Economic and Environmental Performance of the Firm: Synergy or Trade-Off? Insights from the EOQ Model

Abstract

Over the last decades, corporations are increasingly expected to perform well on the triple bottom line: People, Planet and Profit. However, both in academia and in practice, there is no consensus on the feasibility of doing good and doing well simultaneously. The traditional view is that there is an unavoidable trade-off between the social and environmental performance of an organisation and its profitability. The other school of thought claimed that breaking the trade-off and creating a synergy, is not only desired but actually feasible. In this chapter, the validity of both views is tested by using a multi-objective approach to a variant of the well-known EOQ model. It is demonstrated that both views are not contradictory but valid under different conditions. As such this chapter helps to reach a better understanding of the factors that drive trade-offs and synergy behaviour of the triple bottom line measures.

Appendix

In this appendix the formal proofs of the various observations will be given. To do so, the partial derivatives of ${\text{TC}}\left( {Q_{\text{W}}^{*} } \right)$ and ${\text{TE}}\left( {Q_{\text{W}}^{*} } \right)$ with respect to the parameter at hand will be determined. Note that when either both partial derivates are positive or both are negative, the two objectives are aligned and if both partial derivatives have opposite signs, the objectives are conflicting. Throughout this appendix, the assumptions $\Updelta \ne 0$, 0 < α < 1, and c _e, γ _H, γ _O, E _O, E _H > 0 will be used.

Proof of Observation 1

Note that

$$ \frac{\partial }{\partial \alpha }{\text{TC}}\left( {Q_{\text{W}}^{*} } \right) = \sqrt {\tfrac{1}{8}D} \left( {\frac{{C_{\text{H}} }}{{B_{\text{H}} }} - \frac{{C_{\text{O}} }}{{B_{\text{O}} }}} \right)\left( {\frac{{\gamma_{\text{O}} \sqrt {B_{\text{H}} } }}{{\sqrt {B_{\text{O}} } }} - \frac{{\gamma_{\text{H}} \sqrt {B_{\text{O}} } }}{{\sqrt {B_{\text{H}} } }}} \right). $$

It is easy to see that

$$ \left( {\frac{{C_{\text{H}} }}{{B_{\text{H}} }} - \frac{{C_{\text{O}} }}{{B_{\text{O}} }}} \right) > 0 \;{\text{iff}}\;\left( {1 - \alpha } \right)\Updelta < 0, $$

and

$$ \left( {\frac{{\gamma_{\text{O}} \sqrt {B_{\text{H}} } }}{{\sqrt {B_{\text{O}} } }} - \frac{{\gamma_{\text{H}} \sqrt {B_{\text{O}} } }}{{\sqrt {B_{\text{H}} } }}} \right) > 0\;{\text{iff}}\;\Updelta > 0. \, $$

Similarly, note that

$$ \frac{\partial }{\partial \alpha }{\text{TE}}(Q_{\text{W}}^{*} ) = \sqrt {\tfrac{1}{8}D} \left( {\frac{{E_{\text{H}} }}{{B_{\text{H}} }} - \frac{{E_{\text{O}} }}{{B_{\text{O}} }}} \right)\left( {\frac{{\gamma_{\text{O}} \sqrt {B_{\text{H}} } }}{{\sqrt {B_{\text{O}} } }} - \frac{{\gamma_{\text{H}} \sqrt {B_{\text{O}} } }}{{\sqrt {B_{\text{H}} } }}} \right), $$

and

$$ \left( {\frac{{E_{\text{H}} }}{{B_{\text{H}} }} - \frac{{E_{\text{O}} }}{{B_{\text{O}} }}} \right) > 0\;{\text{iff}}\;\alpha \Updelta > 0 \, $$

It follows that $\frac{\partial }{\partial \alpha }{\text{TE}}\left( {Q_{\text{W}}^{*} } \right) > 0$ and $\frac{\partial }{\partial \alpha }{\text{TC}}\left( {Q_{\text{W}}^{*} } \right) < 0$ for all 0 < α < 1. ■

Proof of Observation 2

Using

$$ \frac{\partial }{{\partial E_{O} }}{\text{TC}}\left( {Q_{\text{W}}^{*} } \right) = c_{\text{e}} \sqrt {\tfrac{1}{8}D} \left( {\frac{{C_{\text{H}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} + 2\frac{{\sqrt {B_{\text{H}} } }}{{\sqrt {B_{\text{O}} } }} - \frac{{C_{\text{O}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{O}}^{2} }}} \right), $$

it is easy to verify that this partial derivate is positive iff Condition (13) is fulfilled. Similarly,

$$ \frac{\partial }{{\partial E_{H} }}{\text{TC}}\left( {Q_{\text{W}}^{*} } \right) = c_{\text{e}} \sqrt {\tfrac{1}{8}D} \left( {\frac{{C_{\text{O}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} + 2\frac{{\sqrt {B_{\text{O}} } }}{{\sqrt {B_{\text{H}} } }} - \frac{{C_{\text{H}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{H}}^{2} }}} \right), $$

which is positive iff condition (14) is fulfilled. Furthermore,

$$ \frac{\partial }{{\partial E_{\text{O}} }}{\text{TE}}\left( {Q_{\text{W}}^{*} } \right) = \sqrt {\tfrac{1}{8}D} \left( {\frac{{c_{\text{e}} E_{\text{H}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} + 2\frac{{\sqrt {B_{\text{H}} } }}{{\sqrt {B_{\text{O}} } }} - \frac{{c_{\text{e}} E_{\text{O}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{O}}^{2} }}} \right), $$

and

$$ \frac{\partial }{{\partial E_{\text{H}} }}{\text{TE}}\left( {Q_{\text{W}}^{*} } \right) = \sqrt {\tfrac{1}{8}D} \left( {\frac{{c_{\text{e}} E_{\text{O}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} + 2\frac{{\sqrt {B_{\text{O}} } }}{{\sqrt {B_{\text{H}} } }} - \frac{{c_{\text{e}} E_{\text{H}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{H}}^{2} }}} \right). $$

Note that

$$ \left( {\frac{{c_{\text{e}} E_{\text{H}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} + 2\frac{{\sqrt {B_{\text{H}} } }}{{\sqrt {B_{\text{O}} } }} - \frac{{c_{\text{e}} E_{\text{O}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{O}}^{2} }}} \right) < 0\quad {\text{iff}} $$

$$ E_{\text{O}} < \frac{{ - \alpha \gamma_{\text{O}} }}{{c_{\text{e}} }}\left( {\frac{{2B_{\text{H}} + c_{\text{e}} E_{\text{H}} }}{{B_{\text{H}} + c_{\text{e}} E_{\text{H}} }}} \right) < 0 $$

and

$$ \left( {\frac{{c_{\text{e}} E_{\text{O}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} + 2\frac{{\sqrt {B_{\text{O}} } }}{{\sqrt {B_{\text{H}} } }} - \frac{{c_{\text{e}} E_{\text{H}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{H}}^{2} }}} \right) < 0\quad {\text{iff}} $$

$$ E_{\text{H}} < \frac{{ - \alpha \gamma_{\text{H}} }}{{c_{\text{e}} }}\left( {\frac{{2B_{\text{O}} + c_{\text{e}} E_{\text{O}} }}{{B_{\text{O}} + c_{\text{e}} E_{\text{O}} }}} \right) < 0, $$

from which Observation 2 follows. ■

Proof of Observation 3

Note that

$$ \frac{\partial }{{\partial \gamma_{\text{O}} }}{\text{TC}}\left( {Q_{\text{W}}^{*} } \right) = \sqrt {\tfrac{1}{8}D} \left( {\frac{{\alpha C_{\text{H}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} + 2\frac{{\sqrt {B_{\text{H}} } }}{{\sqrt {B_{\text{O}} } }} - \frac{{\alpha C_{\text{O}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{O}}^{2} }}} \right), $$

which is negative iff

$$ \gamma_{\text{O}} < \frac{{ - c_{\text{e}} E_{\text{O}} \left( {\alpha C_{\text{H}} + \left( {2 - \alpha )B_{\text{H}} } \right)} \right)}}{{\alpha \left( {\alpha C_{\text{H}} + B_{\text{H}} } \right)}} < 0. $$

Similarly,

$$ \frac{\partial }{{\partial \gamma_{\text{H}} }}{\text{TC}}\left( {Q_{\text{W}}^{*} } \right) = \sqrt {\tfrac{1}{8}D} \left( {\frac{{\alpha C_{\text{O}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} + 2\frac{{\sqrt {B_{\text{O}} } }}{{\sqrt {B_{\text{H}} } }} - \frac{{\alpha C_{\text{H}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{H}}^{2} }}} \right), $$

is negative iff

$$ \gamma_{\text{H}} < \frac{{ - c_{\text{e}} E_{\text{H}} \left( {\alpha C_{\text{O}} + \left( {2 - \alpha )B_{\text{O}} } \right)} \right)}}{{\alpha \left( {\alpha C_{\text{O}} + B_{\text{O}} } \right)}} < 0. $$

Also,

$$ \frac{\partial }{{\partial \gamma_{\text{O}} }}{\text{TE}}\left( {Q_{\text{W}}^{*} } \right) = \alpha \sqrt {\tfrac{1}{8}D} \left( {\frac{{E_{\text{H}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} - \frac{{E_{\text{O}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{O}}^{2} }}} \right), $$

is positive iff $\alpha \Updelta > 0$, and

$$ \frac{\partial }{{\partial \gamma_{\text{H}} }}{\text{TE}}\left( {Q_{\text{W}}^{*} } \right) = \alpha \sqrt {\tfrac{1}{8}D} \left( {\frac{{E_{\text{O}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} - \frac{{E_{\text{H}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{H}}^{2} }}} \right), $$

is positive if $\alpha \Updelta < 0$. Observation 3 follows immediately. ■

The proofs of Observation 4 and 5(ii) are trivial and will be omitted.

Proof of Observation 5 (i)

Note that

$$ \begin{aligned} \frac{\partial }{{\partial c_{\text{e}} }}{\text{TC}}\left( {Q_{\text{W}}^{*} } \right) = & \,\sqrt {\tfrac{1}{8}D} \left( {\frac{{C_{0} E_{\text{H}} + C_{\text{H}} E_{\text{O}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} + 2E_{\text{O}} \frac{{\sqrt {B_{\text{H}} } }}{{\sqrt {B_{\text{O}} } }}} \right) + \\ & \sqrt {\tfrac{1}{8}D} \left( {2E_{\text{H}} \frac{{\sqrt {B_{\text{O}} } }}{{\sqrt {B_{\text{H}} } }} - \frac{{C_{\text{O}} E_{\text{O}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{O}}^{2} }} - \frac{{C_{\text{H}} E_{\text{H}} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{H}}^{2} }}} \right). \\ \end{aligned} $$

Careful analysis shows that this is positive iff

$$ 2B_{\text{O}} B_{\text{H}} \left( {E_{\text{O}} B_{\text{H}} + E_{\text{H}} B_{\text{O}} } \right) + \alpha \left( {1 - \alpha } \right)\Updelta^{2} > 0. $$

Furthermore,

$$ \frac{\partial }{{\partial c_{\text{e}} }}{\text{TE}}\left( {Q_{\text{W}}^{*} } \right) = \sqrt {\tfrac{1}{8}D} \left( {\frac{{2E_{\text{H}} E_{\text{O}} }}{{\sqrt {B_{\text{H}} B_{\text{O}} } }} - \frac{{E_{\text{O}}^{ 2} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{O}}^{2} }} - \frac{{E_{\text{H}}^{ 2} \sqrt {B_{\text{H}} B_{\text{O}} } }}{{B_{\text{H}}^{2} }}} \right), $$

which is negative iff $\alpha^{2} \Updelta^{2} > 0$. Together these results constitute Observation 5(i). ■