A new supply chain distribution network design for two classes of customers using transfer recurrent neural network

Abstract

Supply chain management integrates planning and controlling of materials, information, and finances in a process which begins from suppliers and ends with customers. Optimal planning decisions made in such a distribution network usually include transportation, facilities location, and inventory. This study presents a new approach for considering customers’ differentiation in an integrated location-allocation and inventory control model using transfer recurrent neural network (RNN). In this study, a location and allocation problem is integrated with inventory control decisions considering two classes of strategic and non-strategic customers. For the first time, a novel transfer RNN is applied to estimate parameters in order to reach to a near optimal solution. The proposed mathematical model is multi-product, single-period, multi-transportation mode, and with multilevel capacity warehouses with two classes of customers based on a critical level policy. The transfer RNN approach is used to transfer knowledge from a similar domain to the problem domain in this study. The performance result is compared with the condition when no transfer learning approach is applied. The exact calculation method is demonstrated for small scale instances while hybrid meta-heuristic algorithms (Genetic and Simulated Annealing) developed for real size samples. Finally, a sensitivity analysis is carried out for different instances to evaluate the effect of different indexes on the running time and total cost value of the objective function.

Appendix

In this Appendix the property which used to derive Q_jl is explained where the total inventory cost is calculates as follows:

Total inventory System Costs = Order Cost + Reserve Cost + Shortage Cost + Purchase Cost.

According to Fig. 1:

$${\text{t}}_{{1}} = \frac{{Q_{jl} - \left( {C_{2} + B_{jl} } \right)}}{{D_{jl} }}$$

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$${\text{t}}_{{2}} = \frac{{C_{2} }}{{D_{jl} }}$$

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$${\text{t}}_{{3}} = \frac{{B_{jl} }}{{D_{jl} }}$$

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t₁ is the time of responding to all customers without any shortage. t₂ is the time when inventory reaches the level below critical level of non-strategic customers. Accordingly, in this period, non-strategic customers receive no response. t₃ is time of total inventory system shortage.

For Product I and Warehouse J, so:

$${\text{THC }} = \frac{{\left( {Q_{jl} - C_{2} } \right) \cdot \left( {{\text{t}}1 + {\text{t}}2} \right)}}{2} \cdot HC_{jl} = \frac{{\left( {Q_{jl} - C_{2} } \right)^{2} }}{{2.D_{jl} }} \cdot HC_{jl}$$

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$${\text{Shortage for Class 2 Customers}} = \frac{{\left( {B_{jl} + C_{2} } \right) \cdot \left( {{\text{t}}1 + {\text{t}}2} \right)}}{2} = \frac{{\left( {B_{jl} + C_{2} } \right)^{2} }}{{2 \cdot D_{jl} }}$$

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$${\text{Shortage for Class 1 Customers}} = \frac{{t_{3} \cdot B_{jl} }}{2} = \frac{{B_{jl}^{2} }}{{2 \cdot D_{jl} }}$$

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$${\text{TSC}} = {\text{TSC}}^{{2}} + {\text{TSC}}^{{\text{l}}} = (\frac{{\left( {B_{jl} + C_{2} } \right)^{2} }}{{2 \cdot D_{jl} }} \cdot {\text{P}}^{{2}} ) + (\frac{{B_{jl}^{2} }}{{2 \cdot D_{jl} }} \cdot {\text{p}}^{{1}} )$$

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$${\text{TOC }} = {\text{ OC}}_{{{\text{jl}}}}$$

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$${\text{TMC }} = {\text{ C}}_{{\text{j}}} \cdot {\text{Q}}_{{{\text{jl}}}}$$

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$${\text{N}} = \frac{1}{T} = \frac{{D_{jl} }}{{Q_{jl} }}$$

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$${\text{TIC}} = {\text{N}}*\left( {{\text{One}} - {\text{period Cost}}} \right)$$

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$${\text{TIC}} = {\text{C}}_{{\text{j}}} \cdot {\text{D}}_{{{\text{jl}}}} + \left( {\frac{{D_{jl} }}{{Q_{jl} }} \cdot {\text{OC}}_{{{\text{jl}}}} } \right) + \left( {\frac{{\left( {B_{jl} + C_{2} } \right)^{2} }}{{2Q_{jl} }} \cdot {\text{P}}^{{2}} } \right) + \left( {\frac{{B_{jl}^{2} }}{{2.Q_{jl} }} \cdot {\text{p}}^{{1}} } \right) + \left( {\frac{{\left( {Q_{jl} - C_{2} } \right)^{2} }}{{2 \cdot Q_{jl} }} \cdot HC_{jl} } \right)$$

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For Measuring Q_jl^*and B_jl^*, the above equation can be rewritten as follows:

$${\text{Equation 1}} = \frac{{\partial TIC\left( {Q_{jl} ,B_{jl} } \right)}}{{\partial \left( {Q_{jl} } \right)}} = 0$$

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$$Q_{jl}^{2} = \frac{1}{{HC_{jl} }}({2} \cdot {\text{D}}_{{{\text{jl}}}} \cdot {\text{OC}}_{{{\text{jl}}}} + {\text{ HC}}_{{{\text{jl}}}} \cdot B_{jl}^{2} + \left( {B_{jl} + C_{2} } \right)^{2} \cdot {\text{P}}^{{2}} + B_{jl}^{2} \cdot {\text{p}}^{{1}}$$

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$${\text{Equation 2}} = \frac{{\partial TIC\left( {Q_{jl} ,B_{jl} } \right)}}{{\partial \left( {B_{jl} } \right)}} = 0$$

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$${\text{Q}}_{{{\text{jl}}}} = {\text{ B}}_{{{\text{jl}}}} + \frac{{B_{jl} + C_{2} }}{{HC_{jl} }} \cdot {\text{P}}^{{2}} + \frac{{B_{jl} }}{{HC_{jl} }} \cdot {\text{p}}^{{1}}$$

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Q_jl^*and B_jl^*are calculated by means of above equations and insertion of Eq. 2 in Eq. 1 results in:

$$\begin{gathered} B_{jl}^{2} (P^{b2} + P^{l2} + {2} \cdot {\text{P}}^{{2}} \cdot {\text{p}}^{{1}} + {2} \cdot {\text{P}}^{{2}} + {2} \cdot {\text{p}}^{{1}} - {\text{HC}}_{{{\text{jl}}}} \cdot {\text{P}}^{{2}} - {\text{HC}}_{{{\text{jl}}}} \cdot {\text{p}}^{{1}} ) \hfill \\ \;\; + {2} \cdot {\text{B}}_{{{\text{jl}}}} ({\text{C}}_{{2}} \cdot P^{b2} + {\text{C}}_{{2}} \cdot {\text{P}}^{{2}} \cdot {\text{p}}^{{1}} + {\text{C}}_{{2}} \cdot {\text{P}}^{{2}} - {\text{HC}}_{{{\text{jl}}}} \cdot {\text{C}}_{{2}} \cdot {\text{P}}^{{2}} ) \hfill \\ \;\; + (C_{2}^{2} \cdot P^{b2} - {2} \cdot {\text{D}}_{{{\text{jl}}}} \cdot {\text{OC}}_{{{\text{jl}}}} \cdot {\text{HC}}_{{{\text{jl}}}} - {\text{HC}}_{{{\text{jl}}}} \cdot C_{2}^{2} \cdot {\text{ P}}^{{2}} ) \, = \, 0 \hfill \\ \end{gathered}$$

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By calculating the above second-order equation, B_jl^* can be estimated as:

$$\frac{\begin{gathered} - \left( {C2 \cdot P^{b2} + C2 \cdot P^{b} \cdot P^{l} + C2 \cdot P^{b} - C2 \cdot P^{b} \cdot HC_{jl} } \right) \hfill \\ \;\; + \sqrt {4 \cdot C_{2}^{2} \cdot P^{b2} + 8 \cdot P^{b2} \cdot D_{jl} \cdot OC_{jl} \cdot HC_{jl} + 8 \cdot P^{l2} \cdot D_{jl} \cdot OC_{jl} \cdot HC_{jl} + 4 \cdot C_{2}^{2} \cdot HC_{jl} \cdot P^{b} \cdot P^{l2} + 8 \cdot P^{b} } \hfill \\ \;\; \cdot P^{l} \cdot D_{jl} \cdot OC_{jl} \cdot HC_{jl} + 16 \cdot P^{b} \cdot D_{jl} \cdot OC_{jl} \cdot HC_{jl} + 16 \cdot P^{l} \cdot D_{jl} \cdot OC_{jl} \cdot HC_{jl} \hfill \\ \;\; + 8 \cdot C_{2}^{2} \cdot HC_{jl} \cdot P^{b} \cdot P^{l} + 4 \cdot C_{2}^{2} \cdot HC_{jl} \cdot P^{l} \cdot P^{b2} - 8 \cdot P^{b} \cdot D_{jl} \cdot OC_{jl} \cdot HC_{jl}^{2} \hfill \\ \;\; - 8 \cdot P^{l} \cdot D_{jl} \cdot OC_{jl} \cdot HC_{jl}^{2} - 4 \cdot C_{2}^{2} \cdot HC_{jl}^{2} \cdot P^{b} \cdot P^{l} \hfill \\ \end{gathered} }{{2 \cdot \left( {P^{b2} + P^{l2} + 2 \cdot P^{b} + 2 \cdot P^{l} - HC_{jl} \cdot P^{b} - HC_{jl} \cdot P^{l} } \right)}}$$

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Also this B_jl^* is useless according to increasing size of problem. It is better to consider B_jl as a variable.

Q_jl^* can be calculated in Eq. 18 by using B_jl^*

$${\text{Q}}_{{{\text{jl}}}} = {\text{B}}_{{{\text{jl}}}} + \frac{{B_{jl} + C_{2} }}{{HC_{jl} }} \cdot {\text{P}}^{{2}} + \frac{{B_{jl} }}{{HC_{jl} }} \cdot {\text{p}}^{{1}}$$

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