Optimizing Emergency Logistics Centre Locations: A Multi-Objective Robust Model

3.2.1 Collection

U : the set of emergency material collection points u .

G : the set of emergency supplies type g .

I : set of alternative emergency material centres i .

J : a set of emergency material demand points j .

3.2.2 Variables

f i : the fixed cost of opening i alternative emergency logistics centres.

c u i g : the unit transportation cost of transporting the g th emergency material from the material collection point u to the alternative logistics centre i .

c i j g : the unit transportation cost of the g th emergency material from the alternative logistics centre i to the emergency demand point j .

c u j g : the unit transportation cost of the g th contingency material transported from the material collection point u to the contingency demand point j .

t u i g : transport time of the g th emergency material from the collection point u to the alternative logistics centre so that it is transported.

t i j g : transportation time of the g th emergency material from the alternative logistics centre i to the emergency demand point j .

t u j g : transportation time of the g th emergency material from material collection point u to emergency demand point j .

t i g : turnaround time of the g th emergency material at the alternative emergency logistics centre i ;

a i t : the unit storage cost of the g th emergency material at the alternative emergency logistics centre i .

h u g , max : the maximum quantity of the g th emergency material to be collected at material collection point u .

h j g : the amount of the g th emergency material demanded at the emergency demand point j .

θ j g : the quantity of the g th emergency material at the unsatisfied emergency demand point j .

λ j g : the unit penalty coefficient of unmet emergency demand point j for the g th kind of emergency supplies.

3.2.3 Decision Variables

z i ∈ { 0 , 1 } : decision variable of whether or not to site in alternative emergency logistics centre i , z i = 1 means to site in emergency logistics centre i , otherwise z i = 0 ;

y i g : the storage capacity of the g th emergency material in the alternative emergency logistics centre i ;

w u i g ∈ { 0 , 1 } : whether or not to transport the g th emergency material from collection point u to logistics centre i ;

w u j g ∈ { 0 , 1 } : whether the g th emergency material is transported from collection point u to emergency demand point j ;

w i j g ∈ { 0 , 1 } : whether to transport the g th emergency material from logistics centre i to emergency demand point j ;

x u i g : the quantity of the g th emergency material to be transported from collection point u to logistics centre i ;

x u j g : quantity of type g contingency transported from collection point u to emergency demand point j ;

x i j g : the number of emergency goods g transported from the logistics centre i to the emergency demand point j .

5.2.1 Cost Preference Weights

First, the baseline scenario data are taken into account to confirm the reliability of the multi-objective deterministic model. The hybrid frog jump algorithm is used to optimize and solve the cost preference weights, which are taken at intervals of 0.05 from 0 to 1. Table 4 displays the ideal rescue allocation plan for all types of materials at all demand points. According to the optimal rescue allocation programme 2-4-6-6-2-2-5-6-5, Collection Point 2, Emergency Logistics Centre 4 and Emergency Logistics 6 allocate the three emergency supplies at the first demand point. At the same time, Emergency Logistics Centre 6, Collection Point 2 and Collection Point 2 are responsible for providing the three different types of emergency supplies at the second demand point. Emergency Logistics Centre 6 provides rescue services at the third demand point. The findings in Table 4 demonstrate that initially, the decision-maker prioritizes the emergency response time when performing the rescue. As a result, all three emergency logistics centres are opened to meet the initial rescue’s time urgency requirements. However, as the cost preference weights gradually rise (from 0.35 to 0.6), the number of emergency logistics centres is reduced to two (i.e. the locations of Emergency Logistic Centres 5 and 6); when the cost preference weights further increase, the decision maker prioritizes the emergency response cost (i.e. the locations of Emergency Logistic Centres 5 and 6). As the cost preference weight rises, the decision-maker prioritizes contingency costs (from 0.65 to 1), and the number of emergency logistics centres decreases to one (i.e. site selection for Emergency Logistics Centre 4).

Table 5 shows rescue routes and rescue material transportation results under three cost preference weights in the baseline scenario. When the cost preference weights are 0.35, 0.55, and 0.8, the emergency rescue costs are 557,100, 47,100, and 45,390, and the emergency rescue times are 21,700, 25,000, and 26,400, respectively. When the decision maker assigns different weights to the rescue costs, the emergency rescue costs and times have different optimal values, and it is impossible to achieve the possibility of decreasing both the costs and times of the emergency rescue. When the decision maker assigns different weights to rescue costs, the cost and time of emergency rescue have different optimal values, and it is impossible to achieve the possibility that the cost and time of emergency rescue decrease simultaneously.

Second, the relationship between cost preference weights and emergency response should be further analysed, especially regarding how to set cost preference weights in different emergency response phases. Figure 2 shows the trend of the mean and standard deviation of cost and time for all scenarios as a function of cost preference weights. Table 5 and Figure 2 show that when the decision maker assigns larger and larger values to the cost preference weights, the optimal value of rescue cost decreases gradually, but the emergency rescue time increases slowly, which is consistent with the actual rescue situation; on the other hand, different values of the cost preference weights can correspond to various stages of the emergency rescue, and in the initial stage of the emergency rescue, the primary consideration is the high timeliness. It is acceptable to have a higher emergency rescue cost, which can be regarded as corresponding to a smaller cost preference weight. In the early stage of emergency rescue, the primary consideration is high timeliness, and it is acceptable to have higher emergency rescue cost, which can be regarded as corresponding to smaller cost preference weights (e.g. 0.35 in Table 5); in the middle stage of emergency rescue, both timeliness and cost have to be taken into account, which can be regarded as corresponding to moderate cost preference weights (e.g. 0.55 in Table 5); in the later stage of emergency rescue, timeliness takes a back seat, and the primary consideration is the cost, which can be regarded as corresponding to larger cost preference weights (e.g. 0.8 in Table 5). Analysing the trends of rescue cost and rescue time in Figure 2, the corresponding weights can be roughly divided into three intervals, with the first interval being [0, 0.45], the second [0.45, 0.75], and the third [0.75, 1], which can be regarded as corresponding to the early, middle, late, and final stages of rescue, respectively. Low-cost preference characterizes the first stage; the optimal rescue time is small, but the rescue cost remains high, which is in line with the time urgency requirement of activating the emergency rescue system after a disaster occurs; the second stage is characterized cost preference weights, which takes into account the urgency requirement of the rescue time, but also considers the cost limitation that the emergency rescue system can afford, and the cost of the rescue decreases rapidly with the slow increase of weights, but the required rescue time increases quickly; the third interval segment is [0.75, 1], and the three cost preference weights can be seen as corresponding to the early, middle, late or final stage respectively. The third stage is characterized by cost preference weight, which mainly considers the cost limitations of the emergency rescue system and is in line with the characteristics of the late or end stage of emergency rescue, in which the cost decreases slowly and the time increases a little faster.

Figure 2

Trends in the mean and standard deviation of the deterministic model.

Again, in this research, the robust constraint coefficients of all the scenarios are assumed to be 0.2 to examine the benefits and drawbacks of the multi-objective robust optimization deterministic models. The performance of emergency logistics models is reviewed, and strong constraint coefficients are discussed. It concludes that although lower coefficients enhance cost efficiency but must be prepared for extreme occurrences, more significant coefficients boost conservatism but raise operating expenses. The model’s balance of essential goals is impacted, and for the best emergency reactions, a range of 0.15–0.25 is advised. Figures 3 and 4 show the trend of the mean (time and cost) and standard deviation (time and cost) of the two models with the variation of cost preference weights, respectively. The average results in Figure 3 only indicate little difference between the two models when the cost preference weights are taken at [0, 0.55] and [0.75, 1]. Still, the average cost of the deterministic model is significantly higher than the average cost of the robust optimization when the cost preference weights are taken at [0.6, 0.7]. However, when we focus on the standard deviations (time and cost) of the two models given in Figure 4, We can see that both of the deterministic model’s standard deviations are higher than those of the robust optimization, in particular, when the cost preference weights are taken as [0.55, 0.7], the deterministic model’s standard deviation is about twice as high as that of the robust optimization This suggests that the robust optimization is more resistant to various uncertainties. Specifically, in terms of the general trend, the corresponding cost preference weights of the robust model can be divided into three bands: the first band is [0, 0.45], the second band is roughly [0.45, 0.75], and the third band is roughly [0.75, 1], and the three cost preference weight bands can be regarded as corresponding to the early, middle, and late stages of the rescue, respectively. The improvements are validated by comparing the resilient and deterministic models in the research. The robust model’s higher decision-making abilities and improved uncertainty management are demonstrated by comparison against a baseline benchmark. The comparison uses consistent performance indicators to prove its cost, time, and reliability benefits. The scenario analysis illustrates the robust model’s flexibility and durability in many situations. Setting the stage for future comparisons with more sophisticated models, the comparison places the advances of the robust model within the context of current research.

Figure 3

Trends in the mean values of the two models.

Figure 4

Trend of the standard deviation of the two models.

According to the results of the chart and the previous discussion, we can consider emergency rescue to be divided into three types: the rescue of the early stage of the time urgency, the rescue of the middle stage of the rescue of the cost and the rescue of the late stage of the rescue of the price, decision-making or modelling optimization solution can be chosen to choose a different cost preference weight, and here, according to the discussion of the above will be the cost of the weight of the range of values of the weight of preference for the distinction as [0, 0.45], [0.45, 0.75], [0.45, 0.75], [0.75]. Here, based on the above discussion, the range of cost preference weights is distinguished as [0, 0.45], [0.45, 0.75], and [0.75, 1], or a single value, such as 0.35, 0.55, and 0.8.

5.2.2 Impact Analysis of Robust Constraint Coefficients

To examine the impact of different robust constraint coefficients, we consider five values of robust constraint coefficients, i.e. 0.1, 0.15, 0.2, 0.25, and 0.3. Figure 5 gives the robust objective values obtained by the multi-objective robust optimization with different robust constraint coefficients (left) and the average objective values of all scenarios under the current robust optimal solution (right). The average objective value here means that the optimal robust solution can be obtained for a given robust constraint coefficient, which corresponds to the objective value of each scenario, and the average of these objective values is calculated. As can be seen from Figure 5, when the cost preference weights are taken in [0, 0.45] and [0.75, 1], the robust objective value and the average objective value under different robust constraint coefficients are very stable; in addition, it is noted that when the cost preference weights are taken in [0.75, 1] and all the robust constraint coefficients are taken in 0.1, the robust objective value and the average objective value have large fluctuations, which indicates that the robust coefficients are small (i.e. given the robust constraint coefficient). The average objective value of the cost preference weights could be more stable. The robust optimization offers flexible and adaptable techniques, customized risk levels and event probabilities, making it a more dependable and economical approach to emergency logistics planning. This increases public confidence in emergency management systems. This shows that finding a robust optimal solution is more difficult when the robustness coefficient is small (i.e. the given robustness requirements are more stringent). Even in extreme cases, finding a feasible solution that satisfies all the scenarios may be challenging. When the cost preference weights are within [0.5, 0.75], different robust constraint coefficients have other impacts. When the robust requirements are more stringent (robust coefficient of 0.1), the robust objective value increases rapidly with the increase of the cost preference weights; when the robust requirements are more lenient (robust coefficient of 0.3), the robust objective value fluctuates upward and downward; and when the robust requirements are general (0.15–0.25), the robust objective value is relatively high, and the robust objective value is relatively low. The robustness target value is relatively flat when the robustness requirement is moderate (0.15–0.25). Based on the above analysis, we recommend that the robust constraint coefficients in robust optimization be taken in the range of [0.15, 0.25] or the middle value of 0.2.

Figure 5

Impact analysis of robust constraint coefficients.

5.3.1 Deterministic Model Solution

Based on the parameter setting in 5.1, the deterministic siting model can calculate the demand for emergency materials in the affected area and the transportation time between the affected area and the emergency logistics centre, as shown in Tables 10 and 11.

The weighting of transportation time and cost in the model can be adjusted by adjusting the parameter input, i.e. whether transportation time or transportation cost is more critical to the site selection results in the site selection model. Now, we will take λ = 0.5 care of the calculation.

The solution will be done with the help of a Matlab software toolbox called Robust Stochastic Optimization Made Easy, an algebraic toolbox for general optimization under uncertainty. After Matlab software calculations, the optimal objective function is 25,193,000 yuan, the total cost of a deterministic model site selection. The specific composition is shown in Table 12 and Figure 6.

Figure 6

Total cost of deterministic model site selection.

The optimal choice of the emergency logistics centre is shown in Table 13 and Figure 7, where 1 indicates that the location is chosen to establish an emergency logistics centre, and 0 suggests that it is not chosen.

Figure 7

Graphical representation of deterministic model emergency logistics centre selection.

The proportions of needs being met at each emergency material demand point are shown in Table 14.

This allows us to determine the volume of emergency supplies the emergency logistics centre will transfer to each location where they are needed and the volume of supplies the centre will stock, as shown in Table 15.

5.3.2 Robust Optimization Solution

The parameters the model requires are given in 5.1 and 5.2.1, which are also solved by Matlab programming. Different r is set to adjust the degree of conservatism of the robust optimization so that Γ = 5, the optimal objective function value of 2572.02, robust optimization of the total cost of the site selection of the specific composition of the model, as shown in Table 16, and its graphical representation is shown in Figure 8.

Figure 8

Graphical representation of robust optimization.

The optimal selection of the emergency logistics centre in the robust optimization is shown in Table 17, with 1 indicating that the location is selected to establish the emergency logistics centre and 0 suggesting that it is not specified.

Table 18 shows the percentage of satisfied demand at each emergency material demand point.

From this, the inventory quantity of the emergency logistics centre can be calculated, as shown in Table 19.

Robust Optimal Site Selection Model The optimal site selection scheme and transportation routes are shown in Figure 9.

Figure 9

Graphical representation of inventory quantity of the emergency logistics centre.