Definition Of Multi Objective Decision Making

Published Date: 02 Nov 2017

One of the critical challenges in incident management is to provide timely response. Therefore, in this paper, a modified PROMETHEE II (Preference Ranking Organization Method for Enrichment Evaluation) is proposed to improve the efficiency and response time in incident management. Specifically, since the computing time and computation complexity of PROMETHEE II increase with the increase of the number of alternative incident management plans and evaluation criteria, in this paper, multiple steps of traditional PROMETHEE II are integrated into one formula to simplify the computational process, reduce the number of comparisons and database visits, and decrease disc space occupation. An experiment is designed to test and compare the traditional PROMETHEE II method and the modified method using simulated earthquake data. The results show that the modified method improves the efficiency of PROMETHEE II and is able to provide timely evaluation of incident management plans.

Keywords： Emergency management; PROMETHEE II; Efficiency of the algorithm

1. Introduction

In recent years, the frequency of various large-scale emergencies has increased and caused aggravated losses. In order to strengthen the emergency response system and enhance the public information management to improve the government’s ability of preventing and handling emergencies, many special emergency government institutions and research centers were initiated all over the world, e.g. the Japanese disaster's board was established in 1960, the American disaster research center was created in 1963. Emergency management is a very popular research field related to information science, psychology, management and other disciplines. .Emergence management is also a discipline that can deal with the probability of injury or property damages and reduce bad impacts of unexpected events which seriously affect the normal life of community, aiming to effectively reduce the negative consequences through the comprehensive application of many related knowledge science, technologies, plans and management science etc. [1]. In brief, emergency management is an integrated system [2].

Currently, the world's typical emergency management research institutions include: the National Emergency Management Association, the International Association of Emergency Managers, the American Psychological Association's Disaster Research Network, etc. Under the impetus of these research institutions, great achievements on theoretical study in emergency management have been made, including: the theory on how to improve the decision-making efficiency when related information is absent [3]; the multi-level and multi-stage mechanism analysis according to the characteristics of unexpected events [4]; the proposition of sudden mass incidents evolution model [5]; the analysis of emergency response plans initiation strategy [6]; the construction of emergency management "interrupted-continue to " random sort model [7] and the study of emergency chain based on disaster theory [8]; the two-stage of emergency supplies distribution strategy [9] and emergency resource layout [10]. Besides, there are many great achievements on methods including the dynamic penalty strategy for environmental pollution problems in evolutionary game based on dynamics system[11]; the application of analytical network process in emergency situation when related criteria are not completely independent [12]; the combination of the heterogeneous data integration, algorithms of data mining and multi-objective decision making methods to construct the emergencies integrated assessment models [13] etc. The achieved results have shown that the rapid emergency response and the appropriate emergency decision are two key points of emergency management. The most important approach to reach these two key points is to increase the efficiency of emergency decision.

Tufekci and Wallace, two distinguished emergency management experts pointed out that the emergency management is in essence a complex multi-objective optimization problem [14]. With the development of information technology, especially the computer science and internet technique, the multi-criteria decision making (MCDM) technique has been extensively studied and employed in emergency management. As one of the most widely used MCDM methods, the PROMETHEE II method has also been studied by scholars from different angles since it was developed by Nadeau and Landry in 1982 [15]. For instance, the rank reversal analysis of PROMETHEE II [16], the investigation of the selection of preference function and the influence on rank effect by the selection of parameter [17], the semantic variables decision making based on fuzzy theory [18], the evolution model construction associated with AHP method [19], the analysis of the decision of Cyprus energy resource by the application of PROMETHEE II. [20], the water resources decision [21] etc. However, to the best of our knowledge, few methods related to PROMETHEE II have been used in emergency management area, because emergency management is usually a MCDM problem with multitudinous alternatives and criteria, and it is difficult to meet the timeliness requirements in emergency management for so many steps of traditional algorithm and the large number of calculation and comparisons. Therefore, it is necessary to reduce the many steps of traditional algorithm and the large number of calculation and comparisons to propel the application of PROMETHEE II in emergency management since it has the following advantages in emergency management: the simplicity in principle, no limit about the number of alternatives and no restriction about the size of evaluation indexes system. In this paper, based on the preference function of Usual Criterion and U-Shape criterion, the calculating process of PROMETHEE II is reduced and improved by integrating the multiple steps. The experimental result indicates that the modified PROMETHEE II method not only can simplify the computational process, reduce the number of comparison and database visiting , but also promote the efficiency in emergency management.

The remaining of this paper is organized as follows: Section 2 briefly reviews the relevant algorithm concepts and operational process. The revising process, the relevant mathematical theorems and corollary proof are described in Section 3. Section 4 addresses the promotion influence in computational process, comparisons and database visits through experimental design. Section 5 concludes the paper and shows the future study directions.

2. Formulation of the PROMETHEE II [22]

2.1 Definition of multi-objective decision making

The multi-criteria decision problem can be presented as follows:

（2.1）

where A= is a finite set of n proposed alternatives, and C= is a set of k evaluation criteria. The symbolis the corresponding observation value based on i criterion under the j alternative. Each criterion is given a certain particular weight and the weights set is defined as. The weight is used to measure the relative importance of the criterion which is non-negative numbers, independent from the measurement unit of the criteria. The lower the weight is, the less important the criterion. The sum of all weights should be 1, that is, and

2.2 Algorithm procedure of PROMETHEE â…¡

The algorithm procedure of the PROMETHEE II includes five stages, which will further be demonstrated by a case based on Usual Criterion. The case describes a disaster relief plan for a city under a typhoon attack. There are three relief plans named A1，A2，A3, and the experts were asked to score three evaluation indices such as the personnel treatment (c1), the housing reconstruction (c2) and the restore of communication and power (c3) according to their contribution to the plan. The greater score indicates the larger contribution to the plan, and the

Table 1 Evolution table of relief plans

c1

c2

c3

A1

4

3

2

A2

3

2

4

A3

5

1

3

Table 3 Outranking flow

Φ+

Φ-

Φ

Rank

A1

0.55

0.45

0.1

2

A2

0.35

0.65

-0.3

3

A3

0.6

0.4

0.2

1

obtained result is shown in Table 1. Meanwhile, experts were also asked to assess the importance of the three evolution indices and the obtained weights are 0.5, 0.3 and 0.2, respectively.

Table 2 pair-wise comparisons of division, preference function and aggregated preference Indices

d

P

Ï€

c1

c2

c3

c1

c2

c3

A1,A2

1

1

-2

1

1

0

0.8

A1,A3

-1

2

-1

0

1

0

0.3

A2,A1

-1

-1

2

0

0

1

0.2

A2,A3

-2

1

1

0

1

1

0.5

A3,A1

1

-2

1

1

0

1

0.7

A3,A2

2

-1

-1

1

0

0

0.5

The calculating steps of the PROMETHEE II method are as follows:

Step 1: Comparing the alternatives and finding the amplitude of the deviations.

The amplitude of the deviations between the compared alternatives with respect to each criterion can be calculated as:

(2.2)

For instance, with respect to the index c1 , the amplitudes of the deviation of A1 and A2 , A2 and A3 are 1, -2 respectively. Likewise, these pair-wise contrast amplitudes of the deviation can be obtained and shown in Table 2.

Step 2: Selecting a preference function.

Generally, the preference function is a function associated to deviation and is defined as:

（2.3）

Specifically, six different preference functions are defined in PROMETHEE II, which covers almost all the possible criteria. In this case, the Usual Criterion is used to demonstrate the calculating processes of PROMETHEE II, that is, 1 is used to replace the positive deviation and the rest is replaced with 0. Thus, the preference coefficient can be obtained as shown in Table 2.

Step 3: Calculating the aggregated preference index

For alternatives ai, aj, the aggregated preference index is defined as

（2.4）

where

and indicate the preference degree of the former over the latter respectively (from 0 to 1).

is the preference function associated with criterion ct.

is the weight associated with criterion ct.

Obviously, the aggregated preference index is equal to the weighted average number of the compared preference index, e.g. . Then the other aggregated preference index can be obtained and as shown in Table 2.

Step 4: Calculating the outranking flow.

The outranking flow can be divided into positive and negative outranking flow. The positive outranking flow of ai measures the preference over all the other alternatives and is defined by

（2.5）

where h is the total number of alternatives. Similarly, the negative outranking flow of ai is given by

（2.6）

According to (2.5) and (2.6), the outranking flow for each alternative can be calculated and as shown in Table 3.

Step 5: Aggregating the net outranking flow and comparing the scores.

For the PROMETHEE II, the net outranking is used to denote complete ranking. Take alternative ai as example, the net outranking flow is calculated by

（2.7）

where :

（2.8）

The higher the net flow is, the better the alternative will be. Next, the score of the net outranking flow is used to rank the alternatives from the largest value to the lowest value.

The traditional method is used to calculate the aforementioned case and the result shows that the third plan A3 is the best and the second A2 is the worst.

3. Process improvement

The case mentioned above shows that the traditional method totally acquires 9 times ergodic dataset, 68 times comparison and calculation and increase 7% of the storage space. However, with the increase of the dataset size, the comparisons and calculations increase linearly simultaneously, which make the computing process more complicate. The modified process, which embedded five steps (see equations 2.2~2.7) to one (see equations 3.1), can be easily obtained within programming, as well as the ability to greatly simplify the computing process and the operational efficiency, so that the method can better fit the needs of the rapid assessment emergency management.

，where，m=1,2,…,h （3.1）

The modified algorithm only needs to get the pair-wise comparison and goes through the data set n times to obtain the net outranking flow instead of the aggregated preference index calculation and the positive and negative outranking flow computing.

In the six possible preference functions, the first two are binary structure, whose values are either 0 or 1:

（1） Usual Criterion

（2） U-Shape Criterion

Where q is the argument, and when q=0, the U-Shape Criterion is the same with the Usual Criterion.

Contrasting the Usual Criterion and U-Shape Criterion, we have following theorem:

Theorem 1，

Proof: Define wt as the weight of the k evaluation factors in each group and ，where , q is the argument. Assume the w=, whose elements w1，w2,…,wm , are the weights within ,s=1,2,…,m; Similarly, the wm+1, wm+2, …,wk represent the remaining weights. Obviously, the functions can be the Usual Criterion when the q=0, and we can get:

Similarly， so

Because

and

So:

Corollary：

Proof: From equation 1.7 and 1.8, it can be deduced that:，; We can also get ，and from theorem 1, then

，Since , we can get

（3.2）

Combining equation 2.6, 2.8 and 3.2 together, the following equation can be obtained for the net outranking flow for am.

（3.3）

4. Experiments

The occurrence and development of unexpected events are dynamic evolution process. The evolution process was divided into four stages by Steven Fink including sings, attack, extended and continuation period [23]. In the corresponding emergency management, similar stage theories were proposed. For instance, Mitroff suggested that the emergency management should include signal detection, prevention, control, recovery and hindsight learning phases, and put forward five stages theory [24], while Augustin put forward six stages theory such as evasion, preparation, confirmation, control, solve, and benefit stages. In fact, emergency cannot be easily predicted, thus the most favorable time to reduce the negative consequences is the episode period and extension period [25]. If the emergency management is divided into 3 periods such as predict in advance period, course control period in an event and recovery period after an event, the strategic point in reducing the catastrophic loss is the course control period. The course control requires a timely, objectively and roundly evaluation on the events to improve the efficiency of the management and to establish a high efficiency management. It is hard for the conventional PROMETHEE II method to realize a quick evaluation once the data increase in both quantities and directions because the data collection in this period is in a dynamic updated, diverse and increasing situation. Therefore, we attempt to focus on the course control of the emergency management and facilitate the efficiency of the emergency assessment to provide decision support promptly.

The data used in this experiment contains 31 data elements obtained by the command center to simulate the data in the initial earthquake. The command center determines the seriousness of the disaster based on the 31 real-time data sets, thereby providing support for emergency decision in time. The evaluation data contains six indices i.e. direct economic losses, traffic damage, emergency relocation number, house collapsed condition, injured number and deaths number. The corresponding weights are 0.071, 0.0251, 0.1235, 0.0425, 0.2656 and 0.4723 respectively. Applying the PROMETHEE II to these dataset, the rank of the 31 places can be obtained and shown in Figure 1. The lower position of the curve indicates the more serious of the disaster and the more front of the rank. The result shows that the first ten hardest disaster areas are the number 2, 10, 1, 11, 15, 29, 9, 17 and 13.

Fig.1 The ranking result of PROMETHEE II

Experimental data comparison in the experiment between traditional and modified method of PROMETHEE II is shown in Table 4, where:

Relief plans in a typhoon attack within specification of 3*3;

Simulating data of the earthquake within specification of 31*6;

The third data is expressed by n alternatives and m evaluation indexes which within specification of n*m.

The comparison demonstrates that, all the ergodic process, calculation number and storage space increase with the growth of the specifications. Especially, take data specification 3*3 for example, we only need to go through the data set 3 times instead of 9 times after the revision. Similarly, the calculation number reduces to 18 from 68 and the storage space is reduced to 8.75k from 9.82k.

Table 4 Comparison of the classical and modified methods

Data specification

modify

Ergodic process/times

Calculation number/times

Data space/KB

Computation complexity

3*3

Before

9

68

9.82

O(n2)

31*6

37

17949

2443

O(n2)

n*m

n+6

---------

O(n2)

3*3

After

3

18

8.75

O(n2)

31*6

31

5580

41

O(n2)

n*m

n

mn(n-1)

----------

O(n2)

All the comparison results indicate that when the alternatives and the evolution index increase, the modified method can improve the assessment efficiency and bring convenience for emergency decision by simplifying the computational process, reduce the number of comparisons as well as database visits, and decrease disc space occupation.

5. Conclusion

In this paper we modified the PROMETHEE II to deduce a universal formula that is appropriate for different preference function. According to the preference function of USUAL Criterion and U-Shape Criterion, a unified computing formula, a theorem and a corollary were derived and proved mathematically. Simulating experiments of seismic data set compared the computational process, the number of comparisons and database visits, and the disc space occupation. The results demonstrate that the modified method not only can simplify the evolution process of PROMETHEE II, but also can improve the efficiency of the emergency management.

Whether the size of storage space will be influenced by the size of data or not is not proved practically or theoretically in present work, we will explore this issue deeply through a large number of experiments by integrating PROMETHEE II with other methods to construct the program evaluation model for emergency management, and provide support for emergency management decision in the future.

6. Acknowledgments

This research is partially supported by the National Natural Science Foundation of China (70901011, 71173028).