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In developing the models for different inventory systems, most of the researchers assumed that the manufacturer/retailer pays the amount to the supplier immediately as soon as the items are received. This may not be true in today’s business transactions, since suppliers frequently offer their customers a fixed period known as trade-credit period. This trade-credit period allows the customer to settle the account for payment of the amount owed to the supplier without charged any interest. As a matter of fact, the permissible delay in payments gives a benefit to the supplier because this policy attracts to the new customers for increase the sale. Therefore, the existence of the credit period serves to reduce the cost to the purchaser for holding stock because it reduces the amount of capital invested in stock for the duration of the credit period. Based on this phenomenon, Goyal (1985) established a single item inventory model under allowable delay in payments.

Chand and Ward (1987) developed a model that differs from that of Goyal, they modeled the cost of funds tied up in inventory consistent with the assumptions of the classical EOQ model. Chung (1998) developed an alternative approach to determine the economic order quantity under the condition of permissible delay in payments. Jaggi and Aggarwal (1994) considered the inventory model with an exponential deterioration rate under the condition of permissible delay in payments. Hwang and Shinn (1997) modeled an inventory system for retailer’s pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments. Teng (2002) amended the Goyal model (1985) to consider the difference between the selling price and the purchasing cost. Other related articles can be found in Sana and Chaudhuri (2008), Chung and Huang (2009), Singh and Singh (2009), Teng and Chang (2009), Tsao and Sheen (2008), Singh and Sharma (2013, 2014) and their references.

In the real-world application the demand is influenced by the selling price. The dependence of the sale of any item on its selling price is not a new concept, but a common sense conclusion. Demand decrease as selling price increase. This concept is known as downward sloping price. Many of the physical good undergo decay or deterioration over time. Commodities such as fruits, vegetables, foodstuffs are subject to direct spoilage while kept in store. Highly volatile liquids such as gasoline, alcohol, turpentine undergo physical depletion over time through the process of evaporation. Datta and Paul (2001) developed a model with stock and price sensitive demand. Giri et al. (1995) developed deteriorated items with stock dependent demand. Hariga (1996) proposed deteriorating items with time varying demand. Padmanabhan and Vrat (1995) developed an EOQ model with perishable items and stock dependent selling rate. Ray et al. (1998) developed a two storage model with stock dependent demand. Teng and Chang (2005) developed an EPQ model with price and stock dependent demand. Singh and Singh (2007) developed an Singh et al. (2010) proposed a model with deterioration, stock dependent demand under inflation. Kumar et al. (2012) proposed three echelon supply chain inventory model for deteriorating items with limited storage facility and lead–time under inflation. Singh and Sharma (2013) discussed an integrated model with variable production and demand rate under inflation. Dem and Singh (2013) derived a production model for ameliorating items with quality consideration. Young Wu-Zhou et al. (2015) developed a single period inventory and payment model with partial trade credit.  

Most of the previous researchers considered demand rate as a function of time and deterioration at a constant rate, which is unrealistic. The rate of deterioration increases with the increase of time. On the other hand, many convenient experiences disclose that selling price of a product influences the demand rate. Consequently, we have developed this model with these realistic features. It is presumed that production depends on the demand rate and trade-credit facility is also available. The effect of inflation is also considered. The concept of the model is exemplified with numerical examples and sensitivity analysis with respect to the system parameters is performed. 

2.  ASSUMPTIONS

1. The planning horizon is infinite.

2. The production rate depends on demand rate.

3. Deterioration rate is a function of time.

4. Demand rate is price dependent.

5. Production rate will always be greater than the demand rate.

6. A permissible delay period is provided to the manufacturer by the supplier.

3. NOTATIONS

I (t)          =              Inventory level at any time t

D (p)       =              Demand rate and given as,

R             =              Production rate and defined as, where α > 1

                =              Selling price within the inventory cycle

T              =              Cycle time

            =              The time upto which production occurs

r               =              Inflation rate

M             =              Permissible delay period  

θ(t)          =              Kt where K is constant (K<<1)

C₁            =              Set up cost per production run

C             =              Purchasing cost per unit item

C₂            =              Production cost per unit item

C₃            =              Inventory holding cost/ unit/ period

C₄            =              Deterioration cost per unit item

Q             =              Total amount of production

I_e              =              Interest earned per unit time

I_c              =              Interest charged per unit time

TP_i          =              Total average profit where i = 1, 2

4.  MATHEMATICAL MODEL

Fig. 1: Production system with allowable delay period.

The production inventory system with the facility of allowable delay in payments is depicted in Fig. 1. The production starts at t = 0 and it continue up to . During this period the inventory level increases due to the combined effect of the production, demand and deterioration. The inventory level I(t) at any time t during the period (0, β) is governed by the following differential equation

At time t = β production stops and then inventory depletes due to the combined effect of demand and deterioration. At   t = T   the inventory level becomes zero. The differential equation governing the instant state of the inventory level during the period (β, T)is given by:

Solution of equation (1) is given by

Since k is very small, so omitting the higher powers of k, we get

                                             ....(3)

Using boundary condition I (0) = 0,   C = 0, so

                  ..(4)

Solution of equation (2) is given by 

Omitting the higher powers of k, we get

                  ..(5)

Using boundary condition I (T) = 0

    ..(6)

Using expressions (4) and (6) at , we get

                ...(7)

Set up cost PS is given by

                           ...(8)

Raw material is purchased initially at time t = 0. So, purchasing cost of raw material RC is given by

                ..(9)

Present worth of production cost PC is given by

             ..(10)

Present worth of holding cost HC is given by               

            ..(11)

Present worth of deterioration cost DC is given by

              ..(12)

Present worth of sales revenue SR is given by

                  ..(13)

Now two cases may arise according to the position of permissible delay period: (1) M ≤ T and (2) M ≥ T.

We shall discuss these cases simultaneously one by one as follow:

4.1 Case (1): M ≤ T

In this case, present worth of interest payable IC is given by

..(14)

Present worth of interest earned IE is given by

        ..(15)

Consequently, the present worth of total average profit of the system for Case (1) can be formulated as

     ..(16)

4.2 Case (2): M ≥ T

Since, delay period is larger than T so there is no interest charge in this situation.

Present worth of interest payable IC is given by

          ..(17)

Present worth of interest earned IE is given by

           ..(18)

Consequently, the present worth of total average profit of the system for Case (2) can be formulated as

        ..(19)

5. NUMERICAL EXAMPLES

Example 1: To exemplify the model we have considered the data in appropriate units as follows: 

Then, the optimal solution in this case with the help of computational software MATHEMATICA 5.2 is given as: β = 0.2772, T = 0.5627, TP₁ = 23572.6 and the behavior of the profit function w.r.t. time T is shown in Fig. 2

Fig. 2: Total average profit vs. cycle time T (Case (1)).

Example 2: We have considered the data as follows:

Then, the optimal solutions in this case with the help of computational software MATHEMATICA 5.2 is given as: β = 0.3814, T = 0.6729, TP₁ = 32517.6 and the behavior of the profit function w.r.t. time T is shown in Fig. 3

Fig. 3: Total average profit vs. cycle time T (Case (2).

Sensitivity Analysis

We now study the sensitivity analysis of different parameters of the inventory system on the basis of Example 1. The optimal values of the total average system profit (TP₁) change significantly with changes (−15%, −10%, -5%, +5%, +10%, 15%) of different parameters value in Tables 1-2.

Table 1: Variation in the value of β, T and TP₁ with the variation in K.

S.No.	% Change in K	β	T	TP1
1	-15	0.2928	0.5812	23575.84
2	-10	0.2869	0.5748	23574.19
3	-5	0.2811	0.5697	23573.38
4	0	0.2772	0.5627	23572.60
5	+5	0.2708	0.5583	23571.74
6	+10	0.2647	0.5530	23570.96
7	+15	0.2569	0.5476	23569.49

Table 2: Variation in the value of β, T and TP₁ with the variation in h.

S.No.	% Change in h	β	T	TP1
1	-15	0.3089	0.5935	23581.39
2	-10	0.2984	0.5822	23578.32
3	-5	0.2882	0.5734	23575.28
4	0	0.2772	0.5627	23572.60
5	+5	0.2676	0.5518	23569.85
6	+10	0.2573	0.5429	23566.86
7	+15	0.2469	0.5324	23563.52

1. From sensitivity Table 1 it has been observed that as the deterioration parameter K increases the total average profit decreases; which is according to the realism.

2. From Table 2 it has been observed that the total average profit decreases with an increase in holding cost parameter h, which is realistic.

Fig. 4: Behavior of production time β w.r.t.  the variation in system parameters.

Fig. 5: Behavior of cycle time T w.r.t.  the variation in system parameters.

Fig. 6: Behavior of total average profit TP₁ w.r.t.  the variation in system parameters.

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