A multi-warehouse partial backlogging inventory model for deteriorating items under inflation when a delay in payment is permissible

Abstract

In this paper we develop a multi-item multi-warehouse inventory model for deteriorating items for m secondary warehouses (SWs) and one primary warehouse (PW) with displayed stock and price dependent demand under permissible delay in payment. Items are sold from PW which is located at the main market and due to large stock and insufficient space of existing PW, excess items are stored at m SWs of finite capacity. Due to different preserving facilities and storage environment, inventory holding cost is considered to be different in different warehouses. Here the demand of items is a deterministic function of corresponding selling price and the displayed inventory. Shortages are allowed and partially backlogged. The items of SWs are transported to the PW in continuous release pattern and associated transportation cost is proportional to the distance from PW to SWs. Here $M_{i} (<T_{i}$, cycle time) be the period of permissible delay in settling account for ith item, without the interest charges. But if the retailer settles the account after $M_{i}$, he will have to pay with interest per cycle for the inventory not sold after the due date $M_{i}$. A single objective inventory problem is solved numerically by developing Genetic algorithm and the maximum average profit and the corresponding optimum decision variables are evaluated. Finally the model is illustrated using a numerical example. A sensitivity analysis of the optimal solution with respect to the parameters of the system is carried out.

Appendix

1.1 Calculations for Case-II and Case-III with allowable shortage and no inflation

Case-II: ($t_{1i}\le M_{i}< t_{2i}$).

The interest payable per cycle for the inventory not sold after the due date $M_{i}$ is given by

$$\begin{aligned} P_{T}&= I_{p}C_{i}\bigg [\int _{M_{i}}^{t_{2i}}q_{2i}(t)\,dt+\int _{t_{2i}}^{T_{i}}q_{2i}(t)\,dt\bigg ]\nonumber \\&= I_{p}C_{i}\int _{M_{i}}^{t_{2i}}\frac{K_{2i}}{K_{1i}}\bigg \{e^{K_{1i}(t_{2i}-t)}-1\bigg \}\,dt+0\bigg ], \,\,since\,\,q_{2i}(t)=0, \,\,for\,\,t_{2i}\le t\le T_{i}\nonumber \\&= I_{p}C_{i}\frac{K_{2i}}{K_{1i}^{2}}\bigg [\bigg \{e^{K_{1i}(t_{2i}-M_{i})}-1\bigg \}-K_{1i}\bigg (t_{2i}-M_{i}\bigg )\bigg ]. \end{aligned}$$

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The interest earned at time t during the positive inventory is given by

$$\begin{aligned} I_{T}&= p_{i}I_{e}\int _{0}^{t_{2i}}D_{i}t\,dt\nonumber \\&= p_{i}I_{e}\bigg [\int _{0}^{t_{1i}}D_{i}t\,dt + \int _{t_{1i}}^{t_{2i}}D_{i}t\,dt\bigg ]\nonumber \\&= p_{i}I_{e}\bigg [\int _{0}^{t_{1i}}f(p_{i},W_{i})t\,dt + \int _{t_{1i}}^{t_{2i}}f(p_{i},q_{i}(t))t\,dt\bigg ]\nonumber \\&= p_{i}I_{e}\bigg [\int _{0}^{t_{1i}}\frac{\alpha _{i}+\gamma W_{i}}{p_{i}^{\beta }}t\,dt+\int _{t_{1i}}^{t_{2i}}\frac{\alpha _{i}+\gamma q_{2i}(t)}{p_{i}^{\beta }}t\,dt\bigg ]\nonumber \\&= p_{i}I_{e}\bigg [\frac{\alpha _{i}+\gamma W_{i}}{2p_{i}^{\beta }}t_{1i}^{2}+\frac{K_{2i}}{2}\bigg (t_{2i}^{2}-t_{1i}^{2}\bigg ) -\frac{\gamma K_{2i}}{K_{1i}^{3}p_{i}^{\beta }}\bigg \{K_{1i}\bigg (t_{2i}-t_{1i}e^{K_{1i}(t_{2i}-t_{1i})}\bigg )\nonumber \\&\qquad +\bigg (1-e^{K_{1i}(t_{2i}-t_{1i})}\bigg )+\frac{K_{1i}^{2}}{2}\bigg (t_{2i}^{2}-t_{1i}^{2}\bigg )\bigg \}\bigg ]. \end{aligned}$$

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Case-III: ($t_{2i}\le M_{i}< T_{i}$).

The interest earned at time t during the positive inventory period plus the interest earned from the cash invested during the time period $(t_{2i}, M_{i})$ after the inventory is exhausted at time $t_{2i}$, and it is given by

$$\begin{aligned} I_{T}&= p_{i}I_{e}\bigg [\int _{0}^{t_{2i}}D_{i}t\,dt+(M_{i}-t_{2i})\int _{0}^{t_{2i}}D_{i}\,dt\bigg ]\nonumber \\&= p_{i}I_{e}\bigg [\int _{0}^{t_{1i}}D_{i}t\,dt + \int _{t_{1i}}^{t_{2i}}D_{i}t\,dt\nonumber \\&\quad +(M_{i}-t_{2i})\bigg \{\int _{0}^{t_{1i}}\frac{\alpha _{i}+\gamma W_{i}}{p_{i}^{\beta }}\,dt + \int _{t_{1i}}^{t_{2i}}\frac{\alpha _{i}+\gamma q_{2i}(t)}{p_{i}^{\beta }}\,dt\bigg \}\bigg ]\nonumber \\&= p_{i}I_{e}\bigg [\int _{0}^{t_{1i}}f(p_{i},W_{i})t\,dt + \int _{t_{1i}}^{t_{2i}}f(p_{i},q_{i}(t))t\,dt+(M_{i}-t_{2i})\bigg \{\int _{0}^{t_{1i}}\frac{\alpha _{i}+\gamma W_{i}}{p_{i}^{\beta }}\,dt\nonumber \\&+ \int _{t_{1i}}^{t_{2i}}\frac{\alpha _{i}}{p_{i}^{\beta }}\,dt+ \frac{\gamma }{p_{i}^{\beta }}\int _{t_{1i}}^{t_{2i}}\frac{K_{2i}}{K_{1i}}\bigg (e^{K_{1i}(t_{2i}-t)}-1\bigg )\,dt \bigg \}\bigg ]\nonumber \\&= p_{i}I_{e}\bigg [\int _{0}^{t_{1i}}\frac{\alpha _{i}+\gamma W_{i}}{p_{i}^{\beta }}t\,dt+\int _{t_{1i}}^{t_{2i}}\frac{\alpha _{i}+\gamma q_{2i}(t)}{p_{i}^{\beta }}t\,dt\bigg ]+(M_{i}-t_{2i})I_{e}\bigg [K_{4i}t_{1i}\nonumber \\&\quad +p_{i}K_{2i}(t_{2i}-t_{1i})+\frac{K_{2i}p_{i}\gamma }{K_{1i}^{2}p_{i}^{\beta }}\bigg \{\bigg (e^{K_{1i}(t_{2i}-t_{1i})}-1\bigg )- K_{1i}(t_{2i}-t_{1i})\bigg \}\bigg ]\nonumber \\&= p_{i}I_{e}\bigg [\frac{\alpha _{i}+\gamma W_{i}}{2p_{i}^{\beta }}t_{1i}^{2}+\frac{K_{2i}}{2}\bigg (t_{2i}^{2}-t_{1i}^{2}\bigg ) -\frac{\gamma K_{2i}}{K_{1i}^{3}p_{i}^{\beta }}\bigg \{K_{1i}\bigg (t_{2i}-t_{1i}e^{K_{1i}(t_{2i}-t_{1i})}\bigg )\nonumber \\&\quad +\bigg (1-e^{K_{1i}(t_{2i}-t_{1i})}\bigg )+\frac{K_{1i}^{2}}{2}\bigg (t_{2i}^{2}-t_{1i}^{2}\bigg )\bigg \}\bigg ]+(M_{i}-t_{2i})I_{e}\bigg [K_{4i}t_{1i}+p_{i}K_{2i}(t_{2i}-t_{1i})\nonumber \\&\quad +\frac{K_{2i}p_{i}\gamma }{K_{1i}^{2}p_{i}^{\beta }}\bigg \{\bigg (e^{K_{1i}(t_{2i}-t_{1i})}-1\bigg )- K_{1i}(t_{2i}-t_{1i})\bigg \}\bigg ]. \end{aligned}$$

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1.2 Calculations for Case-II and Case-III with allowable shortage and inflation

Case-II: ($t_{1i}\le M_{i}< t_{2i}$).

The interest payable rate at time t is $(e^{i_{p}t}-1)$ dollars per dollar, so the present value (at t=0) of interest payable rate at time t is $I_{p}(t)=(e^{i_{p}t}-1)e^{-rt}$dollars per dollar. Therefore, the interest payable per cycle for the inventory not sold after the due date $M_{i}$ is given by

$$\begin{aligned} P_{T}^{f}&= C_{i}\bigg [\int _{M_{i}}^{t_{2i}}q_{2i}(t)I_{p}(t)\,dt+\int _{t_{2i}}^{T_{i}}q_{2i}(t)I_{p}(t)\,dt\bigg ]\nonumber \\&= C_{i}\bigg [\int _{M_{i}}^{t_{2i}}\frac{K_{2i}}{K_{1i}}\bigg \{e^{K_{1i}(t_{2i}-t)}-1\bigg \}(e^{i_{p}t}-1)e^{-rt}\,dt+0\bigg ], \nonumber \\&\quad since\,\,q_{2i}(t)=0, \,\,for\,\,t_{2i}\le t\le T_{i}\nonumber \\&= C_{i}\frac{K_{2i}}{K_{1i}}\bigg [\frac{1}{l_{p}-K_{1i}}\bigg (e^{l_{p}t_{2i}}-e^{K_{1i}t_{2i}+(l_{p}-K_{1i})M_{i}}\bigg )-\frac{1}{l_{p}}\bigg (e^{l_{p}t_{2i}}-e^{l_{p}M_{i}}\bigg )\nonumber \\&\quad +\frac{1}{K_{1i}+r}\bigg (e^{-rt_{2i}}-e^{K_{1i}t_{2i}-(K_{1i}+r)M_{i}}\bigg )-\frac{1}{r}\bigg (e^{-rt_{2i}}-e^{-rM_{i}}\bigg )\bigg ]. \end{aligned}$$

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The present value of the interest earned at time t, $I_{e}(t)$ is $(e^{i_{e}t}-1)e^{-rt}$. The interest earned during the positive inventory, is given by

$$\begin{aligned} I_{T}^{f}&= p_{i}\int _{0}^{t_{2i}}D_{i}t(e^{i_{e}t}-1)e^{-rt}\,dt\nonumber \\&= p_{i}\bigg [\int _{0}^{t_{1i}}D_{i}t(e^{i_{e}t}-1)e^{-rt}\,dt + \int _{t_{1i}}^{t_{2i}}D_{i}t(e^{i_{e}t}-1)e^{-rt}\,dt\bigg ]\nonumber \\&= p_{i}\bigg [\int _{0}^{t_{1i}}f(p_{i},W_{i})t(e^{l_{e}t}-e^{-rt})\,dt +\int _{t_{1i}}^{t_{2i}}f(p_{i},q_{i}(t))t(e^{l_{e}t}-e^{-rt})\,dt\bigg ]\nonumber \\&= p_{i}\bigg [\int _{0}^{t_{1i}}\frac{\alpha _{i}+\gamma W_{i}}{p_{i}^{\beta }}t(e^{l_{e}t}-e^{-rt})\,dt+\int _{t_{1i}}^{t_{2i}}\frac{\alpha _{i}+\gamma q_{2i}(t)}{p_{i}^{\beta }}t(e^{l_{e}t}-e^{-rt})\,dt\bigg ]\nonumber \\&= p_{i}\bigg [\frac{\alpha _{i}+\gamma W_{i}}{p_{i}^{\beta }}\bigg \{\bigg (\frac{t_{1i}e^{l_{e}t_{1i}}}{l_{e}}-\frac{e^{l_{e}t_{1i}}}{l_{e}^{2}}\bigg )+\frac{1}{l_{e}^{2}}+\bigg (\frac{t_{1i}e^{-rt_{1i}}}{r}+\frac{e^{-rt_{1i}}}{r^{2}}\bigg )-\frac{1}{r^{2}}\bigg \}\nonumber \\&\quad +\frac{\alpha _{i}}{p_{i}^{\beta }}\bigg \{\bigg (\frac{t_{2i}e^{l_{e}t_{2i}}}{l_{e}}-\frac{e^{l_{e}t_{2i}}}{l_{e}^{2}}\bigg )-\bigg (\frac{t_{1i}e^{l_{e}t_{1i}}}{l_{e}}-\frac{e^{l_{e}t_{1i}}}{l_{e}^{2}}\bigg )+\bigg (\frac{t_{2i}e^{-rt_{2i}}}{r}+\frac{e^{-rt_{2i}}}{r^{2}}\bigg )\nonumber \\&\quad -\bigg (\frac{t_{1i}e^{-rt_{1i}}}{r}+\frac{e^{-rt_{1i}}}{r^{2}}\bigg )\bigg \}+\frac{\gamma K_{2i} }{p_{i}^{\beta }K_{1i}}\bigg \{\frac{t_{2i}e^{l_{e}t_{2i}}-t_{1i}e^{K_{1i}t_{2i}+(l_{e}-K_{1i})t_{1i}}}{l_{e}-K_{1i}}\nonumber \\&\quad -\frac{e^{l_{e}t_{2i}}-e^{K_{1i}t_{2i}+(l_{e}-K_{1i})t_{1i}}}{(l_{e}-K_{1i})^{2}}-\frac{t_{2i}e^{l_{e}t_{2i}}-t_{1i}e^{l_{e}t_{1i}}}{l_{e}}+\frac{e^{l_{e}t_{2i}}-e^{l_{e}t_{1i}}}{l_{e}^{2}}\nonumber \\&\quad +\frac{t_{2i}e^{-rt_{2i}}-t_{1i}e^{K_{1i}t_{2i}-(K_{1i}+r)t_{1i}}}{K_{1i}+r}+\frac{e^{-rt_{2i}}-e^{K_{1i}t_{2i}-(K_{1i}+r)t_{1i}}}{(K_{1i}+r)^{2}}\nonumber \\&\quad -\frac{t_{2i}e^{-rt_{2i}}-t_{1i}e^{-rt_{1i}}}{r}-\frac{e^{-rt_{2i}}-e^{-rt_{1i}}}{r^{2}}\bigg \}\bigg ]. \end{aligned}$$

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Case-III: ($t_{2i}\le M_{i}< T_{i}$).

The interest earned per cycle is the interest earned during the positive inventory period plus the interest earned from the cash invested during the time period ($t_{2i}, M_{i}$) after the inventory is exhausted at time $t_{2i}$, is given by

$$\begin{aligned} I_{T}^{f}&= p_{i}\bigg [\int _{0}^{t_{2i}}D_{i}t(e^{i_{e}t}-1)e^{-rt}\,dt+\bigg (e^{l_{e}(M_{i}-t_{2i})}-1\bigg )\int _{0}^{t_{2i}}D_{i}\,dt\bigg ]\nonumber \\&= p_{i}\bigg [\int _{0}^{t_{1i}}D_{i}t(e^{i_{e}t}-1)e^{-rt}\,dt + \int _{t_{1i}}^{t_{2i}}D_{i}t(e^{i_{e}t}-1)e^{-rt}\,dt\nonumber \\&\quad +\bigg (e^{l_{e}(M_{i}-t_{2i})}-1\bigg )\bigg \{\int _{0}^{t_{1i}}D_{i}\,dt+\int _{t_{1i}}^{t_{2i}}D_{i}\,dt\bigg \}\bigg ]\nonumber \\&= p_{i}\bigg [\int _{0}^{t_{1i}}f(p_{i},W_{i})t(e^{l_{e}t}-e^{-rt})\,dt + \int _{t_{1i}}^{t_{2i}}f(p_{i},q_{i}(t))t(e^{l_{e}t}-e^{-rt})\,dt\bigg ]\nonumber \\&\quad +\bigg (e^{l_{e}(M_{i}-t_{2i})}-1\bigg )p_{i}\bigg [\int _{0}^{t_{1i}}f(p_{i},W_{i})\,dt+ \int _{t_{1i}}^{t_{2i}}f(p_{i},q_{i}(t))\,dt\bigg ]\nonumber \\&= p_{i}\bigg [\int _{0}^{t_{1i}}\frac{\alpha _{i}+\gamma W_{i}}{p_{i}^{\beta }}t(e^{l_{e}t}-e^{-rt})\,dt+\int _{t_{1i}}^{t_{2i}}\frac{\alpha _{i}+\gamma q_{2i}(t)}{p_{i}^{\beta }}t(e^{l_{e}t}-e^{-rt})\,dt\bigg ]\nonumber \\&\quad +\bigg (e^{l_{e}(M_{i}-t_{2i})}-1\bigg )p_{i}\bigg [\int _{0}^{t_{1i}}\frac{\alpha _{i}+\gamma W_{i}}{p_{i}^{\beta }}\,dt + \int _{t_{1i}}^{t_{2i}}\frac{\alpha _{i}+\gamma q_{2i}(t)}{p_{i}^{\beta }}\,dt\bigg ]\nonumber \\&= p_{i}\bigg [\frac{\alpha _{i}+\gamma W_{i}}{p_{i}^{\beta }}\bigg \{\bigg (\frac{t_{1i}e^{l_{e}t_{1i}}}{l_{e}}-\frac{e^{l_{e}t_{1i}}}{l_{e}^{2}}\bigg )+\frac{1}{l_{e}^{2}}+\bigg (\frac{t_{1i}e^{-rt_{1i}}}{r}+\frac{e^{-rt_{1i}}}{r^{2}}\bigg )-\frac{1}{r^{2}}\bigg \}\nonumber \\&\quad +\frac{\alpha _{i}}{p_{i}^{\beta }}\bigg \{\bigg (\frac{t_{2i}e^{l_{e}t_{2i}}}{l_{e}}-\frac{e^{l_{e}t_{2i}}}{l_{e}^{2}}\bigg )-\bigg (\frac{t_{1i}e^{l_{e}t_{1i}}}{l_{e}}-\frac{e^{l_{e}t_{1i}}}{l_{e}^{2}}\bigg )+\bigg (\frac{t_{2i}e^{-rt_{2i}}}{r}+\frac{e^{-rt_{2i}}}{r^{2}}\bigg )\nonumber \\&\quad -\bigg (\frac{t_{1i}e^{-rt_{1i}}}{r}+\frac{e^{-rt_{1i}}}{r^{2}}\bigg )\bigg \}+\frac{\gamma K_{2i} }{p_{i}^{\beta }K_{1i}}\bigg \{\frac{t_{2i}e^{l_{e}t_{2i}}-t_{1i}e^{K_{1i}t_{2i}+(l_{e}-K_{1i})t_{1i}}}{l_{e}-K_{1i}}\nonumber \\&\quad -\frac{e^{l_{e}t_{2i}}-e^{K_{1i}t_{2i}+(l_{e}-K_{1i})t_{1i}}}{(l_{e}-K_{1i})^{2}}-\frac{t_{2i}e^{l_{e}t_{2i}}-t_{1i}e^{l_{e}t_{1i}}}{l_{e}}+\frac{e^{l_{e}t_{2i}}-e^{l_{e}t_{1i}}}{l_{e}^{2}}\nonumber \\&\quad +\frac{t_{2i}e^{-rt_{2i}}-t_{1i}e^{K_{1i}t_{2i}-(K_{1i}+r)t_{1i}}}{K_{1i}+r}+\frac{e^{-rt_{2i}}-e^{K_{1i}t_{2i}-(K_{1i}+r)t_{1i}}}{(K_{1i}+r)^{2}}\nonumber \\&\quad -\frac{t_{2i}e^{-rt_{2i}}-t_{1i}e^{-rt_{1i}}}{r}-\frac{e^{-rt_{2i}}-e^{-rt_{1i}}}{r^{2}}\bigg \}\bigg ] +\bigg (e^{l_{e}(M_{i}-t_{2i})}-1\bigg )\bigg [K_{4i}t_{1i}\nonumber \\&\quad +p_{i}K_{2i}(t_{2i}-t_{1i})+\frac{K_{2i}p_{i}\gamma }{K_{1i}^{2}p_{i}^{\beta }}\bigg \{\bigg (e^{K_{1i}(t_{2i}-t_{1i})}-1\bigg )- K_{1i}(t_{2i}-t_{1i})\bigg \}\bigg ]. \end{aligned}$$

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