A robust possibilistic programming approach for blood supply chain network design in disaster relief considering congestion

Abstract

We propose a fuzzy-robust multi-objective optimization model for blood supply chain network design in disaster relief. The problem aims to minimize (1) the expected total cost of the system, (2) the implicit cost associated with patients’ waiting in hospitals, and (3) the unsatisfied demands. To model patients’ waiting and to improve the effectiveness of relief activities by decreasing congestion in hospitals, a queuing framework is utilized. A robust possibilistic programming approach is applied to capture the epistemic uncertainty of parameters and provide risk-averse solutions for policymakers. Given the conflicting objectives sought, we use multi-objective decision-making techniques, including the Torabi and Hasini approach and $\varepsilon$-constrained method. Experimentation on a real-life case study confirms that the proposed framework can help decision-makers adopt suitable strategies for planning blood collection and delivery in emergencies. The results obtained show that substantial improvements in the service quality could be achieved at modest increases in the network cost by properly locating facilities using our model. We also observed that both the service level and the service quality remain relatively stable over a wide range of model parameter values, highlighting the robust nature of the proposed approach. Based on our findings, we recommend implementing risk-pooling strategies, setting realistic service level targets, and incorporating service equity as an objective. Although our model accounts for many realistic considerations, it overlooks some relevant issues like network disruptions to remain tractable, which is worthy of further investigation in future research.

Appendices

Appendix 1

See Tables 8, 9, 10, 11 and 12.

Table 8 Comparison of present paper with the most relevant papers in the literature

Table 9 Evaluation of the changes of $\eta$ in TH approach (${\theta _1} = 0.4,\,{\theta _2} = 0.3$ and ${\theta _3} = 0.3,\,\xi = 0.1,\,\tau = 2\,units$)

Table 10 The summary of sensitivity analysis on the average blood quantity required per victim ($\tau$), ($\xi =0.1$)

Table 11 Sensitivity analysis on the parameter values of average injury rate, capacity of temporary and permanent facilities and maximum blood supply of donor groups

Table 12 Sensitivity analysis on the parameter values of average injury rate, capacity of temporary and permanent facilities, maximum blood supply of donor groups and hospital service rates

Appendix 2

To reformulate the expected values in (14), we use the approach proposed by Dubois and Prade (1987), and chance constraints in (14) are reformulated based on the approaches proposed by Liu and Iwamura (1998); Inuiguchi and Ramık (2000). Hence, (14) can be reformulated as follows:

$$\begin{aligned} \min&\,\,\,E[{Z_1}] = \frac{{({C_{(1)}} + {C_{(2)}} + {C_{(3)}} + {C_{(4)}})}}{4}y + \frac{{({W_{(1)}} + {W_{(2)}} + {W_{(3)}} + {W_{(4)}})}}{4}x \nonumber \\ \min&\,\,\,E[{Z_2}] = x \nonumber \\ \min&\,\,\,E[{Z_3}] = \frac{{({V_{(1)}} + {V_{(2)}} + {V_{(3)}} + {V_{(4)}})}}{4} - x \nonumber \\ \text {s.t.}\;&Ax = 0 \nonumber \\&Bx \le [\alpha {O_{(1)}} + (1 - \alpha ){O_{(2)}}]y \nonumber \\&Lx \le \beta {D_{(1)}} + (1 - \beta ){D_{(2)}} \nonumber \\&Gy \le \psi {S_{(1)}} + (1 - \psi ){S_{(2)}} \nonumber \\&y \in \{ 0,\,1\},\,x \ge 0{.} \end{aligned}$$

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Appendix 3

Formulation (15) includes the product of binary (y) and continuous ($\alpha$) variables, which makes the model nonlinear. The nonlinear term can be converted into the linear one by defining a new variable and adding some constraints to the model. By defining the auxiliary variable $\kappa = \alpha \times y$, the abovementioned nonlinear model could be converted to an equivalent linear one as follows:

$$\begin{aligned} \min \,\,\,&E[{Z_1}] + \delta _{1} ({Z_{1(Max)}} - E[{Z_1}]) + {\varphi _1}[\kappa {O_{(1)}} + (y - \kappa ){O_{(2)}} - y{O_{(1)}}]+ {\varphi _2}[\beta {D_{(1)}} + (1 - \beta ){D_{(2)}} - {D_{(1)}}] \nonumber \\&+ {\varphi _3}[\psi {S_{(1)}} + (1 - \psi ){S_{(2)}} - {S_{(1)}}]\nonumber \\ \min \,\,\,&E[{Z_2}] + {\varphi _1}[\kappa {O_{(1)}} + (y - \kappa ){O_{(2)}} - y{O_{(1)}}]+ {\varphi _2}[\beta {D_{(1)}} + (1 - \beta ){D_{(2)}} - {D_{(1)}}]+ {\varphi _3}[\psi {S_{(1)}} + (1 - \psi ){S_{(2)}}\nonumber \\&- {S_{(1)}}]\nonumber \\ \min \,\,\,&E[{Z_3}] + {\delta _3}({Z_{3(Max)}} - E({Z_3})) + {\varphi _1}[\kappa {O_{(1)}} + (y - \kappa ){O_{(2)}} - y{O_{(1)}}]+ {\varphi _2}[\beta {D_{(1)}} + (1 - \beta ){D_{(2)}} - {D_{(1)}}] \nonumber \\&+ {\varphi _3}[\psi {S_{(1)}} + (1 - \psi ){S_{(2)}} - {S_{(1)}}]\nonumber \\ \text {s.t.}\;&Ax = 0\nonumber \\&Bx \le \kappa {O_{(1)}} + (y - \kappa ){O_{(2)}}\nonumber \\&Lx \le \beta {D_{(1)}} + (1 - \beta ){D_{(2)}}\nonumber \\&Gy \le \psi {S_{(1)}} + (1 - \psi ){S_{(2)}}\nonumber \\&\kappa \, \le M\,y\nonumber \\&\kappa \,\, \ge M\,(y\, - 1) + \alpha \nonumber \\&\kappa \le \,\alpha \nonumber \\&y \in \{ 0,\,1\},\,x,\,\kappa \ge 0,\,\,0.5 \le \alpha ,\,\beta ,\,\psi \le 1. \end{aligned}$$

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Here, the scalar M is a big number.