Joint pricing and inventory decisions with carbon emission considerations, partial backordering and planned discounts

Abstract

Typical economic order quantity models of inventory feature demand rate as a constant parameter and do not allow for backordering. Furthermore, the purchasing cost of the ordered materials is considered constant. In reality, the demand rate is related to the unit purchasing cost and other factors, such as time and availability of products in the market. A quantity discount is regularly applied to encourage ordering more products by decreasing the price. In some situations, carbon dioxide emissions are carefully scrutinized and a program to handle these. Greenhouse gases are put in place. Hence, for this research, the rate of demand in the model was assumed proportional to the unit purchasing cost and partial backordering was allowed as a fixed parameter. Because plants emit greenhouse gases (carbon dioxide), we considered mitigation efforts. A mathematical model and computational procedures are shown with the solution algorithms that demonstrate the capability of the model. An example problem was solved with the model and sensitivity analysis was conducted to inform the managerial insights offered.

Appendix

There is an analogous formula for polynomials of degree three: The solution of

$$ ax^{3} + bx^{2} + cx + d = 0 $$

That is

$$\begin{aligned}& x = \sqrt[3]{{\left( {\frac{{ - b^{3} }}{{27a^{3} }} + \frac{bc}{{6a^{2} }} - \frac{d}{2a}} \right) + \sqrt {\left( {\frac{{ - b^{3} }}{{27a^{3} }} + \frac{bc}{{6a^{2} }} - \frac{d}{2a}} \right)^{2} + \left( {\frac{c}{3a} - \frac{{b^{2} }}{9a}} \right)^{3} } }}\\ &\quad + \sqrt[3]{{\left( {\frac{{ - b^{3} }}{{27a^{3} }} + \frac{bc}{{6a^{2} }} - \frac{d}{2a}} \right) - \sqrt {\left( {\frac{{ - b^{3} }}{{27a^{3} }} + \frac{bc}{{6a^{2} }} - \frac{d}{2a}} \right)^{2} + \left( {\frac{c}{3a} - \frac{{b^{2} }}{9a}} \right)^{3} } }} - \frac{b}{3a}\end{aligned}$$

This can be briefly written as

$$ x = \sqrt[3]{{q + \sqrt {q^{2} + \left( {r - p^{2} } \right)^{3} } }} + \sqrt[3]{{q - \sqrt {q^{2} + \left( {r - p^{2} } \right)^{3} } }} + p $$

where

$$ p = \frac{ - b}{3a} $$

$$ q = p^{3} + \frac{bc - 3ad}{{6a^{2} }} $$

$$ r = \frac{c}{3a} $$

This formulation is used to calculate the optimal value for $T$, which is a polynomials of degree three.