Table 2 Indices, parameters and decision variables.
From: Designing an integrated blockchain-enabled supply chain network under uncertainty
| Â | Descriptions |
|---|---|
Indices | |
 \(i\in I\) | Index of plants |
 \(j\in J\) | Index of warehouses |
 \(k\in K\) | Index of customers |
 \(t\in T\) | Index of iterations |
 \(q\in Q\) | Index of inputs used at each Decision-Making Unit (DMU) |
 \(p\in P\) | Index of outputs produced at each DMU |
 \(b\in B\) | Index of blocks generated by blockchain technology in the supply chain |
Parameters | |
 \({p}_{i}^{u}\) | Upper bound of produced quantities at plant i |
 \({p}_{i}^{l}\) | Lower bound of produced quantities at plant i |
 \({q}_{ij}^{u}\) | Upper bound of transported quantities from plant i to warehouse j |
 \({q}_{jk}^{u}\) | Upper bound of transported quantities from warehouse j to customer k |
 \({w}_{j}^{u}\) | Upper capacity of warehouse j |
 \({\beta }_{j}\) | Coefficient relating quantity at capacity at warehouse j |
 \({I}_{j}^{0}\) | Inventory level stored at warehouse j |
 \({c}_{i}^{p}\) | Production cost at plant i |
 \({c}_{ij}^{v}\) | Unit transportation cost of products transported from plant i to warehouse j |
 \({c}_{ij}^{f}\) | Route transportation cost of products transported from plant i to warehouse j |
 \({c}_{jk}^{v}\) | Unit transportation cost of products transported from warehouse j to customer k |
 \({c}_{jk}^{f}\) | Route transportation cost of products transported from warehouse j to customer k |
 \({f}_{j}^{c}\) | Installation cost of warehouse j |
 \({d}_{k}^{R}\) | Demand of customer k |
 \(\upeta\) | Level of service |
 \({f}_{b}^{pr}\) | Measure of progress made by the attacker in terms of its failure based on the number of blocks type b generated by independent warehouses |
 \({B}_{j}^{A}\) | Blockchain technology adoption parameter in warehouse j in order to create transparency in the supply chain |
 \(\underline{B}, \overline{B }\) | Minimum and maximum level of transparency expected by the supply chain manager |
 \({B}^{N}\) | Minimum number of warehouses participating in the blockchain |
 \(\rho\) | Conversion factor of the installation cost to the cost of using the blockchain |
 \(\gamma\) | Conversion factor of the variable transportation cost to the cost of using the blockchain |
 \({a}_{j}\) | Solutions with efficiency score greater than or equal to the threshold of the supply chain manager that are selected with \({\upxi }_{j}\) |
 \({I}_{jq}^{D}\) | Amount of input q for DMU j |
 \({O}_{jp}^{D}\) | Amount of output q for DMU j |
 \(\varepsilon\) | Non-Archimedean infinitesimal epsilon |
 \(E\left(\widetilde{\xi }\right), Var\left(\widetilde{\xi }\right)\) | Expected value and variance of random parameter |
 \(1-{\alpha }^{s}\) | Confidence level for chance constraint |
 \({Z}_{1-{\alpha }^{s}}\) | Inverse function of the standard normal cumulative distribution function |
 \({f}^{I,tr},{f}^{I,co}\) | Aspiration levels (ideal solutions) |
 \({f}^{N,tr},{f}^{N,co}\) | Aspiration levels (nadir solutions) |
 \({\theta }^{tr},{\theta }^{co}\) | Weight associated with each fuzzy goal |
Decision variables | |
 \({p}_{i}\) | Production quantity at plant i |
 \({q}_{ij}^{1\to 2}\) | Transported quantity from plant i to warehouse j |
 \({q}_{jk}^{2\to 3}\) | Transported quantity from warehouse j to customer k |
 \({w}_{j}\) | Capacity of warehouse j |
 \({g}_{k}\) | Percentage of unmet demand of customer k |
 \({x}_{ij}^{1\to 2}\) | 1 if the connection between plant i and warehouse j exists, 0 otherwise |
 \({x}_{jk}^{2\to 3}\) | 1 if the connection between warehouse j and customer k exists, 0 otherwise |
 \({y}_{j}\) | 1 if warehouse j will be installed, 0 otherwise |
 \({B}_{b}^{T}\) | Total number of blocks type b (related to the second layer of the supply chain that are extracted from independent warehouses |
 \({B}_{j}^{DN}\) | 1 if warehouse j is equipped with an IoT tool to produce the block, 0 otherwise |
 \({B}_{ij}^{1\to 2}\) | 1 if warehouse j and plant i participate to form the blockchain, 0 otherwise |
 \({B}_{jk}^{2\to 3}\) | 1 if warehouse j and customer k participate to form the blockchain, 0 otherwise |
 \({\upxi }_{j}\) | 1 if warehouse j will be installed under efficiency level a %, 0 otherwise |
 \({\vartheta }_{jq}\) | Weight assigned to input q for DMU j |
 \({\mu }_{jp}\) | Weight assigned to output p for DMU j |
 \({d}_{j}\) | Level of inefficiency of DMU j |
 \({\omega }_{j}\) | Level of efficiency of DMU \(j\) |
\({F}^{tr},{F}^{co}\) | Objective functions related to transparency maximization and costs minimization |
