A fuzzy multi-objective programming optimization model for emergency resource dispatching under equitable distribution principle

Abstract

In the initial stage of emergency rescue for major railway emergencies, there may be insufficient emergency resources. In order to ensure that all the emergency demand points can be effectively and fairly rescued, considering the fuzzy properties of the parameters, such as the resource demand quantity, the dispatching time and the satisfaction degree, the railway emergency resources dispatching optimization model is studied, with multi- demand point, multi-depot, and multi-resource. Based on railway rescue features, it was proposed that the couple number of relief point - emergency point is the key to affect railway rescue cost and efficiency. Under the premise of the maximum satisfaction degree of quantity demanded at all emergency points, a multi-objective programming model is established by maximizing the satisfaction degree of dispatching time and the satisfaction degree of the couple number of relief point - emergency point. Combined with the ideal point method, a restrictive parameter interval method for optimal solution was designed, which can realize the quick seek of Pareto optimal solution. Furthermore, an example is given to verify the feasibility and effectiveness of the method.

Keywords

Emergency resource dispatching equitable distribution principle couple number of relief point - emergency point satisfaction degree restrictive parameter interval method

1 Introduction

Railway emergency rescue resources mainly comprises the following equipment and tools: rescue trains (including the rescue crane, rescue tool lorry, railcar), special rescue supplies (rail, turnout, pushdozer, auxiliary beam for construction, etc.) and professional rescue personnel, with obvious professionalism and irreplaceability. The reasonable dispatching of the limited emergency resource is not only a technical problem, but also a complex social ethics problem [10]. Fairness is the cornerstone of society and the essence requirement of emergency rescue. Aristotle, the ancient Greek philosopher, argued that social justice and distributive justice would maximize the total social utility. In order to maintain the harmony and fairness of the society, as well as to guarantee the stability of the whole railway system, the principle of equitable distribution for resource scheduling has caught the scholars’ attention [7 , 13]. According to sociology, fairness has two dimensions: the equity of system arrangement and the approval fairness from the people. The former is reflected the rationality in resource distribution and the obtained differences, and the latter is embodied in subjective common identity of people, i.e., satisfaction degree.

Scholars at home and abroad have achieved some results in the equitable distribution principle of emergency resources. Fianu [11] developed a Markov decision model to realize the fair distribution of emergency resources in each disaster area. Yeboah [2] studied the balance between resource shortage and the equity requirements for emergency fire truck distribution in each block based on the concept of survival equity. Kao [4] considered many factors affecting patients’ survival, and optimized spatial allocation of emergency medical resources based on the perspective of perceived fairness. Gipe [6] conducted a research on the fairness of scarce emergency medical resources allocation among different age groups according to the principle of fair allocation psychological perception. Zhang [14] proposed that demand points should adopt the proportional distribution when they were seriously lacking in resources on the early stage of the rescue; however, in the later stage, the Talmud distribution principle should be adopted.

For the railway resource dispatching problem, under static rescue requirement, assumed that the resources can fully meet the total demand, Tian [16] established a model of resource fair dispatching and designed a solution algorithm. By comparing with some attributes in the existing cases, such as the types of locomotives and vehicles, the situation of railway line and railroad sleepers, the damage of vehicle body as well as the distance from accident point to residential area, Li [8] dispatched resource by a method Case-Based Reasoning (CBR). Wang [12] researched the fair allocation optimization of railway emergency resource in the goal of the earliest rescue start time, the fewest number of used rescue points and the least cost of emergency resource scheduling.

In addition, the uncertainty of emergency environment is not only the key problem in emergency resource scheduling, but also is the research focus in the field. Li [3] established a renovated on-schedule supply regulation model based on a fuzzy multi-objective programming method. Tallon [5] researched the resource allocation plan for emergency medical services systems under the uncertainty of emergency environment. Tang [15] aimed at the railway emergency resources dispatching problems with uncertain demand, based on the fuzzy evaluation and GC-TOPSIS, allocated and scheduled the emergency resources. Al-Refaie [1] established a fuzzy programming model with the goal of minimum overtime hours of medical staff and maximum satisfaction degree of patients on the specified operation date, and optimized the scheduling of medical rescue resources.

These research effectively solve the uncertainty problems of emergency resource dispatching time or demand quantity, however, it still fails to consider the perception of subjective fairness of the disaster-affected people on the allocation, as well as the correlation between the perception of subjective fairness and the obtained resources quantity and time.

Much more importantly, for the railway emergency, the rescue train is key equipment. And each relief point is usually equipped with only one rescue train. Therefore, railway rescue must decrease the couple number of relief point - emergency point for low rescue cost and high rescue efficiency. It is the characteristics of railway rescue and there is a lack of relevant research. Assumed that each emergency point is equally important and needs to be effectively rescued, based on the principle of equitable distribution, considering lots of uncertainties, using fuzzy uncertainty processing method, by satisfaction degree that depicts big or small of uncertainty in the membership between uncertain factors and the fuzzy set, the emergency resource scheduling problem was researched with multi- demand point, multi-depot, and multi-resource in this paper.

2 The related concepts

2.1 Problem description

There are m emergency points (i.e. demand points) E₁, E₂, ⋯ , E_m; n relief points (i.e. rescue depots) S₁, S₂, ⋯ , S_n; and w kinds of emergency resources. For the k^th (k = 1, 2, ⋯ , w) resource, the reserve in S_i is $p_{i}^{k}$ , and the resource demand of E_j is the trapezoidal fuzzy number ${\tilde{d}}_{j}^{k}$ . The delivery time from S_i to E_j is t_ij. How can it realize resources equilibrium distribution among emergency points?

2.2 Scheme representation

Suppose φ is any a rescue scheme, and φ^k is the dispatching scheme for the k^th resource in the scheme φ. So φ^k can be expressed in a matrix form: $φ^{k} = [\begin{matrix} x_{11}^{k} & x_{12}^{k} & \dots & x_{1 j}^{k} & \dots & x_{1 n}^{k} \\ x_{21}^{k} & x_{22}^{k} & \dots & x_{2 j}^{k} & \dots & x_{2 n}^{k} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ x_{i 1}^{k} & x_{i 2}^{k} & \dots & x_{ij}^{k} & \dots & x_{in}^{k} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ x_{m 1}^{k)} & x_{m 2}^{k} & \dots & x_{mj}^{k} & \dots & x_{mn}^{k} \end{matrix}]$ (1)

In the formula (1), $x_{ij}^{k}$ is the actual acquired resource quantity of the k^th resource from S_i to E_j. So the φ can be expressed as: $φ = {[φ^{1}, φ^{2}, \dots, φ^{k}, \dots, φ^{w}]}^{T}$ (2)

2.3 Definition of related concepts

(1) The satisfaction degree of scheduling resource quantity based on equitable distribution principle.

For any one emergency point E_j under any one scheme φ, it corresponds to a satisfaction degree value under demand constraint. Given trapezoidal fuzzy number ${\tilde{d}}_{j}^{k} = (b 1_{j}^{k}, b 2_{j}^{k}, b 3_{j}^{k}, b 4_{j}^{k})$ as the fuzzy demand quantity of E_j in the k^th resource, and its acquired quantity by dispatching from all rescue depots is $x_{j}^{k}$ , if the dispatching resource quantity is too little or too much, it is easy to cause the lack or waste, as shown in Fig. 1(a). So the satisfaction degree function $μ (x_{j}^{k})$ of dispatching quantity can be expressed as $μ (x_{j}^{k}) = {\begin{matrix} 0 x_{j}^{k} < b 1_{j}^{k} \\ \frac{x_{j}^{k} - b 1_{j}^{k}}{b 2_{j}^{k} - b 1_{j}^{k}} b 1_{j}^{k} ⩽ x_{j}^{k} < b 2_{j}^{k} \\ 1 b 2_{j}^{k} ⩽ x_{j}^{k} < b 3_{j}^{k} \\ \frac{b 3_{j}^{k} - x_{j}^{k}}{b 4_{j}^{k} - b 3_{j}^{k}} b 3_{j}^{k} ⩽ x_{j}^{k} < b 4_{j}^{k} \\ 0 b 4_{j}^{k} ⩽ x_{j}^{k} \end{matrix}$ (3)

Fig. 1

Satisfaction degree functions ((a) satisfaction degree of dispatching quantity; (b) satisfaction degree of dispatching time; (c) satisfaction degree of the couple number of relief point - emergency point involved in the rescue).

Considering the rescue resources shortage in the initial stage, in the allocation of the same kind of resources, there is mutual competition among all emergency points. If certain an emergency point is allocated more resources, it may cause that other emergency points are allocated less resources.

In order to ensure the fairness of distribution and the effective rescue of each demand point, the related concepts are defined as follows.

Definition 1. Under the principle of equitable distribution, the satisfaction degree of dispatching quantity of the k^th resource in the dispatching scheme φ^k is μ (x (φ^k)), and it is the minimum value in the satisfaction degree of all demand points to the dispatching quantity of the resource, that is: $μ (x (φ^{k})) = \min {μ (x_{1}^{k}), μ (x_{2}^{k}), \dots}$ (4)

Definition 2. Because there is no competition between the distributions of different kinds of resources, the comprehensive satisfaction degree of dispatching quantity for multiple kinds of resources is defined as the weighted average of the satisfaction degree of all single resource dispatching quantity.

Suppose the satisfaction degree of dispatching quantity for various resources are respectively μ (x (φ¹)) , μ (x (φ²)) , ⋯ , μ (x (φ^w)), the satisfaction degree weight of each resource dispatching quantity are respectively γ¹, γ², ⋯ , γ^w, and $\sum_{k = 1}^{w} γ^{k} = 1$ . So the satisfaction degree μ (x (φ)) of dispatching quantity in the scheme φ is $μ (x (φ)) = \sum_{k = 1}^{w} γ^{k} μ (x (φ^{k}))$ (5)

(2) Satisfaction degree of dispatching time and the couple number of relief point - emergency point involved in the rescue.

The dispatching time is related to the rapidity and effectiveness of rescue, in addition, according to the railway rescue features, it is the key to affect the cost and efficiency of rescue that the couple number of relief point - emergency point involved in the rescue. And to define the related concepts as follows:

Definition 3. Dispatching time t (φ) is the longest of all resources delivery time from relief point to emergency point in the scheme φ, i.e., $t (φ) = max {(t (φ^{1}), t (φ^{2}), \dots, t (φ^{w})}$ (6)

Definition 4. For the rescue problem with multi- demand point, multi-depot, and multi-resource, if a certain rescue depot (relief point) provide rescue of any kind resources for a certain emergency point, then they formed a couple of relief point - emergency point, otherwise there is no such association between them. All the number of such associations are called the couple number of relief point - emergency point.

Under satisfying the constrained conditions, in a high quality rescue scheme, the dispatching time should be as early as possible and the couple number of relief point - emergency point participating in the rescue should be as few as possible. Therefore, the right trapezoidal fuzzy number is adopted to express the corresponding satisfaction degree information, as shown in Fig. 1(b) and 1(c).

In Fig. 1(b) and 1(c), t (φ), N (φ) and μ (t (φ)), μ (N (φ)) is respectively the dispatching time, the couple number of relief point - emergency point, as well as their corresponding satisfaction in scheme φ. And a₁, a₂, n₁, n₂ is respectively their corresponding fuzzy parameter. So the satisfaction degree of dispatching time and the couple number of relief point - emergency point in scheme φ can be respectively expressed as: $μ (t (φ)) = {\begin{matrix} 1 \begin{matrix} 0 ⩽ t (φ) ⩽ a_{1} \end{matrix} \\ \frac{a_{2} - t (φ)}{a_{2} - a_{1}} a_{1} < t (φ) < a_{2} \\ 0 t (φ) ⩾ a_{2} \end{matrix}$ (7) $μ (N (φ)) = {\begin{matrix} 1 \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} & 0 ⩽ N (φ) ⩽ n_{1} \end{matrix} \\ \frac{n_{2} - N (φ)}{n_{2} - n_{1}} \begin{matrix} n_{1} < N (φ) < n_{2} \end{matrix} \\ 0 \begin{matrix} \begin{matrix} \end{matrix} & N (φ) ⩾ n_{2} \end{matrix} \end{matrix}$ (8)

3 The research idea and modelling

3.1 The actual resource allocation quantity for each emergency point under the fair principle

According to definition 1 and formula (4), with a certain supply of resources, in order to maximize the satisfaction degree of the scheduling quantity of a certain resource, it is necessary to make all emergency points to achieve the same satisfaction degree on the resource allocation. For the k^th resource, the reserve of S_i (i = 1, 2, ⋯ , n) is $p_{i}^{k}$ , and the resource demand ${\tilde{d}}_{j}^{k}$ of E_j cannot be directly regarded as its actual resource allocation quantity $x_{j}^{k}$ . So $x_{j}^{k}$ is unknown. However, the actual supply and demand relationship of resources is only two types as follow.

(1) The available resources quantity is sufficient to meet the actual requirements of all demand points.

In this case, $\sum_{i = 1}^{n} p_{i}^{k} ⩾ \sum_{j = 1}^{m} b 2_{j}^{k}$ . As shown in Fig. 1(a), for any emergency point E_j, when the scheduling quantity reaches $b 2_{j}^{k}$ , the satisfaction degree of the resource scheduling quantity reaches the maximum ( $μ (x_{j}^{k}) = 1$ ). When the number of scheduling exceeds $b 2_{j}^{k}$ , satisfaction cannot be improved. Satisfaction even drops when $b 3_{j}^{k}$ is exceeded. Therefore, when all other conditions are the same, choosing a small scheduling quantity is conducive to the decrease of emergency dispatching time and the couple number of relief point - emergency point. In this case, the best scheduling quantity is only $b 2_{j}^{k}$ to E_j. That is, let: $x_{j}^{k} = b 2_{j}^{k}$ (9)

(2) The available resources quantity is not able to completely meet the requirements of all demand points; it is divided into two categories:

$➀ \sum_{j = 1}^{m} b 1_{j}^{k} < \sum_{i = 1}^{n} p_{i}^{k} < \sum_{j = 1}^{m} b 2_{j}^{k}$ . As shown in Fig. 1(a), satisfaction degree increases with the increase of allocation quantity, so all resource must be supplied unreservedly, i.e., $\sum_{j = 1}^{m} x_{j}^{k} = \sum_{i = 1}^{n} p_{i}^{k}$ . In order to meet the fairness, the satisfaction degree of each emergency point must be equal. Taking the sector $b 1_{j}^{k} ⩽ x_{j}^{k} < b 2_{j}^{k}$ and giving $μ (x_{1}^{k}) = μ (x_{2}^{k}) =, \dots, = μ (x_{m}^{k}) = μ (x^{k})$ , according to formula (3), then: ${\begin{matrix} x_{1}^{k} = μ (x^{k}) • b 2_{1}^{k} + (1 - μ (x^{k})) • b 1_{1}^{k}, \\ x_{2}^{k} = μ (x^{k}) • b 2_{2}^{k} + (1 - μ (x^{k})) • b 1_{2}^{k}, \\ ⋮ \\ x_{m}^{k} = μ (x^{k}) • b 2_{m}^{k} + (1 - μ (x^{k})) • b 1_{m}^{k} \end{matrix}$ (10) $μ (x^{k}) = \frac{\sum_{i = 1}^{n} p_{i}^{k} - \sum_{j = 1}^{m} b 1_{j}^{k}}{\sum_{j = 1}^{m} b 2_{j}^{k} - \sum_{j = 1}^{m} b 1_{j}^{k}}$ (11)

Substitute the calculated μ (x^(k)) from formula (11) into formula (10), then the actual resource allocated quantity $x_{1}^{(k)}, x_{2}^{(k)}, \dots, x_{m}^{(k)}$ of each demand point can be calculated.

$➁ \sum_{i = 1}^{n} p_{i}^{k} ⩽ \sum_{j = 1}^{m} b 1_{j}^{k}$ . The allocation quantity of each demand is in the interval $[0, b 1_{j}^{k}]$ , so $μ (x_{j}^{k}) = 0$ and $x_{j}^{k} = 0$ .

3.2 Single objective model of the satisfaction degree of dispatching quantity

From the above analysis, the single objective model of the satisfaction degree of dispatching quantity can be expressed as: $max z_{x} = μ (x (φ))$ (12) $s . t . φ \in Ω$ (13)

In the model, Ω is the set of all feasible dispatching schemes, x (φ) is the resource dispatching quantity corresponding to scheme φ.

Based on the above method, the satisfaction degree of the resource allocation quantity for each emergency point is equal, and the allocation quantity satisfaction degree can reach the maximum.

3.3 Single objective dispatching time satisfaction degree

The single objective of the dispatching time satisfaction degree can be expressed as: $max z_{1} = μ (t (φ))$ (14)

μ (t (φ)) is a piecewise function, so the solution of formula (14) is relatively complex. However, μ (t (φ)) is are the decreasing functions of t (φ), so the higher satisfaction degree is corresponding to the shorter dispatching time, and so $max z_{1} = μ (t (φ)) \Leftrightarrow min z_{1}^{'} = t (φ)$ (15)

According to the formula (6), introduce 0-1 logical variable $y_{ij}^{k}$ and y_ij ( $y_{ij}^{k}$ and y_ij respectively indicates that the participating status which the rescue depot S_i provides the k^th resource and any kind of resources for the emergency point E_j. If S_i provide the k^th resource to E_j (i.e. $x_{ij}^{k} > 0$ ), then $y_{ij}^{k} = 1$ ; otherwise, $y_{ij}^{k} = 0$ . If y_ij = 1, it represents that S_i provided at least a resource to E_j, and y_ij = 0 represents that S_i provided nothing to E_j. So the single objective for shortest dispatching time is: $min z_{1}^{'} = max {y_{11} t_{11}, y_{12} t_{12}, . . ., y_{nm} t_{nm}}$ (16)

Since formula (16) is an objective function in a non-linear form, it can be transformed into a linear form through variables $y_{ij}^{k}$ and y_ij. The model is expressed as follows: $min z_{1}^{'} = t (φ)$ (17) $s . t . y_{ij} ⩾ y_{ij}^{k}$ (18) $t (φ) ⩾ y_{ij} t_{ij}$ (19) $x_{ij}^{k} ⩽ {My}_{ij}^{k}$ (20) $y_{ij}^{k} ⩽ x_{ij}^{k}$ (21) $\sum_{j = 1}^{m} x_{ij}^{k} ⩽ p_{i}^{k}$ (22) $\sum_{i = 1}^{n} x_{ij}^{k} = x_{j}^{k}$ (23) $\sum_{i = 1}^{n} \sum_{j = 1}^{m} y_{ij} = N (φ)$ (24)

$\begin{matrix} y_{ij}^{k} = 0 or 1 . x_{ij}^{k} ⩾ 0, N (φ) ⩾ 0 and all of \\ them integers, as well as t (φ) ⩾ 0 \end{matrix}$ (25)

In the model, i = 1, 2, ⋯ , n ; j = 1, 2, ⋯ , m ; k = 1, 2, ⋯ , w, and the positive number M can be arbitrary large. The constraints (18)–(21) can realize the transformation the objective function from the complex nonlinear objective (16) to the integer linear objective (17).

Formula (18) indicates that if the rescue depot supplies any kind of resource to the emergency point, then the depot is involved in the rescue. Formula (19) represents the resource dispatching time t (φ) is the maximum of the resource delivery time in scheme φ. At the same time, by the constraint (formula (19)), the nonlinear objective function (formula (16)) can be transformed into linear form (formula (17)).

Formula (20) and (21) are special constraints to deal with the logical relationship between $x_{ij}^{(k)}$ and $y_{ij}^{(k)}$ . Formula (20) indicates that once $x_{ij}^{(k)}$ is greater than 0, then the rescue depot S_i must participate in the rescue and provide the k^th resource to the emergency point E_j, that is, $y_{ij}^{(k)} = 1$ . Formula (21) indicates that once the rescue depot S_i participates and provides the k^th resource for emergency point E_j, that is $y_{ij}^{(k)} = 1$ , then $x_{ij}^{(k)}$ must be greater than or equal to 1.

Equation (22) indicates that the total dispatched resources amount from any a rescue depot to all emergency points is less than or equal to the amount of resources owned by the rescue depots. Equation (23) indicates that the total obtained amount of any kind of resources for any emergency points from all rescue depots, is equal to the allocated resources amount by depots. Equation (24) represents that the total correlations participating in rescue between rescue depots and emergency points is the couple number of relief point - emergency point. Equation (25) is the value constraint of the variable.

In this way, the complex nonlinear problem with the greatest satisfaction degree of dispatching time can be transformed into an integer linear programming problem.

3.4 Single objective satisfaction degree of the couple number of relief point - emergency point

Moreover, the single objective of the maximum satisfaction degree of the couple number of relief point - emergency point can be expressed as: $max z_{2} = μ (N (φ))$ (26)

See from the formula (8), μ (N (φ)) also decreases with N (φ). In the same way, the objective of the maximum satisfaction degree of the couple number of relief point - emergency point can be equivalent to the fewest couple number of relief point - emergency point. That is: $max z_{2} = μ (N (φ)) \Leftrightarrow min z_{2}^{'} = N (φ)$ (27)

So, the single objective model of the fewest couple number of relief point - emergency point is: $min z_{2}^{'} = N (φ)$ (28) $s . t . φ \in Ω$ (29)

In the model, the Ω is the set of all feasible dispatching schemes. The constraint conditions are the same as the single objective model with those of the shortest scheduling time model.

4 Method for solution

It is difficult to all single objectives achieve the best at the same time. So it usually needs to coordinate all objectives, and select the excellent in non-inferior solutions (also called Pareto solution).

In this paper, on the basis of MATLAB solution, a search method for the optimal solution with limited parameter interval is proposed to realize the fast search for the Pareto optimal solution. Combined with the ideal point method, the optimal scheme is selected for the multi-objective programming problem with the greatest satisfaction of resource dispatching time and the couple number of relief point - emergency point.

4.1 The ideal point method

To assume that $φ_{t}^{'}$ , $φ_{t}^{″}$ , $μ (φ_{t}^{'})$ and $μ (φ_{t}^{″})$ are respectively the optimal solution, the worst solutions, the best and the worst objective value in dispatching time satisfaction degree model. Besides, $φ_{N}^{'}$ , $φ_{N}^{″}$ , $μ (N (φ_{N}^{'}))$ and $μ (N (φ_{N}^{″}))$ are respectively those in the model of the couple number of relief point - emergency point. As well as R_a and r_a are respectively the approach degree between non inferior solutions φ_a and positive or negative ideal point. $R_{a} = β_{1} \frac{μ (φ_{a})}{μ (φ_{t}^{'})} + β_{2} \frac{μ (N (φ_{a}))}{μ (N (φ_{N}^{'}))}$ (30) $r_{a} = β_{1} \frac{μ (φ_{t}^{″})}{μ (φ_{a})} + β_{2} \frac{μ (N (φ_{N}^{″}))}{μ (N (φ_{a}))}$ (31)

β₁ and β₂ are respectively weight of the two objectives, and β₁ + β₂ = 1. In formula (30) and (31), the worst objective value is easy to get 0. If $μ (φ_{t}^{″}) = 0$ , $μ (N (φ_{N}^{″})) = 0$ , and μ (φ_a) (or μ (N (φ_a))) is also just 0, then 0/0 happens. At this time, in order to enable the calculation to continue, considering that the value is completely consistent with the worst solution, that is the closest, so 0/0 =1 is given. The relative approach degree of φ_a to the ideal point is: $ɛ_{a} = \frac{R_{a}}{R_{a} + r_{a}}$ (32)

And it is the best scheme which has the largest ɛ_a.

4.2 Restrictive parameter interval method for searching Pareto optimal solution

A non-inferior solution is a vector in a solution set which satisfies Pareto optimal conditions. The emergency dispatching problem with multi- demand point, multi-depot, and multi-resource is a multi-objective optimization problem with a complex and discontinuous Pareto front. The number of non-inferior solutions will increase exponentially with the increase of rescue depots, emergency points and resources. In order to reduce the computational workload, a Pareto optimal solution search method with limited parameter interval is proposed in this paper.

Suppose the emergency dispatching time is selected as the limited parameter, and if the scheduling time is limited to no more than a certain value, then the value is defined as the limited emergency scheduling time tt. Suppose that the optimal solution in the single-objective model of scheduling time, the shortest emergency scheduling time is tt₁, and that in the single objective model of the couple number of relief point - emergency point is tt₂, then the dispatching time of non-inferior solutions can be limited to the interval [tt₁, tt₂]. If tt < tt₁, it is not feasible. If tt > tt₂, in the corresponding scheme, the couple number of relief point - emergency point participated in the rescue must be no less than that of the single objective model solution, which is obviously worse. Therefore, the shortest dispatching time is limited as tt₁, ⋯ , tt₂ in turn. If the t_ij > tt, then x_ij = 0. I.e., the rescue depot S_i is not allowed to rescue the emergency point E_j. And add this constraint into the single objective model of the couple number of relief point - emergency point. To solve the non-inferior solution that satisfies the constraint t (φ) ⩽ t_ij and the fewest couple number of relief point - emergency point participating in the rescue, to ensure that the obtained Pareto optimal solutions are all in the Pareto front, and the optimal scheme is determined by comparing its relative proximity to the ideal point. Obviously, the method can reduce the search work of solutions in large-scale rescue problems.

5 Example and result analyses

5.1 Case description

Assuming that a certain railway network is affected by natural disasters, there are 4 emergency points E₁, E₂, E₃, E₄, as well as 4 depots S₁, S₂, S₃ and S₄. Each emergency point needs two kinds of resources. The parameters in the function μ (x (φ)) and μ (N (φ)) are respectively given a₁ = 8, a₂ = 12, n₁ = 6 and n₂ = 12. And β₁ = 0.6, β₂ = 0.4. The details of delivery time, the quantity of resources are shown in Tables 1 and 2.

Table 1
Delivery time for the first kind of resource

The first kind of resource

E ₁ E ₂ E ₃ E ₄ Reserve

S ₁ 3 11 3 10 50

S ₂ 1 8 2 8 60

S ₃ 7 4 12 5 40

S ₄ 5 6 9 12 50

Requirement $\begin{matrix} (30, 40, 50, 60) \end{matrix}$ (60, 70, 80, 90) (50, 60, 70, 80) (40, 50, 60, 70)

	The first kind of resource
S ₁	3	11	3	10	50
S ₂	1	8	2	8	60
S ₃	7	4	12	5	40
S ₄	5	6	9	12	50
Requirement	$\begin{matrix} (30, 40, 50, 60) \end{matrix}$	(60, 70, 80, 90)	(50, 60, 70, 80)	(40, 50, 60, 70)

Table 2

Delivery time for the second kind of resource

	The second kind of resource
	E ₁	E ₂	E ₃	E ₄	Reserve
S ₁	3	11	3	10	100
S ₂	1	8	2	8	80
S ₃	7	4	12	5	90
S ₄	5	6	9	12	110
Requirement	(20, 30, 40, 50)	(50, 60, 70, 80)	(40, 50, 60, 70)	(60, 70, 80, 90)

5.2 The quantity of actual distribution resources for each emergency point

For the first resource, $\sum_{j = 1}^{4} b 2_{j}^{1} = 220$ , $\sum_{j = 1}^{4} b 1_{j}^{1} = 180$ , $\sum_{i = 1}^{4} p_{i}^{1} = 200$ and $\sum_{j = 1}^{4} b 1_{j}^{1} ⩽ \sum_{i = 1}^{4} p_{i}^{1} ⩽ \sum_{j = 1}^{4} b 2_{j}^{1}$ . According to formula (11), μ (x¹) =0.5. And then, it is substituted to into formula (10), the quantity of actual distribution resource for each emergency point respectively is: $x_{1}^{1} = 35, x_{2}^{1} = 65, x_{3}^{1} = 55$ and $x_{4}^{1} = 45$ .

For the second resource, $\sum_{j = 1}^{4} b 2_{j}^{2} = 210$ and $\sum_{i = 1}^{4} p_{i}^{2} = 380$ . Obviously, $\sum_{i = 1}^{4} p_{i}^{2} > \sum_{j = 1}^{4} b 2_{j}^{2}$ .

According to formula (9), μ (x²) =1 and $x_{j}^{2} = b 2_{j}^{2}$ . I.e., $x_{1}^{2} = b 2_{1}^{2} = 30$ , $x_{2}^{2} = b 2_{2}^{2} = 60$ , $x_{3}^{2} = b 2_{3}^{2} = 50$ , $x_{4}^{2} = b 2_{4}^{2} = 70$ .

So, the satisfaction degree of resources dispatching quantity will reach the maximum: max μ (x (φ¹)) =0.5 and max μ (x (φ²)) =1. And according to formula (5), the comprehensive satisfaction degree of dispatching quantity for two kinds of resources is: max μ (x (φ)) =0.75.

5.3 Dispatching scheme and result analysis

(1) the best and the worst solution

➀ Using the function intlinprog of MATLAB2016B, the optimal solution $φ_{t}^{'}$ (called as the 1st scheme, φ1) of dispatching time satisfaction degree model, and the optimal solution $φ_{N}^{'}$ of the couple number satisfaction degree model (called as the 2nd scheme, φ2) respectively are: $φ_{t}^{'} = φ 1 = [\begin{matrix} 35 & 0 & 15 & 0 \\ 0 & 0 & 40 & 20 \\ 0 & 15 & 0 & 25 \\ 0 & 50 & 0 & 0 \\ 30 & 0 & 50 & 0 \\ 0 & 0 & 0 & 70 \\ 0 & 60 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$ (33) $φ_{N}^{'} = φ 2 = [\begin{matrix} 0 & 0 & 50 & 0 \\ 0 & 15 & 0 & 45 \\ 35 & 0 & 5 & 0 \\ 0 & 50 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 70 \\ 30 & 0 & 50 & 0 \\ 0 & 60 & 0 & 0 \end{matrix}]$ (34)

In the 1st scheme, for the first kind of resource, E₁ gets 35 resources from S₁; E₂ gets 15 resources from S₃ and 50 resources from S₄; E₃ gets 15 resources from S₁ and 40 resources from S₂; E₄ gets 20 of resources from S₂ and 25 resources from S₃; for the second kind of resource, E₁ gets 30 resources from S₁; E₂ gets 60 resources from S₃; E₃ gets 50 resources from S₁; E₄ gets 70 resources from S₂. So the dispatching time t (φ1) =8, the dispatching time satisfaction degree μ (t (φ1)) =1, the couple number of relief point - emergency point N (φ1) =7, the couple number satisfaction degree μ (N (φ1)) =5/6 and the worst value of the dispatching time satisfaction model $μ (φ_{μ}^{″})) = 0$ .

In the 2nd scheme, t (φ2) =12, μ (t (φ2)) =0, N (φ2) =6, μ (N (φ2)) =1 and $μ (φ_{N}^{'}) = 1$ .

➁ From Tables 1 and 2, it can be seen that the longest dispatching time is t₄₄ = 12, and the corresponding satisfaction is μ (t (φ)) =0, so the worst value of the dispatching time satisfaction model is $μ (φ_{μ}^{″})) = 0$ .

➂ This example has 4 depots and 4 emergency points, so the couple number can reaches the maximum:16. However, to overestimate the couple number will make the couple number satisfaction degree on negative ideal point to the worst: 0. According to formula (30), if $μ (φ_{t}^{″})$ and $μ (N (φ_{N}^{″}))$ are all 0, then r_a = 0. To bring it into formula (31), and if R_a ≠ 0, it always has ɛ_a = 1. I.e., the relative closeness to the ideal points is 1. In this case, it is impossible to choose the best from the non -inferior solution. In fact, according to the number of basic feasible solutions of the transportation problem is m + n - 1 (m is the number of depots, and n is the number of emergency points), so the maximum couple number of relief point - emergency point is set as $N (φ_{N}^{″}) = 4 + 4 - 1 = 7$ . I.e., $μ (N (φ_{N}^{″})) = 5 / 6$ .

(2) To search non-inferior scheme by the restrictive parameter interval method

According to the two single-objective models, tt₁ = 8 and tt₂ = 12. In Tables 1 and 2, the maximal t_ij is 12, so tt can be limited in the interval [8, 12). In Tables 1 and 2, there are only four numerical values as 8,9,10 and11 in the interval. Therefore, tt = 8, 9, 10, 11, and the example only needs to search the non-inferior scheme for 4 times.

When tt = 8, tt = 9, tt = 10 and tt = 11, the Pareto optimal solutions are respectively called as the 3th (φ3), 4th (φ4), 5th (φ5) and 6th scheme (φ6), shown as follows: $φ 3 = [\begin{matrix} 0 & 0 & 50 & 0 \\ 10 & 0 & 5 & 45 \\ 0 & 40 & 0 & 0 \\ 25 & 25 & 0 & 0 \\ 0 & 0 & 40 & 0 \\ 0 & 0 & 10 & 70 \\ 0 & 0 & 0 & 0 \\ 30 & 60 & 0 & 0 \end{matrix}]$ (35) $φ 4 = [\begin{matrix} 35 & 0 & 15 & 0 \\ 0 & 15 & 0 & 45 \\ 0 & 40 & 0 & 0 \\ 0 & 10 & 40 & 0 \\ 30 & 0 & 50 & 0 \\ 0 & 0 & 0 & 70 \\ 0 & 60 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$ (36) $φ 5 = [\begin{matrix} 0 & 0 & 5 & 45 \\ 0 & 60 & 0 & 0 \\ 35 & 5 & 0 & 0 \\ 0 & 0 & 50 & 0 \\ 0 & 0 & 30 & 70 \\ 0 & 60 & 0 & 0 \\ 30 & 0 & 0 & 0 \\ 0 & 0 & 20 & 0 \end{matrix}]$ (37) $φ 6 = [\begin{matrix} 35 & 15 & 0 & 0 \\ 0 & 0 & 55 & 5 \\ 0 & 0 & 0 & 40 \\ 0 & 50 & 0 & 0 \\ 30 & 60 & 0 & 0 \\ 0 & 0 & 50 & 0 \\ 0 & 0 & 0 & 70 \\ 0 & 0 & 0 & 0 \end{matrix}]$ (38)

In the 3th scheme, for the first kind of resource, E₁ gets 10 resources from S₂ and 25 resources from S₄; E₂ gets 40 resources from S₃ and 25 resources from S₄; E₃ gets 50 resources from S₁ and 5 resources from S₂; E₄ gets 45 resources from S₂; for the second kind of resource, E₁ gets 30 resources from S₄; E₂ gets 60 resources from S₄; E₃ gets 40 resources from S₁ and 10 resources from S₂; E₄ gets 70 resources from S₂. So the dispatching time t (φ3) =8, the dispatching time satisfaction degree μ (t (φ3)) =1, the couple number of relief point - emergency point N (φ3) =7 and the couple number satisfaction degree μ (N (φ3)) =5/6.

In the 4th scheme, the dispatching time t (φ4) =9, the dispatching time satisfaction degree μ (t (φ4)) =0.75, the couple number of relief point - emergency point N (φ4) =7 and the couple number satisfaction degree μ (N (φ4)) =5/6.

In the 5th scheme, the dispatching time t (φ5) =10, the dispatching time satisfaction degree μ (t (φ5)) =0.5, the couple number of relief point - emergency point N (φ5) =6 and the couple number satisfaction degree μ (N (φ5)) =1.

In the 6th scheme, the dispatching time t (φ6) =11, the dispatching time satisfaction degree μ (t (φ6)) =0.25, the couple number of relief point - emergency point N (φ6) =6 and the couple number satisfaction degree μ (N (φ6)) =1.

(3) The best from non-inferior solution

According to formula (30), (31) and (32), R1_a, r1_a, ɛ1_a, ... R6_a, r6_a can be calculated, as shown in Table 3.

Table 3

The ideal point proximity of different schemes

Scheme No.	R _a	r _a	ɛ _a
1	0.933333	0.4	0.7
2	0.4	0.933333	0.3
3	0.933333	0.4	0.7
4	0.45	0.4	0.574468
5	0.7	0.333333	0.677419
6	0.55	0.333333	0.622642

According to the ideal point method, the 1st scheme (φ1) and the 3th scheme (φ3) are all the best. Their ɛ_a are all equal to 0.7.

6 Conclusions

When a large-scale railway emergency occurs, in order to ensure the rapid recovery of the system as a whole and consider the fairness and harmony of the rescue, each emergency point should be provided with effective emergency rescue. In view of the possible shortage of emergency resources in the early stage of rescue, considering the fuzziness of resource demands in each emergency point, this paper proposes to maximize the satisfaction degree of resource dispatching quantity for each emergency point based on the principle of equitable distribution. At the same time, it is believed that the couple number of relief point - emergency point is the key to affect the cost and effect of railway rescue. On this basis, a multi-objective mathematical programming model was established with the highest satisfaction degree of emergency time and the highest satisfaction degree in the couple number of relief point - emergency point. In this paper, an optimal solution search method with the restrictive parameter interval is designed. Furthermore, the feasibility and effectiveness of the model and solution method are verified by an example, which provides a reference for large-scale railway emergency resources rescue dispatching and social resources equitable distribution.

Footnotes

Acknowledgments

This project was supported by the key research and development project of Jiangxi Science and Technology Department (No. 20192BBG70075) and the Humanities and Social Sciences Planning Foundation of Ministry of Education (No. 17YJAZH073).

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