it reaches zero.During that time, the inventory in OW decreases due to deterioration only.The stock in OW depletes due to the combined effect of demand and deterioration during time T 2 .During the time T 3 , both warehouses are empty, and part of the shortage is backordered in the next replenishment.
The OW inventory system can be represented by the following differential equations

3.1
The first-order differential equations can be solved using the boundary conditions, I 01 0 We −αT β 1 , and I 03 0 0, one has 3.3 The RW inventory system can be represented by the following differential equation The first-order differential equation can be solved using the boundary condition, I r 0 I r , one has Figure 1: Graphical representation of the OW inventory system.where 1 From Figure 1, replenishment is made at t 1 0, the present worth ordering cost is OR A.

3.8
Advances in Operations Research 7 2 Inventory occurs during T 1 and T 2 time periods.The present worth inventory cost in OW is 3.9 3 Shortages occur during T 3 time period.The present worth shortage cost in OW is SC s 4 Lost sales occur during T 3 time period.The present worth lost sale cost in OW is 5 Replenishment occurs at t 0 and t T 1 T 2 T 3 T .The item cost includes loss due to deterioration as well as the cost of the item sold.The present worth item cost in OW is 6 From Figure 2, inventory occurs during T 1 time periods.The present worth inventory cost in RW is

3.13
7 Replenishment occurs at t 0. The item cost therefore includes loses due to deterioration as well as the cost of the item sold.The present worth cost in RW is The present value of the total relevant cost during the cycle is

3.16
The optimization problem can be formulated as: 3.17 In the above expression, TUC is the objective function which we have to minimize, which is the function of T 1 , T 2 , and T 3 .To minimize the objective function, the solution methodology is presented in Section 6.

When the System Has Only Rented Warehouse (RW)
Figure 3 shows the graphical representation of the RW inventory system having only rented warehouse.The RW inventory level function during T 1 and T 2 time periods is similar to 3.6 and 3.4 .The ordering cost and holding cost of the system are similar to 3.8 and 3.13 .
Shortages occur during T 2 time period.The present worth shortage cost in RW is SC s Lost sales occur during T 2 time period.The present worth lost sales cost in RW is Replenishment occurs at t 0 and t T 1 T 2 T .The item cost therefore includes loss due to deterioration as well as the cost of the item sold.The present worth item cost in RW is

10 Advances in Operations Research
The order quantity in RW per order is Noting that T T 1 T 2 , the total present value of the total relevant cost per unit time during the cycle is the sum of ordering cost, holding cost, shortage cost, lost sale cost, and item cost.

4.5
In the above expression, TUC r is the objective function which we have to minimize, which is the function of T 1 and T 2 .To minimize the objective function, the solution methodology is presented in Section 6.

When the System Has Only Own Warehouse (OW)
Figure 4 shows the graphical representation of the OW inventory system having only own warehouse.The OW inventory level function during T 1 and T 2 time periods is similar to 3.6 and 3.4 .The ordering cost, holding cost, shortage cost, and lost sale cost of the system are similar to 3.8 , 3.9 , 3.10 , and 3.11 .The order quantity in OW per order is

5.1
Replenishment occurs at t 0 and t T 1 T 2 .The item cost therefore includes loss due to deterioration as well as the cost of the item sold.The present worth item cost in OW is Noting that T T 1 T 2 , the total present value of the total relevant cost per unit time during the cycle is the sum of ordering cost, holding cost, shortage cost, lost sale cost, and item cost

5.3
In the above expression, TUC 0 is the objective function which we have to minimize, which is the function of T 1 and T 2 .To minimize the objective function, the solution methodology is presented in Section 6.

Solution Procedure
To derive the optimal solutions, the following classical optimization technique was used.
Step 1.Take the partial derivatives of TUC T 1 , T 2 , T 3 with respect to T 1 , T 2 , and T 3 and equate the results to zero.The necessary conditions for optimality are

Advances in Operations Research
Step 2. The simultaneous equations above can be solved for T * 1 , T * 2 , and T * 3 . Step

Numerical Example
Optimal replenishment policy to minimize the total present value cost is derived by using the methodology given in the preceding section.The following parameters are assumed: a 400, b 0.05, ordering cost 100, holding cost in OW 2, holding cost in RW 25, shortage cost 25, lost sale cost 10, item cost 10, inflation rate 0.06, own warehouse capacity 100, fraction backordered 0.8, and the deterioration parameters α 0.05, β 1.8, g 0.02, and h 1.8.
From the below Table 1 and Figure 5, the main observations drown from the numerical example are as follows: For Model 1 1 From Table 1, when all the given conditions and constraints are satisfied, the optimal solution is found.In this example, the minimal value of the total present value cost per unit time is $1006.31,while the respective optimal values of T * 1 , T * 2 , T * 3 , and T * are 0.53, 4.31, 3.41, and 8.25, respectively.2 When there is only rented warehouse, the minimal value of the total present value cost per unit time is $2112.41,while the respective optimal period of positive and negative inventory levels are 0.04 and 0.11, respectively.
3 When there is only own warehouse with fixed capacity W units, the minimal value of the total present value cost per unit time is $3378.63,while the respective optimal periods of positive and negative inventory levels are 5.52 and 3.67, respectively.The system has no space to store excess unit and its TUC is higher than our example due to holding cost and shortage cost.
For Model 2 1 From Table 1, when all the given conditions and constraints are satisfied, the optimal solution is found.In this example, the minimal value of the total present value cost per unit time is $1786.38,while the respective optimal values of T * 1 , T * 2 , T * 3 , and T * are 0.67, 6.42, 4.11, and 11.20, respectively.2 When there is only rented warehouse, the minimal value of the total present value cost per unit time is $3755.68,while the respective optimal periods of positive and negative inventory levels are 0.58 and 0.12, respectively.
3 When there is only own warehouse with fixed capacity W units, the minimal value of the total present value cost per unit time is $5302.62,while the respective optimal periods of positive and negative inventory levels are 6.27 and 4.23, respectively.The system has no space to store excess unit and its TUC is higher than our example due to holding cost and shortage cost.
For Model 3 1 From Table 1, when all the given conditions and constraints are satisfied, the optimal solution is found.In this example, the minimal value of the total present 2 When there is only rented warehouse, the minimal value of the total present value cost per unit time is $4912.22,while the respective optimal periods of positive and negative inventory levels are 0.09 and 0.13, respectively.
3 When there is only own warehouse with fixed capacity W units, the minimal value of the total present value cost per unit time is $6078.92,while the respective optimal periods of positive and negative inventory levels are 8.52 and 5.76, respectively.The system has no space to store excess unit and its TUC is higher than our example due to holding cost and shortage cost.

Sensitivity Analysis
In order to study how the parameters affect the optimal solution, the sensitivity analysis is carried out with respect to the various parameters.The results of the sensitivity analysis are presented in Table 2, Figure 6 and Figure 7.
The main observations drawn from the sensitivity analysis are as follows: 1 the value of PCI is the most sensitive to the shape parameter β of the deterioration rate and the inflation rate r .When β increases by 20%, the value of PCI increases by over 135%.When r increases by 20%, the value of PCI decreases by over 61%.
2 the values of PCI are quite sensitive to the parameters α and are not so sensitive to the parameters h, g, and W.
3 the parameters α, β, g, h, W, and B increase proportional to the value of PCI.The PCI is inversely proportional to the parameters r.
4 a high parameter value of β results in a high value of T 1 and T 3 .A high parameter value of r results in a small value of T 1 , T 2 , and T 3 .Change in the total cost with respect to α (%) Change in the total cost with respect to β (%) Change in the total cost with respect to g (%) Change in the total cost with respect to h (%)

Conclusion
In this study, an inventory model is presented to determine the optimal replenishment cycle for two-warehouse inventory problem under inflation, varying rate of deterioration, and partial backordering.The model assumes limited warehouse's capacity of the distributors.Here, shortages are allowed and partially backlogged.The holding cost at RW is higher as compared to OW.The rate of deterioration in both warehouses is different and follows a two-parameter Weibull distribution.The DCF approach permits a proper recognition of the financial implication of the lost sale in inventory analysis.Some items such as fashionable goods, luxury items, and electronic products are easily identifiable with such kind of a setup, the demand rate is exponentially increasing with time and shortages are allowed and partially Advances in Operations Research backlogged with exponential backlogging rate.The discounted cash flow DCF and classical optimization technique are used to derive the optimal replenishment policy.A numerical example and sensitivity analysis are implemented to illustrate the model with the help of MATHEMATICA-5.2.When there is only rented or own warehouse in the inventory system, the total present values of the total relevant cost per unit time are higher than the twowarehouse model.From the numerical example, we could finally conclude that model 1 is more profitable in comparison to model 2 and model 3 for two-warehouse inventory system.As the backlogging and deterioration rate increases, the total cost of the system also increases.But as the inflation rate increases, the total cost of the system decreases.From the sensitivity analysis, it is evident that the deterioration rate, the inflation, and the backordering rate affect the total cost of the system.In order to optimize the system, the decision maker must develop the most economical replenishment strategy.
There is ample scope for further extension of the present research study in fuzzy environments, trade credits, and situations of stock-dependent demand.

Figure 2 :
Figure 2: Graphical representation of the RW inventory system.

Figure 3 :
Figure 3: Graphical representation of the RW inventory system one warehouse-rented .

Figure 4 :
Figure 4: Graphical representation of the OW inventory system one warehouse-own .

Figure 6 :
Figure 6: Percentage change in the total cost with respect to the parameters W and r.

Figure 7 :
Figure 7: Percentage change in the total cost with respect to the parameters α, β, g, and h.
I 0i t i : Inventory level in OW at time t i , 0 ≤ t i ≤ T i , i 1, 2, 3 I r t 1 : Inventory level in RW at time t 1 , 0 ≤ t 1 ≤ T 1 TUC:The present value of the total relevant cost per unit time.
9 Lead time is zero and initial inventory level is zero.10Thereplenishment rate is infinite.The following notations were used throughout the paper.W: Capacity of OW α: Scale parameter of the deterioration rate in OW β: Shape parameter of the deterioration rate in OW g: Scale parameter of the deterioration rate in RW, α > g Advances in Operations Research h: Shape parameter of the deterioration rate in RW r: Inflation rate A: Ordering cost per order $/order H: Holding cost per unit per unit time in OW $/unit/unit time F: Holding cost per unit per unit time in RW $/unit/unit time , F > H s: Shortage cost per unit time $/unit/unit time 3. With T * 1 , T * 2 , and T * 3 found in Step 2, derive TUC * T * 1 , T * 2 , T * 3 .

Table 1 :
Comparison of models with different demand patterns.

Table 2 :
The sensitivity analysis data showing various parameters.