Robust optimization for a multiple-priority emergency evacuation problem under demand uncertainty
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Abstract
This study examines a multiple-priority emergency evacuation optimization problem with time-dependent demand uncertainty. A multiple-priority dynamic traffic model—namely, the multiple-priority cell transmission model (MPCTM) —is developed to simulate the priority of network flows for emergency evacuation response. Moreover, a robust optimization approach is applied to formulate such an emergency evacuation response problem. The robust counterpart solutions of the proposed uncertainty model have been shown to be tractable, using the duality theorem. Finally, a real example of Ya’an earthquake emergency evacuation planning verifies the effectiveness of the proposed MPCTM.
Keywords
Emergency evacuation Multiple priority Cell transmission model Demand uncertainty Robust optimization1 Introduction
Today, large-scale disasters are occurring more frequently than in the past: since 1980, hundreds of natural catastrophes have occurred worldwide each year, and these disasters have claimed over two million lives and caused property losses worth 3,000 billion dollars (Wirtz et al. 2014). To ultimately mitigate the loss of human life and property, there are growing concerns about emergencyevacuation planning, which envisions the evacuation of people and materials from dangerous areas to safe locations during the initial phase of a disaster. In emergency evacuation planning, it is assumed that severely wounded people and important commodities will be moved as soon as possible. As such, they require priority treatment, which differs from general evacuation. This study focuses on a multiple-priority emergency evacuation problem.
Following large-scale disasters, the affected area generates an enormous evacuation traffic demand within a short time period; these typically comprise the wounded and various commodities. However, in reality, due to the lack of historical information, the evacuation traffic demand possesses highly dynamic uncertainty. Robust optimization method is based on a minmax formulation and produces robus solutions that are immune against data uncertainty. Furthermore, this uncertainty may not be addressed by general decisions, and this can result in reduced viability among those affected and induce serious secondary disasters. It is important to study a problem in evacuation traffic planning that is robust to uncertainty. Thus, our research adopts the robustoptimization approach to address the uncertain demand in emergency evacuation planning.
In this study, we examine a multiple-priority cell transmission model (MPCTM), based on a robust optimization approach that looks to provide a robust and computationally tractable solution to an emergency evacuation problem that features demand uncertainty. Meanwhile, the MPCTM—which provides a priority hierarchy—describes uncertain demand by using given uncertainty sets, and minimizes total transit cost in the worst-case-oriented decisions derived from the uncertainty sets. Numerical experiments relating to the Ya’an earthquake are shown to demonstrate the superior system performance of the MPCTM in emergency evacuation planning, and a robust solution outperforms a deterministic solution in coping with environmental uncertainty.
This study considers a multiple-priority network flows model for emergency evacuation response. The model incorporates sets of priorities into the cell transmission model (CTM), such that emergency vehicles are given higher priority for evacuation.
This study applies the robust optimization approach to a multiple-priority emergency evacuation problem with demand uncertainty, which belongs to a polyhedral uncertainty set. The approach demonstrates the realizations of demand uncertainty in the MPCTM.
This study proposes a real-world example (i.e., the Ya’an earthquake). The computational results illustrate the advantage of the proposed MPCTM and robust optimization. First, the MPCTM can minimize the transit time of the higher priority, thereby reducing the total transit cost. Second, compared to the deterministic solution, the robust solution may effectively reduce the cost. Third, the demand uncertainty level and budget level can impact the total cost, and the MPCTM can perform much better.
2 Literature review
This section presents a review of emergency evacuation network literature that establishes a framework for this research. The literature in this paper can be categorized into four streams: the emergency evacuation problem, the emergency evacuation problem with priority, the emergency evacuation problem in an uncertainty environment, and the application of robust optimization approach.
One stream is the research on the emergency evacuation problem. Emergency evacuation is in the response phase of disaster management, the purpose of which is to reduce the impact of disasters through timely evacuation. In an evacuation network model, the traffic flow is usually treated as a fluid for describing the traffic propagation phenomenon. The CTM (Daganzo 1993, 1995) was used to modeling the traffic propagation by many researchers. Chiu et al. (2007) proposed a multi-dimensional evacuation CTM model, which integrated the optimal evacuation destination, traffic assignment and evacuation departure schedule decisions into a unified evacuation network. Zhao et al. (2015) formulated an evacuation optimization model based on CTM which aimed to minimize the network clearance time. Hadiguna et al. (2014) built an innovative decision-support system to assess the feasibility of public facilities for evacuation after a disaster. To seek the optimal emergency evacuation routing, Liu et al. (2016) proposed a direction guidance system that guided passengers to escape from the optimal exits. Duan et al. (2016) proposed a personalized route planning system based on the Wardrop Equilibrium model for pedestrian-vehicle mixed evacuation ,which focused on the shortest clearance time. Asfaw et al. (2019) studied a wildfire emergency evacuation problems of an Indigenous community by a lack of community evacuation plan. Taneja and Bolia (2018) presented a bi-level model to optimize the evacuation strategy during mass gatherings. To minimize the quantities of unsatisfied demand, unserved wounded, and nontransferred workers, Al Theeb and Murray (2017) addressed a complex post-disaster humanitarian relief problem requiring the coordination of multiple heterogenous vehicles to facilitate three logistics operations.
The second stream is the research on the evacuation transportation planning with priority, which aims at minimizing evacuation transit time (or transit cost). Human disaster decision makers want to serve heavily injured people or other affected residents and materials evacuating as quickly as possible during the emergency response phase. Chiu and Zheng (2007) proposed a simultaneous mobilization destination, traffic assignment, and departure schedule for multi-priority groups (SMDTS-MPG) model in response to a no-notice disaster response. The objective of this model was to solve the optimal strategies for simultaneously mobilizing multiple priority groups with different types of intended destinations and priorities. Yi and Özdamar (2007) described an integrated location-distribution model that considered the priority among the injured people and all commodities, where the heavily injured people and urgent medicine held the highest priority. Parr et al. (2013) applied a transit signal priority in an urban evacuation that can save lives by reducing bus travel time. Moreover, Li et al. (2016) presented a travel itinerary problem which aimed to minimize travel cost for traveling multiple destinations. For our highlight work, we develop a novel linear programming formulation based on the CTM, named MPCTM. The model deploys the multiple priority emergency evacuation resources and achieves the minimization of both of the transit time of the higher priority resource and total transit cost.
The third stream is the research on the emergency evacuation problem with uncertainty which is characterized by random and fuzzy variables. The theoretical researches and applications mentioned above are deterministic demand. However, in real-world evacuation network, the demand is mostly highly uncertain. So what should the network system do when more demand is realized than used for prediction? In order to reduce the unnecessary cost, several researches take into account the well-known tendency of realized demand uncertainty as documented in the following. In stochastic environment, Sumalee et al. (2011) and Zhong et al. (2013) extended the stochastic cell transmission model to simulate traffic flow on networks with stochastic demand and supply. Do Chung et al. (2012) provided a chance-constrained programming approach for a joint chance-constrained cell transmission model based SO-DTA where uncertain demand was only partial distributional information known. Ghasemi et al. (2019) presented an uncertain multi-objective multi-commodity multi-period multi-vehicle location-allocation mixed-integer mathematical programing model, where the uncertainty was modeled using a scenario-based probability approach. Wang et al. (2016) investigated a stochastic optimization model to generate the evacuation plans based on different evaluation criteria. To address stochastic demand, Levin and Boyles (2016) proposed a cell transmission model for dynamic lane reversal with a Markov decision process formulation. Ukkusuri et al. (2017) proposed a stochastic linear mixed-integer programming model that considers three key areas of emergency logistics: facility and stock pre-positioning, evacuation planning and relief vehicle planning. The objective of the model is to minimize the total cost of opening distribution centre, transporting evacuees and shipping relief supplies. In fuzzy environment, Zheng and Ling (2013) considered three correlated fuzzy ranking criteria in an emergency transportation planning of disaster relief supply chain. Yang et al. (2015) used multi-objective biogeography-based optimization algorithm to solve supply chain network design with uncertain transportation cost and uncertain customer demand, which were characterized by continuous possibility distributions. To evaluate preparedness and response ability, Celik and Gumus (2016) proposed a hybrid approach based on interval type-2 fuzzy sets, which aimed to minimize the losses and number of affected people.
The last stream is the research on the robust optimization approach for dealing with optimization problems with the underlying uncertain data. Robust optimization is a specific methodology that may be outperformed with an unknown or partial known probability distributions, which more likely true in real network. First of all, Soyster (1973) introduced the idea of robust optimization. He considered a linear optimization problem that the solution was feasible for a given convex set, however, the solution was too conservative. After that, a prosperous development of robust optimization was put forward by Bertsimas and Sim (2003) and Ben-Tal et al. (2009) and so on. The approach could handle a variety of optimization problems, such as linear programming (LP), conic-quadratic programming (CQP) and semi-definite programming (SDP). The robust optimization approach has been applied in various fields, such as inventory management (See and Sim 2010), supply chain network (Bai and Liu 2016), and project portfolio (Liu and Liu 2017). For our related emergency evacuation network, Karoonsoontawong and Waller (2007) presented a robust optimization model for the dynamic traffic assignment based continuous network design problem, which accounted for a bilevel objective and long-term origin-destination demand uncertainty. The model also embedded the Daganzo’s cell transmission model. Ben-Tal and Do Chung (2011) applied an affinely adjustable robust counterpart approach for dynamically assigning emergency response and evacuation traffic flow planning with time dependent demand uncertainty. Rezaei-Malek et al. (2016) used a scenario-based robust stochastic approach for a disaster relief logistics network with perishable commodities. In order to address the hospital evacuation problem under uncertainty, Rabbani et al. (2016) developed a bi-objective programming model using a robust possibilistic programming approach. In this paper, we apply robust optimization approach for MPCTM with demand uncertainty.
For our paper differing from the aforementioned literature, no work has been done in multiple priority dynamic traffic assignment based CTM formulation with uncertain demand. In this paper, our effort is given to study the MPCTM based on a robust optimization method. Then we verify the performance of the MPCTM in a real-world evacuation transportation network.
3 Multiple-priority cell transmission model for emergency evacuation
Notations
Symbol | Description |
---|---|
I | set of discrete time intervals |
C | set of cells |
C_{R} | set of source cells |
C_{S} | set of sink cells |
A | adjacency matrix representing transportation network connectivity |
L | set of priority |
\({\rho _{l}^{t}}\) | transit cost coefficient |
\({Q_{i}^{t}}\) | maximum number of vehicles that can flow into or out of cell i during time interval t |
\({N_{i}^{t}}\) | maximum number of vehicles in cell i at time interval t |
\({\delta _{i}^{t}}\) | traffic flow parameter in cell i at time interval t |
\(d_{li}^{t}\) | demand of priority l in cell i at time interval t |
\(x_{li}^{t}\) | number of vehicles of priority l in cell i at time interval t |
\(y_{lij}^{t}\) | number of vehicles of priority l moving from cell i to cell j at time interval t |
4 Robust Optimization Formulation of the Multiple-Priority Cell Transmission Model
In this section, we apply the MPCTM to study an emergency evacuation flow network; in this way, we can depict the dynamic evacuation traffic flow. In the deterministic CTM, the demand quantity is known; however, in real emergency events, it is very difficult to accurately forecast real demand information. A small perturbation of uncertain demand information may lead to severe suboptimality or even infeasibility. To overcome evacuation demand information uncertainty, we explore the robust optimization approach (Bertsimas and Sim 2003) to convert a deterministic linear programming model into a robust counterpart model.
According to the polyhedral set U_{pol}, the robust counterpart with uncertain demand data is equivalent to the following formulation:
Theorem 1
Given the polyhedral uncertainty set U_{pol}for evacuation demand, and due to the duality theorem, the robust counterpart problem is equivalent to the following tractable deterministic formulation. Note thatrandq are dual variables.
For the constraint maximum protected criterion, the Constraint (1) affected by the uncertain demand is equivalent to
Constraints (15) and (16) have forms similar to that in (14), corresponding to a similar dual form. Therefore, based on the strong duality property, the dual linear optimization problem RC_{pol} has an equal optimal solution with RC that becomes tractable.
The model RC_{pol} − MPCTM becomes a deterministic linear programming problem with finite constraints; then, we can process it with some classical algorithms.
5 Numerical Experiments
In this section, to illustrate the performance of the proposed model and its potential achievements, we propose a real-world instance based on the Ya’an earthquake. Moreover, a robust solution can be chosen to guarantee MPCTM optimality. The linear programming problem is solved using CPLEX 12.6 on a personal computer with AMD 1.80 GHz CPU and 4 GB RAM, under Windows 7.
5.1 Problem Instance Design
Time-invariant cell properties
Cell | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
\({N_{i}^{t}}\) | \(\infty \) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | \(\infty \) |
\({Q_{i}^{t}}\) | \(\infty \) | 60 | 60 | 60 | 60 | 60 | 60 | 30 | 30 | 30 | 30 | 30 | 30 | \(\infty \) | |
\(\hat {x}_{i}^{t}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |
\({N_{i}^{t}}\) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 120 | 100 | 100 | 100 | 100 |
\({Q_{i}^{t}}\) | 60 | 60 | 60 | 60 | 60 | 60 | 30 | 30 | 30 | 30 | \(\infty \) | 60 | 60 | 30 | 30 |
\(\hat {x}_{i}^{t}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | ||||||
\({N_{i}^{t}}\) | 100 | 100 | 150 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |||||
\({Q_{i}^{t}}\) | 30 | 30 | \(\infty \) | 60 | 60 | 30 | 30 | 30 | 30 | 30 | |||||
\(\hat {x}_{i}^{t}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Time-dependent cell properties
Time | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
\({Q_{8}^{t}}\) | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 0 |
16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |
\({Q_{8}^{t}}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Time-dependent demand and transit cost coefficient for two priority sets
Time | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ||
Priority set 1 | \(d_{11}^{t}\) | 35 | 20 | 10 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\({\rho _{1}^{t}}\) | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
Priority set 2 | \(d_{21}^{t}\) | 200 | 120 | 80 | 30 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\({\rho _{2}^{t}}\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | ||
Priority set 1 | \(d_{11}^{t}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\({\rho _{1}^{t}}\) | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
Priority set 2 | \(d_{21}^{t}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
\({\rho _{2}^{t}}\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
In this instance, the MPCTM with 40 nodes and 42 time intervals has 123,203 constraints and 106,643 variables; this was solved using CPLEX 12.6.
5.2 Experimental results
In this section, we summarize the experimental results for the network in terms of transit time and transit cost, corresponding to priority and nonpriority cases, respectively.
Table 5 lists the experimental results regarding transit time with priority. In this tabulation, the entries in the total column show the total transit time; additionally, pr.sets refers to the respective transit time of two priority sets of vehicles, where the upper one represents the priority set 1 and the lower one is the priority set 2. The tabulation presents the change in transit time with three different uncertainty levels (i.e., 𝜃 = 0.1, 0.2, and 0.3) and four different budget levels (i.e., Γ = 1, 2, 3, and 4). Then, let \(u_{prl}^{\theta {\Gamma }}\) be the transit time of the l vehicle type in the conditions 𝜃 and Γ with priority (e.g., \(u_{pr1}^{0.11}=803.5\)) and when in the DLP model, \({\Gamma }=0(u_{prl}^{\theta 0})\).
The uncertainty level 𝜃 is a protection parameter. The larger 𝜃 is, the longer the transit time will be, relative to the outlined value. Moreover, the budget level Γ is a conservative degree parameter. As the budget level increases, the RC_{pol} entries will increase accordingly. When Γ = 4—that is, RC_{pol} = RC_{box}—the robust strategy gives the best worst-case objective value that will guarantee 100% immunization against perturbations. In line with various risk attitudes, the decision-maker can trade off between robustness and optimality, based on the budget level. Note that the DLP transit time is equivalent to its robust solution with the zero uncertainty level (𝜃 = 0).
Transit time with priority (\(u_{prl}^{\theta {\Gamma }}\))
𝜃 | DLP | RC_{pol} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Γ = 1 | Γ = 2 | Γ = 3 | Γ = 4(RC_{box}) | |||||||
total | pr.sets | total | pr.sets | total | pr.sets | total | pr.sets | total | pr.sets | |
0.1 | 7145.0 | 770.0 | 7699.0 | 803.5 | 8030.0 | 830.5 | 8229.5 | 843.0 | 8292.5 | 847.0 |
6375.0 | 6895.5 | 7199.5 | 7386.5 | 7445.5 | ||||||
0.2 | 7145.0 | 770.0 | 8263.0 | 838.0 | 8930.0 | 891.0 | 9347.0 | 916.0 | 9480.0 | 924.0 |
6375.0 | 7425.0 | 8039.0 | 8431.0 | 8556.0 | ||||||
0.3 | 7145.0 | 770.0 | 8827.5 | 874.5 | 9875.0 | 951.5 | 10537.0 | 989.0 | 10757.5 | 1001.0 |
6375.0 | 7953.0 | 8923.5 | 9548.0 | 9756.5 |
Transit time with nonpriority (\(u_{nprl}^{\theta {\Gamma }}\))
𝜃 | DLP | RC_{pol} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Γ = 1 | Γ = 2 | Γ = 3 | Γ = 4(RC_{box}) | |||||||
total | npr.sets | total | npr.sets | total | npr.sets | total | npr.sets | total | npr.sets | |
0.1 | 7145.00 | 1077.50 | 7699.00 | 1181.25 | 8030.00 | 1242.25 | 8229.50 | 1269.25 | 8292.50 | 1278.25 |
6067.50 | 6617.75 | 6787.75 | 6960.25 | 7014.25 | ||||||
0.2 | 7145.00 | 1077.50 | 8263.00 | 1273.50 | 8930.00 | 1360.50 | 9347.00 | 1420.50 | 9480.00 | 1440.00 |
6067.50 | 6989.50 | 7569.50 | 7926.50 | 8040.00 | ||||||
0.3 | 7145.00 | 1077.50 | 8827.50 | 1341.00 | 9875.00 | 1505.25 | 10537.00 | 1596.50 | 10757.50 | 1631.75 |
6067.50 | 7486.50 | 8369.75 | 8940.50 | 9125.75 |
In comparing Tables 5 and 6, one can see that the sum of the transit time of the two priority sets remains unchanged over the two different cases. The reason is that the vehicles of priority sets 1 and 2 can travel to a destination along the same route and in different orders. However, the actual travel time of different priority sets will decrease when the transit cost coefficient increases, as the MPCTM may bestow upon emergency vehicles a higher priority.
Total cost with priority (\(z_{pr}^{\theta {\Gamma }}\))
𝜃 | DLP | RC_{pol} | |||
---|---|---|---|---|---|
Γ = 1 | Γ = 2 | Γ = 3 | Γ = 4(RC_{box}) | ||
0.1 | 7915.0 | 8502.5 | 8860.5 | 9072.5 | 9139.5 |
0.2 | 7915.0 | 9101.0 | 9821.0 | 10263.0 | 10404.0 |
0.3 | 7915.0 | 9702.0 | 10826.5 | 11526.0 | 11758.5 |
Total cost with nonpriority (\(z_{npr}^{\theta {\Gamma }}\))
𝜃 | DLP | RC_{pol} | |||
---|---|---|---|---|---|
Γ = 1 | Γ = 2 | Γ = 3 | Γ = 4(RC_{box}) | ||
0.1 | 8222.50 | 8880.25 | 9272.25 | 9498.75 | 9570.75 |
0.2 | 8222.50 | 9536.50 | 10290.50 | 10767.50 | 10920.00 |
0.3 | 8222.50 | 10168.50 | 11380.25 | 12133.50 | 12389.25 |
5.3 Results analysis
To illustrate the performance of our proposed MPCTM and robust optimization approach under uncertainty, we present the analytical results in two parts. First, we compare the transit cost with priority to that with nonpriority. Second, we compare the robust transit cost to the deterministic transit cost.
Cost decrease degree (CDD)
𝜃 | DLP | RC_{pol} | |||
---|---|---|---|---|---|
Γ = 1 | Γ = 2 | Γ = 3 | Γ = 4(RC_{box}) | ||
0.1 | 3.740% | 4.254% | 4.441% | 4.487% | 4.506% |
0.2 | 3.740% | 4.567% | 4.562% | 4.685% | 4.725% |
0.3 | 3.740% | 4.588% | 4.866% | 5.007% | 5.091% |
In summary, the objective function derived by implementing a robust network design solution with uncertain demand generated by a given uncertainty set shows sound performance in the MPCTM. This may be significant, as the decrease in transit cost with priority corresponds to the loss of human life and property during a disaster. Meanwhile, one can employ the polyhedral uncertainty set to obtain a robust solution, which will afford a more realistic choice in emergency evacuation planning.
The multiple-priority emergency evacuation model can not only accurately simulate the real-time road conditions of each section, but also reflect the dynamic characteristics of the evacuation process. At the same time, it can also reflect the impact of congestion and the difference of crowd priority in the actual dispersion process, so as to provide some reference for the formulation of optimal management decision-making scheme.
6 Conclusions
This study applies the robust optimization approach with the cell transmission model -based multiple-priority emergency evacuation model under demand uncertainty. The multiple-priority cell transmission model (MPCTM) develops an efficient emergency evacuation flows network, which deploys higher-priority flows over lower-priority ones. When the roadway’s flow capacity is restricted, higher-priority vehicles will always be first to get through. We also studied a variant of the MPCTM that considers demand uncertainty, and which belongs to a polyhedral set and uses the robust optimization approach to manage it. Integrating with uncertainty renders the original model computationally intractable, since it is a semi-infinite program; additionally, by using the duality theorem, it can be reformulated as an equivalent and tractable optimization problem. By using real-world numerical results, we find that the MPCTM can provide a higher-quality solution than the model with nonpriority. Furthermore, a robust solution improves feasibility compared to a deterministic solution under uncertainty level 𝜃. In addition, the budget level Γ can be developed to reduce the solution conservativeness, in line with the decision preference.
Finally, we would like to suggest future research directions. Other uncertainty sets—including an ellipsoidal uncertainty set or a norm uncertainty set—should be developed in the MPCTM to derive a more optimized solution. Other effective algorithms (such as fuzzy optimization) should be employed with the model when there is more information about the uncertain data.
Footnotes
Notes
Acknowledgments
This work are supported by the National Natural Science Foundation of China (Grant No. 61773150 and Grant No. 71801077), the Top-notch talents of Heibei province (Grant No. 702800118009), Humanities Social Sciences Research Project of Colleges and Universities in Hebei Province (SD171005) and Social Science Foundation of Hebei Province (HB17GL012).
Compliance with Ethical Standards
Conflict of interests
The authors declare that there is no conflict of interest regarding the publication of this article.
Human and animals participants
This article does not contain any studies with human participants or animals performed by any of the authors.
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