Energy considerations for the economic production quantity and the joint economic lot sizing

Abstract

The increased awareness on sustainability, along with several governments’ actions for setting greenhouse gas restrictions, has recently generated a relevant pressure on industries towards the improvement of environmental performances. Among the several aspects, companies’ focus is mainly on energy use, firstly for its relevant and direct impact on the total cost, and secondly for its environmental linkage. The aim of this contribution is the integration of the energy-related objectives in lot sizing, extending the economic production quantity from the manufacturer point of view and extending the joint economic lot size model from the single-vendor single-buyer supply chain perspective, so as to show how this approach can lead to a more sustainable production process. The present work proposes a novel framework for dimensioning production lot sizes, based on both economic and energy implications in processes characterized by a variable production rate. Furthermore, an increased attention for the sustainability of the production–inventory system is introduced by considering energy as a key factor in the lot sizing problem, due to the close link between energy and environmental concerns. A traditional agreement and a vendor managed inventory with consignment stock for the joint economic lot sizing have been investigated explicitly formulating energy in production aspects. A numerical example is also presented to compare the behaviour of the models.

Appendix 1

This section aims to demonstrate the existence of an optimal value for the decision variables which minimizes the average costs in the models developed in Sect. 4. The procedure for the evaluation of the existence of a global optimum is the same: firstly, the partial derivatives with respect to the decision variables are calculated; then, the Hessian matrix is obtained; and finally, the determinant of the Hessian is discussed.

1.1 EPQ model with energy requirement (Sect. 4.1)

The partial derivatives of the average cost defined in Eq. (3) with respect to $q$ and $p$ are hereafter defined:

$$ \frac{{\partial AC_{e} \left( {q,p} \right)}}{\partial q} = - \frac{AD}{{q^{2} }} + \frac{h}{2}\left( {1 - \frac{D}{p}} \right) $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial q^{2} }} = \frac{2AD}{{q^{3} }} $$

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$$ \frac{{\partial AC_{e} \left( {q,p} \right)}}{\partial p} = \frac{hDq}{{2p^{2} }} - \frac{{W_{0} eD}}{{p^{2} }} $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial p^{2} }} = - \frac{hDq}{{p^{3} }} + \frac{{2W_{0} eD}}{{p^{3} }} $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial q\partial p} = \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial p\partial q} = \frac{hD}{{2p^{2} }}. $$

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Then, the Hessian matrix is calculated to determine whether the average cost is convex in the decision variables, thus presenting a minimum:

$$ H = \left( {\begin{array}{*{20}c} {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial q^{2} }}} & {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial p\partial q}} \\ {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial q\partial p}} & {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial p^{2} }}} \\ \end{array} } \right) $$

$$ det\left( H \right) = - \frac{{D^{2} \left[ {h^{2} q^{3} + 8 A p\left( {h q - 2 e W_{0} } \right)} \right]}}{{4 p^{4} q^{3} }}. $$

If the Hessian is positive definite ($det\left( H \right) > 0$), then it is demonstrated that $AC_{e} \left( {q,p} \right)$ presents a global minimum.

1.2 JELS model under a traditional agreement with energy costs (Sect. 4.2)

The partial derivatives of the average costs defined in Eq. (7) with respect to $q$, $n$ and $p$ are:

$$ \frac{{\partial AC_{e} \left( {q,n,p} \right)}}{\partial q} = - \frac{{\left( {A_{1} + nA_{2} } \right)D}}{{nq^{2} }} + h_{1} \left( {\frac{D}{p} + \frac{{\left( {p - D} \right)n}}{2p}} \right) + \frac{{h_{2} - h_{1} }}{2} $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial q^{2} }} = - \frac{{2\left( {A_{1} + nA_{2} } \right)D}}{{q^{3} }} $$

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$$ \frac{{\partial AC_{e} \left( {q,p} \right)}}{\partial p} = h_{1} \frac{Dq}{{p^{2} }}\left( {\frac{n}{2} - 1} \right) - \frac{{W_{0} eD}}{{p^{2} }} $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial p^{2} }} = - h_{1} \frac{Dq}{{p^{3} }}\left( {n - 2} \right) + \frac{{2W_{0} eD}}{{p^{3} }} $$

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$$ \frac{{\partial AC_{e} \left( {q,n,p} \right)}}{\partial n} = - \frac{{A_{1} D}}{{n^{2} q}} + h_{1} \frac{{\left( {p - D} \right)q}}{2p} $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial n^{2} }} = \frac{{2A_{1} D}}{{n^{3} q}} $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial q\partial p} = \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial p\partial q} = h_{1} \frac{D}{{p^{2} }}\left( {\frac{n}{2} - 1} \right) $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial q\partial n} = \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial n\partial q} = \frac{{A_{1} D}}{{n^{2} q^{2} }} + h_{1} \frac{{\left( {p - D} \right)}}{2p} $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial n\partial p} = \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial p\partial n} = h_{1} \frac{Dq}{{2p^{2} }}. $$

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Then, the Hessian matrix is calculated, showing that the average cost curve is convex in the decision variables, thus presenting a global minimum:

$$ H = \left( {\begin{array}{*{20}c} {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial q^{2} }}} & {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial n\partial q}} & {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial p\partial q}} \\ {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial q\partial n}} & {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial n^{2} }}} & {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial p\partial n}} \\ {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial q\partial p}} & {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial n\partial p}} & {\frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial p^{2} }}} \\ \end{array} } \right). $$

If the Hessian is positive definite ($det\left( H \right) > 0$), then it is demonstrated that $AC_{e} \left( {q,n, p} \right)$ it presents a global minimum.

1.3 JELS model under a consignment stock agreement with energy costs (Sect. 4.3)

The partial derivatives of the average costs defined in Eq. (13) with respect to $q$, $n$ and $p$, that differs from the previous scenario, are defined below. The other derivatives are obtained starting from the related derivatives in the JELS model under a traditional agreement with energy costs and substituting $h_{1}$ with $h_{2}^{CS} .$

$$ \frac{{\partial AC_{e} \left( {q,n,p} \right)}}{\partial q} = - \frac{{\left( {A_{1} + nA_{2} } \right)D}}{{nq^{2} }} + h_{2}^{CS} \left( {\frac{D}{p} + \frac{{\left( {p - D} \right)n}}{2p}} \right) - \frac{{\left( {h_{2}^{CS} - h_{1}^{CS} } \right)D}}{2p} $$

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$$ \frac{{\partial AC_{e} \left( {q,p} \right)}}{\partial p} = h_{2}^{CS} \frac{Dq}{{p^{2} }}\left( {\frac{n}{2} - 1} \right) + \frac{{\left( {h_{2}^{CS} - h_{1}^{CS} } \right)qD}}{{2p^{2} }} - \frac{{W_{0} eD}}{{p^{2} }} $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{{\partial p^{2} }} = - h_{2}^{CS} \frac{Dq}{{p^{3} }}\left( {n - 2} \right) - \frac{{\left( {h_{2}^{CS} - h_{1}^{CS} } \right)qD}}{{p^{3} }} + \frac{{2W_{0} eD}}{{p^{3} }} $$

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$$ \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial q\partial p} = \frac{{\partial AC_{e}^{2} \left( {q,p} \right)}}{\partial p\partial q} = h_{2}^{CS} \frac{D}{{p^{2} }}\left( {\frac{n}{2} - 1} \right) + \frac{{\left( {h_{2}^{CS} - h_{1}^{CS} } \right)D}}{{2p^{2} }}. $$

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It should be noted that the Hessian matrix is calculated and discussed as in the previous model.