Two-Warehouse Inventory Systems for Seasonal Deteriorating Products with Permissible Delay in Payments

Abstract

In this paper two inventory systems are presented assuming general ramp type demand rate, constant deterioration rate and partial backlogging of unsatisfied demand. Since the capacity of the Owned Warehouse (OW) is usually limited and, under some special circumstances, the procurement of a large amount of items can be decided, a Rented Warehouse (RW) can be used to store the excess quantity. In addition, we assume, that the supplier offers the retailer a credit scheme, which provides a fixed delay period for settling his account. Sales revenue generated during this credit period is deposited to an interest bearing account. At the end of this period the retailer settles the account. Thereafter, capital opportunity cost for the value of items still in stock is charged. The study of the inventory system requires exploring the feasible ordering relations between the time parameters. Consequently the following cases, which lead to two inventory models, must be examined: (1) the offered credit period is less than the time point when the demand rate is stabilized, (2) the offered credit period is longer than the time point when the demand rate is stabilized and less than the planning horizon. The single warehouse inventory problem is also examined as a special case of the model. The optimal replenishment policy for each model is determined. The results obtained are highlighted by suitably selected examples.

Appendices

Appendix 1

The solution of Eq. (20) is given by ${a}_{1.A.1}\left({t}_1\right)=b\left({t}_1,{x}_1={h}_A\left({t}_1\right)\right)-p{I}_e\left(M-{t}_1\right)=0$. Differentiating a _1. A.1(t ₁) with respect to t ₁ we obtain:

$$ {a_{1.A.1}}^{\prime}\left({t}_1\right)={b}^{\prime}\left({t}_1,{x}_1={h}_A\left({t}_1\right)\right)+p{I}_e>0, $$

where

$$ \begin{array}{l}{b}^{\prime}\left({t}_1,{x}_1={h}_A\left({t}_1\right)\right)=[\left(\frac{c_{1O}}{\theta_2}-\frac{c_{1R}}{\theta_1}+{c}_t\right){\theta}_2\left(1-\frac{d{x}_1}{d{t}_1}\right)+\frac{c_{1R}+{c}_3{\theta}_1}{\theta_1}\left({\theta}_1-{\theta}_2\right){e}^{\theta_1{x}_1}\frac{d{x}_1}{d{t}_1}+\frac{c_{1R}+{c}_3{\theta}_1}{\theta_1}{\theta}_2{e}^{\theta_1{x}_1}]{e}^{\theta_2\left({t}_1-{x}_1\right)}+{c}_2\left[\beta \left(T-{t}_1\right)+\left(T-{t}_1\right){\beta}^{\prime}\left(T-{t}_1\right)\right]-{c}_4{\beta}^{\prime}\left(T-{t}_1\right)>0,\end{array} $$

since from Eq. (17) we obtain $\frac{d{x}_1}{d{t}_1}={e}^{\theta_2\left({t}_1-{x}_1\right)}\frac{f\left({t}_1\right)}{f\left({x}_1\right)}>1$ and by the assumptions we have $\frac{c_{1R}}{\theta_1}>\frac{c_{1O}}{\theta_2}+{c}_t$, ${\theta}_1>{\theta}_2$ and $\beta \left(T-{t}_1\right)+\left(T-{t}_1\right){\beta}^{\prime}\left(T-{t}_1\right)>0$.

Also, we have

$$ {a}_{1.A.1}(0)=\frac{c_{1R}+{c}_3{\theta}_1}{\theta_1}{e}^{-{\theta}_2{x}_1}\left({e}^{\theta_1{x}_1}-1\right)+\frac{c_{1O}+{c}_3{\theta}_2}{\theta_2}\left({e}^{-{\theta}_2{x}_1}-1\right)-{c}_2T\beta (T)-{c}_4\left(1-\beta (T)\right)+{c}_t{e}^{-{\theta}_2{x}_1}-p{I}_eM<0, $$

if ${c}_2T\beta (T)+{c}_4\left(1-\beta (T)\right)\ge {c}_t$ (see Skouri and Konstantaras 2013), and ${a}_{1.A.1}(T)>0$.

Hence Eq. (20) has a unique solution which satisfies the second order conditions for a minimum.

Accordingly from Eqs. (21) to (25) we have

$$ {a}_{1.A.2}\left({t}_1\right) = b\left({t}_1,{x}_1={h}_A\left({t}_1\right)\right)+\frac{C_p{I}_c}{\theta_2}\left({e}^{\theta_2\left({t}_1-M\right)}-1\right)=0\Rightarrow {a_{1.A.2}}^{\prime}\left({t}_1\right)={b}^{\prime}\left({t}_1,{x}_1={h}_A\left({t}_1\right)\right)+{C}_p{I}_c{e}^{\theta_2\left({t}_1-M\right)}>0, $$

$$ {a}_{1.A.3}\left({t}_1\right) = b\left({t}_1,{x}_1={h}_A\left({t}_1\right)\right)+{C}_p{I}_c\left[\frac{1}{\theta_1}{e}^{\theta_2\left({t}_1-{x}_1\right)}\left({e}^{\theta_1\left({x}_1-M\right)}-1\right)+\frac{1}{\theta_2}\left({e}^{\theta_2\left({t}_1-{x}_1\right)}-1\right)\right]\Rightarrow $$

$$ {a_{1.A.3}}^{\prime}\left({t}_1\right)={b}^{\prime}\left({t}_1,{x}_1={h}_A\left({t}_1\right)\right)+{C}_p{I}_c\left[\frac{\theta_2-{\theta}_1}{\theta_1}\left({e}^{\theta_1\left({x}_1-M\right)}-1\right)\left(1-\frac{d{x}_1}{d{t}_1}\right)+{e}^{\theta_1\left({x}_1-M\right)}\right]{e}^{\theta_2\left({t}_1-{x}_1\right)}>0, $$

$$ {a}_{1.B.1}\left({t}_1\right)=b\left({t}_1,{x}_1={h}_B\left({t}_1\right)\right)+\frac{C_p{I}_c}{\theta_2}\left({e}^{\theta_2\left({t}_1-M\right)}-1\right)\Rightarrow {a_{1.B.1}}^{\prime}\left({t}_1\right)={b}^{\prime}\left({t}_1,{x}_1={h}_B\left({t}_1\right)\right)+{C}_p{I}_c{e}^{\theta_2\left({t}_1-M\right)}>0, $$

$$ {a}_{1.B.2}\left({t}_1\right)=b\left({t}_1,{x}_1={h}_B\left({t}_1\right)\right)+{C}_p{I}_c\left[\frac{1}{\theta_1}{e}^{\theta_2\left({t}_1-{x}_1\right)}\left({e}^{\theta_1\left({x}_1-M\right)}-1\right)+\frac{1}{\theta_2}\left({e}^{\theta_1\left({t}_1-{x}_1\right)}-1\right)\right]\Rightarrow $$

$$ {a_{1.B.2}}^{\prime}\left({t}_1\right)={b}^{\prime}\left({t}_1,{x}_1={h}_B\left({t}_1\right)\right)+{C}_p{I}_c\left[\frac{\theta_2-{\theta}_1}{\theta_1}\left({e}^{\theta_1\left({x}_1-M\right)}-1\right)\left(1-\frac{d{x}_1}{d{t}_1}\right)+{e}^{\theta_1\left({x}_1-M\right)}\right]{e}^{\theta_2\left({t}_1-{x}_1\right)}>0, $$

$$ {a}_{1.\varGamma .1}\left({t}_1\right) = b\left({t}_1,{x}_1={h}_{\varGamma}\left({t}_1\right)\right)+{C}_p{I}_c\left[\frac{1}{\theta_1}{e}^{\theta_2\left({t}_1-{x}_1\right)}\left({e}^{\theta_1\left({x}_1-M\right)}-1\right)+\frac{1}{\theta_2}\left({e}^{\theta_1\left({t}_1-{x}_1\right)}-1\right)\right]\Rightarrow $$

$$ {a_{1.\varGamma .1}}^{\prime}\left({t}_1\right)={b}^{\prime}\left({t}_1,{x}_1={h}_{\varGamma}\left({t}_1\right)\right)+{C}_p{I}_c\left[\frac{\theta_2-{\theta}_1}{\theta_1}\left({e}^{\theta_1\left({x}_1-M\right)}-1\right)\left(1-\frac{d{x}_1}{d{t}_1}\right)+{e}^{\theta_1\left({x}_1-M\right)}\right]{e}^{\theta_2\left({t}_1-{x}_1\right)}>0, $$

taking into account the assumptions of Theorem 1.

The optimal value of x ₁ is derived by using the appropriate relation from (17) to (19).

Obviously, the solution of each equation must also satisfy the constraints that are derived by the ordering relations between x ₁, t ₁ , μ, M and T, for each case accordingly.

Appendix 2

The solution of Eq. (35) is given by ${a}_{\mathrm{0.1.1}}\left({t}_1\right)={b}_0\left({t}_1\right)-p{I}_e\left(M-{t}_1\right)=0$. Differentiating a _O.1.1(t ₁) with respect to t ₁ we obtain: ${a_{\mathrm{0.1.1}}}^{\prime}\left({t}_1\right)={b_0}^{\prime}\left({t}_1\right)+p{I}_e>0$, where

$$ {b}_O^{\prime}\left({t}_1\right)=\left({c}_{1O}+{c}_3{\theta}_2\right){e}^{\theta_2{t}_1}+{c}_2\left[\beta \left(T-{t}_1\right)+\left(T-{t}_1\right){\beta}^{\prime}\left(T-{t}_1\right)\right]-{c}_4{\beta}^{\prime}\left(T-{t}_1\right)>0, $$

since $\beta \left(T-{t}_1\right)+\left(T-{t}_1\right){\beta}^{\prime}\left(T-{t}_1\right)>0$.

Also, we have ${a}_{O.\mathrm{1.1}}(0)=-{c}_2T\beta (T)-{c}_4\left(1-\beta (T)\right)-p{I}_eM<0$ and ${a}_{\mathrm{0.1.1}}(T)>0$.

Hence Eq. (35) has a unique solution which satisfies the second order conditions for a minimum.

Accordingly from Eqs. (36) to (37) we have

$$ {a}_{O.\mathrm{1.2}}\left({t}_1\right) = {b}_O\left({t}_1\right)+\frac{C_p{I}_c}{\theta_2}\left({e}^{\theta_2\left({t}_1-M\right)}-1\right)=0\Rightarrow {a_{O.\mathrm{1.2}}}^{\prime}\left({t}_1\right)={b}_0^{\prime}\left({t}_1\right)+{C}_p{I}_c{e}^{\theta_2\left({t}_1-M\right)}>0, $$

$$ {a}_{O.\mathrm{1.3}}\left({t}_1\right)={b}_0\left({t}_1\right)+\frac{C_p{I}_c}{\theta_2}\left({e}^{\theta_2\left({t}_1-M\right)}-1\right)\Rightarrow {a_{O.\mathrm{1.3}}}^{\prime}\left({t}_1\right)={b}_0^{\prime}\left({t}_1\right)+{C}_p{I}_c{e}^{\theta_2\left({t}_1-M\right)}{e}^{\theta_2{t}_1}>0. $$

Obviously, the solution of each equation must also satisfy the constraints that are derived by the ordering relations between t ₁ , μ, M and T, for each case accordingly.