An EOQ model for stochastic demand for limited capacity of own warehouse

Abstract

This paper deals with an economic order quantity (EOQ) model for uncertain demand when capacity of own warehouse (OW) is limited and the rented warehouse (RW) is considered, if needed. The expected average cost function is formulated for both continuous and discrete distributions of demand function by trading off holding costs and stock out penalty. The model is justified by suitable illustrations for various types of distributions.

Appendices

Appendix A: When q≥W(≠φ)

Let I _r (t) is on-hand inventory at RW, I _w (t) is on-hand inventory at OW and I _s (t) is shortage level at time t. Here, two cases may arise for uncertain demand (x):

1.1 A.1 Case I: when shortage does not occur, i.e., q≥x

As the demand over the period [0,T] is x, the demand per unit time is x/T. The stock at Rw is cleared first. Thereafter, the stock at OW is used to adjust the demand of the customers. Now, the on-hand inventories are:

$$I_{r} ( t ) = ( q-W ) - \frac{xt}{T},\quad 0\leq t\leq t_{r}\ \mbox{with } I_{r} ( 0 ) =q-W\ \mbox{and}\ I_{r} ( t_{r} ) =0 $$

and

$$I_{w} ( t ) =W- \frac{x}{T} ( t- t_{r} ),\quad t_{r} \leq t\leq T\ \mbox{with}\ I_{w} (T)\geq0. $$

Using I _r (t _r )=0, we have t _r =(q−W)T/x. Now, I _w (T)≥0 implies q≥x. Therefore, the average inventory cost at RW is $\mathrm{Inv}_{r}^{1} = \frac{c_{r}}{T} \int_{0}^{t_{r}} \{ ( q-W ) - \frac{xt}{T} \} dt = \frac{c_{r}}{2x} ( q-W )^{2}$ and the average inventory cost at OW is

$$\begin{aligned} \mathrm{Inv}_{w}^{1} &= \frac{c_{h}}{T} \biggl[ W t_{r} + \int_{t r}^{T} \biggl\{ W- \frac{x}{T} ( t- t_{r} ) \biggr\} dt \biggr]\\ & = \frac{c_{h}}{T} \biggl[ WT- \frac{x}{2T} \biggl( T- \frac{q-W}{x} T \biggr)^{2} \biggr] = c_{h} \biggl[ W- \frac{x}{2T} \biggl( 1- \frac{q-W}{ x} \biggr)^{2} \biggr]. \end{aligned}$$

1.2 A.2 Case 2: when shortage occurs

In this situation, q≤x, the on-hand inventories and shortage are as follows:

$$\begin{aligned} & I_{r} ( t ) = ( q-W ) - \frac{xt}{T},\quad 0\leq t\leq t_{r}\ \mbox{with } I_{r} ( 0 ) =q-W\ \mbox{and } I_{r} ( t_{r} ) =0;\\ & I_{w} ( t ) =W- \frac{x}{T} ( t- t_{r} ),\quad t_{r} \leq t\leq t_{r} +t_{w}\ \mbox{with } I_{w} ( t_{r} +t_{w} ) =0 \end{aligned}$$

and

$$I_{s} ( t ) = \frac{x}{T} ( t- t_{r} -t_{w} ),\quad t_{r} +t_{w} \leq t\leq T\ \mbox{with } I_{s} ( T ) \leq0. $$

Now, I _r (t _r )=0 implies t _r =(q−W)T/x and I _w (t _r +t _w )=0 implies t _w =WT/x. Therefore, the average inventories and shortage are:

$$\begin{aligned} \mathrm{Inv}_{r}^{2} & = \frac{c_{r}}{T} \int _{0}^{t_{r}} \biggl\{ ( q-W ) - \frac{xt}{T} \biggr\} dt = \frac{c_{r}}{2x} ( q-W )^{2},\\ \mathrm{Inv}_{w}^{2} & = \frac{c_{h}}{T} \biggl[ W t_{r} + \int_{t r}^{t_{r} + t_{w}} \biggl\{ W- \frac{x}{T} ( t- t_{r} ) \biggr\} dt \biggr]\\ & = \frac{c_{h}}{T} \biggl[ W ( t_{r} + t_{w} ) - \frac{x}{2T} ( t_{w} )^{2} \biggr] = c_{h} \biggl( q- \frac{W}{2} \biggr) \frac{W}{x} \end{aligned}$$

and

$$\mathrm{Inv}_{s}^{2} = \frac{c_{s}}{T} \int _{ ( t r +t w )}^{T} \frac{x}{T} ( t- t_{r} -t_{w} ) dt = \frac{c_{s}}{2} x \biggl( 1- \frac{q}{x} \biggr)^{2}. $$

The expected average cost, combining case 1 and case 2, we have

$$\begin{aligned} \mathit{EAC}_{1} ( q ) & = \int_{0}^{q} \bigl( \mathrm{Inv}_{r}^{1} + \mathrm{Inv}_{w}^{1} \bigr) f ( x ) dx + \int_{q}^{\infty} \bigl( \mathrm{Inv}_{r}^{2} + \mathrm{Inv}_{w}^{2} \bigr) f ( x ) dx + \int_{q}^{\infty} \mathrm{Inv}_{s}^{2} f ( x ) dx\\ & = c_{h} \int_{0}^{q} \biggl\{ W- \frac{x}{2} \biggl( 1- \frac{q-W}{x} \biggr)^{2} \biggr\} dx + c_{h} \int_{q}^{\infty} W \biggl( q- \frac{W}{2} \biggr) \frac{f(x)}{x} dx\\ &\quad {}+ \frac{c_{r}}{2} ( q-W )^{2} \int_{0}^{\infty} \frac{f(x)}{x} dx + \frac{c_{s}}{2} \int_{q}^{\infty} x \biggl( 1- \frac{q}{x} \biggr)^{2} f ( x ) dx. \end{aligned}$$

Appendix B: When q≤W(≠φ)

In this case, RW is not needed. Let I _w (t) is on-hand inventory at OW and I _s (t) is shortage level at time t. Here, two cases may arise for uncertain demand (x):

2.1 B.3 Case 1: when shortage does not occur, i.e., q≥x

As the demand over the period [0,T] is x, the demand per unit time is x/T. The stock at Rw is cleared first. Thereafter, the stock at OW is used to adjust the demand of the customers. Now, the on-hand inventory is:

$$I_{w} ( t ) =q- \frac{xt}{T},\quad 0\leq t\leq T\ \mbox{with } I_{w} (T)\geq0. $$

Now, I _w (T)≥0 implies q≥x. Therefore, the average inventory cost at OW is $\mathrm{Inv}_{w}^{1} = \frac{c_{h}}{T} [ \int_{0}^{T} \{ q- \frac{xt}{T} \} dt ] = c_{h} [ q- \frac{x}{2} ]$.

2.2 B.4 Case 2: when shortage occurs

In this situation, q≤x, the on-hand inventory and shortage are as follows:

$$I_{w} ( t ) =q- \frac{xt}{T},\quad 0\leq t\leq t_{w} \ \mbox{with } I_{w} ( t_{w} ) =0 $$

and

$$I_{s} ( t ) = \frac{x}{T} ( t- t_{w} ),\quad t_{w} \leq t\leq T\ \mbox{with } I_{s} ( T ) \leq0. $$

Now, I _w (t _w )=0 implies t _w =qT/x. Therefore, the average inventory and shortage are:

$$\mathrm{Inv}_{w}^{2} = \frac{c_{h}}{T} \biggl[ \int _{0}^{t_{w}} \biggl\{ q- \frac{xt}{T} \biggr\} dt \biggr] = c_{h} \frac{q^{2}}{2x}\quad \mbox{and}\quad \mathrm{Inv}_{s}^{2} = \frac{c_{s}}{T} \int _{t w}^{T} \frac{x}{T} ( t- t_{w} ) dt = \frac{c_{s}}{2} x \biggl( 1- \frac{q}{x} \biggr)^{2}. $$

The expected average cost, combining Case 1 and Case 2, we have

$$\begin{aligned} \mathit{EAC}_{2} ( q ) &= \int_{0}^{q} \bigl( \mathrm{Inv}_{w}^{1} \bigr) f ( x ) dx + \int _{q}^{\infty} \bigl( \mathrm{Inv}_{w}^{2} \bigr) f ( x ) dx + \int_{q}^{\infty} \mathrm{Inv}_{s}^{2} f ( x ) dx\\ & =c_{h} \int_{0}^{q} \biggl( q- \frac{x}{2} \biggr) f ( x ) dx + \frac{c_{h}}{2} q^{2} \int _{q}^{\infty} \frac{f(x)}{x} dx+ \frac{c_{s}}{ 2} \int_{q}^{\infty} x \biggl( 1- \frac{q}{x} \biggr)^{2} f ( x ) dx. \end{aligned}$$