A sustainable supply chain model for time-varying deteriorating items under the promotional cost-sharing policy and three-level trade credit financing

Abstract

This research develops a sustainable supply chain model for time-varying deteriorating items with customers’ credit period, customers’ credit amount, promotional efforts, and selling price-dependent demand. The model incorporates joint policies of promotional cost-sharing, three-level trade credit financing, and carbon tax. Under three-level trade credit financing, the supplier and the wholesaler offer some credit periods to the wholesaler and the retailer, respectively. As a result of this opportunity, the retailer allows customers to delay the payment of some portion of the total purchased amount. Here, shortages are assumed to occur in the form of partial backorder. The main objective of this investigation is to minimize the carbon emissions and maximize the joint profit of the retailer and the wholesaler simultaneously. To achieve this, the model is formulated as a Signomial Geometric Programming problem and solved efficiently using a global optimization method. The performance of the developed model and solution method is evaluated through several numerical examples and sensitivity analysis, providing valuable managerial insights. The computed results reveal that the optimal selling price varies depending on the nature of deterioration rates. Constant functions result in higher prices compared to linear and three-parameter Weibull functions. Coordination strategy and promotional cost-sharing policy among supply chain partners are shown to impact profits positively. Additionally, the wholesaler's credit period is a crucial factor influencing pricing decisions, logistics operations, and carbon emissions. The findings further demonstrate that extending the wholesaler's credit period under a carbon tax policy leads to a 26% increase in total joint profit, a 10% decrease in the wholesaler's carbon emissions, and a 21% decrease in the retailer's carbon emissions.

Appendices

Appendix 1

Referring to Fig. 4, the wholesaler’s initial inventory level for cycle $n$ is:

$$I_{w}^{n} \left( 0 \right) = Q_{w}^{n} = 0$$

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The wholesaler’s initial inventory level for the cycle $(n-1)$ is calculated as follows:

$$I_{w}^{n - 1} \left( 0 \right) = Q_{w}^{n - 1} = q\exp \left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)$$

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Also, the wholesaler’s initial inventory level for the cycle $(n-2)$ is calculated as follows:

$$I_{w}^{n - 2} \left( 0 \right) = Q_{w}^{n - 2} = \left( {q + Q_{w}^{n - 1} } \right)\exp \left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)$$

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By putting (A.2) into (A.3), we have:

$$I_{w}^{n - 2} \left( 0 \right) = Q_{w}^{n - 2} = q\left[ {\exp \left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) + \left( {\exp \left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)} \right)^{2} } \right]$$

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Following this manner, the wholesaler’s initial inventory level for the 1st cycle is calculated as follows:

$$\begin{aligned} I_{w}^{1} \left( 0 \right) = Q_{w}^{1} = & q\left[ {\exp \left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) + ... + \left( {\exp \left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)} \right)^{n - 1} } \right] \\ = & q\exp \left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)\left[ {\frac{{1 - \exp \left( {\left( {n - 1} \right)\left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)} \right)}}{{1 - \exp \left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)}}} \right] \\ \end{aligned}$$

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Appendix 2

As shown in Fig. 4, the deterioration amount during the first cycle can be expressed as follows:

$$C_{1\,} \theta_{w} \left( {T_{w} } \right) = Q_{w}^{1} - Q_{w}^{1} \exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)$$

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The deterioration amount during the second cycle can be calculated as follows:

$$\begin{aligned} C_{2\,} \theta_{w} \left( {T_{w} } \right) = & Q_{w}^{2} - Q_{w}^{2} \exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) \\ = & \left[ {Q_{w}^{1} \exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) - q} \right] - \left[ {Q_{w}^{1} \exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) - q} \right]\exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) \\ = & \left[ {Q_{w}^{1} \exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) - q} \right] - \left[ {Q_{w}^{1} \exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) - q} \right]\exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) \\ \end{aligned}$$

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The deterioration amount during the third cycle can be calculated as follows:

$$\begin{aligned} C_{3\,} \theta_{w} \left( {T_{w} } \right) = & Q_{w}^{3} - Q_{w}^{3} \exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) \\ = & \left[ {Q_{w}^{1} \exp \left( { - \,2\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) - q} \right] - \left[ {Q_{w}^{1} \exp \left( { - \,3\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) - q\exp \left( { - \,2\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)} \right] \\ \end{aligned}$$

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The deterioration amount during the $\left(n-1\right)$ cycle can be calculated as follows:

$$\begin{gathered} C_{n - 1\,} \theta_{w} \left( {T_{w} } \right) = Q_{w}^{n - 1} - Q_{w}^{n - 1} \exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) \hfill \\ \quad \quad \quad \quad \quad = \left[ {Q_{w}^{1} \exp \left( { - \,\left( {n - 2} \right)\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) - q} \right] - \left[ {Q_{w}^{1} \exp \left( { - \,\left( {n - 1} \right)\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)} \right. \hfill \\ \left. {\quad \quad \quad \quad \quad \quad - q\exp \left( { - \left( {n - 2} \right)\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)} \right] \hfill \\ \end{gathered}$$

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Therefore, the wholesaler’s total deterioration amount during $n$ cycles of ${T}_{w}$ can be calculated using the following equation:

$$\begin{aligned} \sum\limits_{j = 1}^{n - 1} {C_{j\,} \theta_{w} \left( {T_{w} } \right)} = & Q_{w}^{1} - \left( {n - 2} \right)q - Q_{w}^{1} \exp \left( { - \left( {n - 1} \right)\left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)} \right) \\ & + \,q\left( {\exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) + \exp \left( { - \,2\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right) + ...\exp \left( { - \left( {n - 2} \right)\left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)} \right)} \right) \\ = & Q_{w}^{1} - \left( {n - 2} \right)q - Q_{w}^{1} \exp \left( { - \left( {n - 1} \right)\left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)} \right) \\ & + \,q\exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)\left[ {\frac{{1 - \exp \left( { - \left( {n - 2} \right)\left( {\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)} \right)}}{{1 - \exp \left( { - \,\theta_{1} \left( {\frac{{T_{w} }}{n}} \right)^{{\theta_{2} }} } \right)}}} \right] \\ \end{aligned}$$

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