Abstract
This work considers a multi-period multi-echelon multi-product dynamic supply chain network design problem for both strategic and tactical decisions. Strategic decisions include the determination of the locations of the facilities, the capacities of the open facilities, and the capacities of the dedicated and the flexible technologies, and the tactical decisions include the determination of the prices of the products, the flows of materials and products among the locations, and the quantities of the products to produce in each plant in each period. The demand at each customer zone is modeled by a logit price-response function and is approximated by a piecewise linear function. A mixed-integer nonlinear programming model is developed to maximize the expected net present value while making these decisions. A robust possibilistic stochastic programming approach is used to deal with price-sensitive demands under hybrid, i.e., disruption and operational, uncertainties. The model considers the effect of robustness level on technology selection and price decisions, and enables tradeoffs between the robustness level and the expected net present value. The applicability of the model and the performance of the solution approach are examined through computational experiments. The results show that the optimal technology investment is a function of the types of uncertainties and the flexible-to-dedicated technology cost ratio. The results also show a significant advantage of the proposed robust possibilistic stochastic programming model over the other models in the simultaneous controllability of the possibilistic and scenario variabilities. The sensitivity of some key parameters in the model are analyzed in the computational experiments.
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Appendices
Appendices
Appendix A: The basic possibilistic stochastic programming (PSP) model
The basic PSP model for the considered SCND problem, formulated in subsection 0, is as follows.
Constraints (17)–(20), (22)–(25) and (28)–(33).
In constraints (A.2)–(A.6), \(\alpha\) indicates the minimum confidence level set by the decision makers for satisfying the possibilistic chance constraints under each scenario. For notational simplicity, the compact form of the basic PSP model stated as follows is used:
The notations in this compact form (A.7) are further explained in the following. The vectors \({{\varvec{\upupsilon}}}^{s}\) and \({\mathbf{y}}^{s}\) represent the continuous variables under each scenario and \({\mathbf{x}}\) represents the binary variables. A scenario is indexed by \(s \in S\) and is defined as a realization of all the fuzzy scenario-based variables. Each scenario \(s\) has an associated probability \(\pi^{s}\), with \(\sum\limits_{s \in S} {\pi^{s} } = 1\).
The coefficients of the constraints are represented by the matrices \({\mathbf{A}}\), \({\mathbf{B}}\) and \({\mathbf{D}}\), and the parameters are represented by the vectors \({\mathbf{c}}\), \({\mathbf{d}}\) and \({\mathbf{f}}\), respectively. The coefficients in \({\mathbf{A}}\) and \({\mathbf{B}}\) are known in advance. The price endpoints are represented by the vector \({\mathbf{Pr}}\). The elements of \({\mathbf{D}}\), \({\mathbf{f}}\) and \({\mathbf{c}}\) are the fuzzy variables describing the capacities, the fixed opening costs, and other costs, respectively, of each plant. Furthermore, the uncertain vector \({\mathbf{d}}\) represents the fuzzy scenario-based demands. Fuzzy coefficients corresponding to \({\mathbf{D}}\), \({\mathbf{f}}\) and \({\mathbf{c}}\) are represented by \({\tilde{\mathbf{D}}}\), \({\tilde{\mathbf{f}}}\) and \({\tilde{\mathbf{c}}}\), respectively, and the fuzzy scenario based variables corresponding to \({\mathbf{d}}\) are denoted by \({\tilde{\mathbf{d}}}^{s}\), where \({\mathbf{\tilde{f} = (f,\underline {f} ,\overline{f})}}_{{{\mathbf{LR}}}}\), \({\mathbf{\tilde{c} = (c,\underline {c} ,\overline{c})}}_{{{\mathbf{LR}}}}\) and \({\tilde{\mathbf{d}}}^{{\mathbf{s}}} {\mathbf{ = (d}}^{{\mathbf{s}}} {\mathbf{,\underline {d} }}^{{\mathbf{s}}} {\mathbf{,\overline{d}}}^{{\mathbf{s}}} {\mathbf{)}}_{{{\mathbf{LR}}}}\) for \(s \in S\) are described by LR trapezoidal fuzzy numbers.
Note that \(\tilde{F}^{s} ({{\varvec{\upupsilon}}}^{s} ,{\mathbf{x}}^{s} ,{\mathbf{y}}^{s} ) = \rho {\tilde{\mathbf{d}}}^{s} {{\varvec{\upupsilon}}}^{s} - {\tilde{\mathbf{f}}\mathbf{x}}^{s} - {\tilde{\mathbf{c}}\mathbf{y}}^{s}\) is a fuzzy profit of the solution \(({{\varvec{\upupsilon}}}^{s} ,{\mathbf{x}}^{s} ,{\mathbf{y}}^{s} )\), under scenario \(s\). The fuzzy profit under each scenario, i.e., \(\tilde{F}^{s}\), is stated in (A.8)
The total fuzzy profit, i.e., \(\tilde{F}\), is stated in (A.9)
Given a scenario \(s\), the possibilistic mean of the profit, denoted by \(F^{s}\), can be computed using (A.10)
Moreover, the weighted possibilistic mean of the objective function over all scenarios, denoted by \(F\), can be calculated using (A.11)
Hence, by assuming \({\tilde{\mathbf{D}}} = ({\mathbf{D}},{\mathbf{\underline {D} }},{\overline{\mathbf{D}}})_{LR}\), the basic crisp non-robust PSP model is constructed as follows:
Appendix B: The RSP and RPP models
The robust stochastic programming (RSP) model, according to Mulvey et al. [45], is stated as follows for the SCND problem:
s.t. Constraints (17)–(20), (22)–(25), (28)–(33), (35)–(40), and (43)–(45).
The robust possibilistic programming (RPP) model, according to Pishvaee et al. [16], is stated as follows:
s.t.Constraints (17)–(20), (22)–(25), (28)–(33), and (35)–(40).
The RSP and the RPP models can control the scenario and possibilistic variabilities in the objective function, respectively.
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Farrokh, M., Ahmadi, E. & Sun, M. A robust stochastic possibilistic programming model for dynamic supply chain network design with pricing and technology selection decisions. OPSEARCH 60, 1082–1120 (2023). https://doi.org/10.1007/s12597-023-00643-2
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DOI: https://doi.org/10.1007/s12597-023-00643-2