Weapon selection problem is a strategic issue and has significant impacts on the efficiency of a defense system. Several alternatives must be considered and evaluated in terms of many different conflicting criteria and sub criteria and therefore an effective evaluation approach is essential to improve decision quality. Analytic hierarchy process (AHP) is one such technique used by the researchers over the years in establishing the relative values of military weapon system. Despite its popularity, some shortcomings of AHP have been reported in the literature, which have limited its applicability. For example, the number of judgments to be elicited in AHP increases as the number of alternatives and criteria increase. The issue of rank reversal is another prominent limitation of traditional AHP. This study, presenting a scientific framework to assess weapon systems, uses a hybrid approach of Data Envelopment Analytic Hierarchy Process (DEAHP). The novelty of this research lies in the application of this hybrid approach to a weapon selection problem and concludes that the DEAHP method outperforms the AHP method and also does not suffer from the rank reversal phenomenon.
Keywords: AHP, DEA, Multi criteria decision making.
INTRODUCTION
Because weapon systems are regarded as crucial to the outcome of war, the selection of weapon systems is a critical national decision. It is an important issue as an improper weapon selection can negatively affect the overall performance and productivity of a defence system. Selecting the new weapon is a time-consuming and difficult process, requiring advanced knowledge and deep experience. The rapid development of military technologies makes weapon systems ever more sophisticated, expensive, and quickly accelerates research on methods for selection of these systems. The Republic of Korea (ROK) Ministry of National Defense [1] (MND) has been raising its force investment budget to more than 30% of its defense budget, most of which is for weapon systems procurement. Finance minister Pranab Mukherjee has announced more than 17% hike in India's defense expenditure for the financial year 2012-13, as the country looks to off-set growing Chinese dominance in Asia. All this reflects the current system demands from the analysts to develop concrete and tangible methods for the selection of weapon systems.
Like most real-world decision making problems, the selection of a weapon systems requires a multiple criteria decision analysis (MCDA). Ho [2] classified MCDAs into two technical categories, multiple objective decision making (MODM) and multiple attribute decision making (MADM). MADM selects the best alternative among the various attributes that are to be considered. One of the most popular MADM techniques includes AHP [3] . AHP structurally combines tangible and intangible criteria with alternatives in decision making and logically integrates the judgment, experience, and intuition of decision makers. Because of its usability and flexibility, AHP has been widely applied to complex and unstructured decision making problems such as military decision making. Several literatures are available which makes use of AHP for weapon system selection. Some of the prominent are by Cheng [4] which proposes performance evaluation and optimal selection of weapon systems having multi-level and multi-factor features. On weapon system projects, some researchers applied combined approaches such as a hybrid AHP-integer programming approach to screen weapon systems projects (Greiner et.al [5]) and an AHP approach based on linguistic variable weights ( [4], [6] ).
Linking AHP with DEA
Despite the usefulness of AHP, some shortcomings of AHP have been reported in the literature, which have limited its applicability. The number of judgments to be elicited in AHP increases as the number of alternatives and criteria increase. This is often a tiresome and exerting exercise for the decision maker. For example, the number of judgments to be elicited in AHP increases as the number of alternatives and criteria increase. The issue of rank reversal [7] is another prominent limitation of traditional AHP. The ranking of alternatives determined by the traditional AHP may be altered by the addition or deletion of another alternative for consideration. Hence linking of AHP with other techniques such as DEA [8] has been suggested to overcome its limitations. One of the earliest attempts to integrate DEA with multi-objective linear programming was provided by Golany [9]. Since then, there have been several attempts to use the principles of DEA and AHP in the MADM literature. For example, some articles [10-12] have used AHP for handling subjective factors and then used DEA to identify efficiency score based on the entire data, including those generated by the AHP. In addition, DEA and AHP have been linked with other techniques for specific applications [13-14]. In this paper, the concepts of efficiency measurement in DEA are integrated with the concepts of weight measurement in AHP and hybrid approach of Data Envelopment Analytic Hierarchy Process [15] has been used for weapon system selection as an alternative to the conventional and singular methods of weight derivation in analytic hierarchy process (AHP). To the best of knowledge, it is by far a first attempt of its kind in the field of defense.
The remainder of the paper is organized as follows. The next section briefly discusses the methodologies of DEA, AHP and explains the DEAHP methodology. This is then followed by the application of the DEAHP method to a hypothetical missile system selection case. Conclusions are presented in the final section.
METHODOLOGY
2.1 Analytic Hierarchy Process
The AHP consists of three main operations, including hierarchy construction, priority analysis and consistency verification. First of all, the decision makers need to break down complex multiple criteria decision problems into its component parts of which every possible attributes are arranged into multiple hierarchical levels. After that, they have to compare each cluster in the same level in a pair wise fashion based on their own experience and knowledge. Since the comparisons are carried out through personal or subjective judgments, some degree of inconsistency may be occurred. To guarantee the judgments are consistent, the final operation called consistency verification is incorporated in order to measure the degree of consistency among the pair wise comparisons by computing the consistency ratio. If it is found that the consistency ratio exceeds the limit (i.e. if CR>0.1), the decision makers should review and revise the pair wise comparisons. Once all pair wise comparisons are carried out at every level, and are proved to be consistent, the judgments can then be synthesized to find out the priority ranking of each criterion and its attributes.
2.2 Data envelopment analysis (DEA)
Data Envelopment Analysis (DEA) is a widely applied non-parametric mathematical programming approach for analyzing the productive efficiency and performance evaluation of Decision Making Units (DMUs) or with the same multiple input and multiple outputs. For each alternative (DMU), the efficiency is measured and an optimization will be proposed according to the below indicated linear program. The DMUs are denoted by . Each DMU i employs m inputs to produce s different outputs. Specifically, DMU j consumes amount of input i and produced amount of output r. It is assumed that and and that each DMU has at least one positive input and one positive output value. Let and denote the input and output weights respectively. Then the inputs and output weights are calculated for the observed DMU0 using the following linear programming problem [16].
Subject to
(P1)
The optimal objective function values of problem (P1), when solved, represent the efficiency score of the observed DMU. This DMU is relatively efficient if their optimal objective function value equals unity. For inefficient units, DEA also provides those efficient units (namely peers or benchmarks), which the inefficient units can emulate to register performances that could improve their efficiency scores.
2.3 Data Envelopment Analytic Hierarchy Process (DEAHP)
Here, it is proposed that DEA concepts can be used in the last two steps of applying AHP to a decision problem-namely, deriving local weights from a given judgment matrix and aggregating local weights to get final weights. Efficiency calculations using DEA require outputs and inputs. Each row of the judgment matrix is viewed as a DMU and each column of the judgment matrix is viewed as an output. Thus a judgment matrix of size NÃ-J will have N DMUs and J outputs [15, 17]. Since DEA models cannot be used entirely with outputs and require at least one input, a dummy input column has been added having input value as 1 for all DMUs. It is proposed that the efficiency scores calculated using DEA models could be interpreted as the local weights of the DMUs.
Traditional AHP View Proposed DEA View
O1
O2
…
On
Dummy input
DMU1
1
a12
a1n
1
DMU2
1/a12
1
a2n
1
……
…
…
…
…
…
DMU N
1/a1n
1/a2n
1
1
Cr1
Cr2
…
Cr n
Dummy input
ALT 1
1
a12
a1n
1
ALT2
1/a12
1
a2n
1
……
…
…
…
…
…
ALT N
1/a1n
1/a2n
1
1 FIGURE 1: A comparison of the traditional AHP view and the proposed DEA view
(here ALT: alternative; Cr: criteria O: output)
While computing the final weights of the main factors as well as the performance rating of employees, following theorem has been used.
Theorem : Let the local weights of alternatives with respect to different criteria be given by the matrix below
Where is the local weight of alternative i (i=1,2,…N) with respect to criterion j. If the importance of criteria is incorporated in the form of multipliers then final weights aggregated using DEA is proportional to the weighted sum for the ith alternative.
3. EMPIRICAL ILLUSTRATION
To illustrate the procedure involved in the proposed hybrid approach and to demonstrate its effectiveness, a hypothetical case on surface-to-air missile system selection is presented.
3.1 Problem setup and data
The problem is designed as a hierarchical structure of four levels: First the goal of the decision problem, followed by the criteria, sub criteria, and alternative levels. As shown in FIGURE 2, to select an optimal alternative, six candidate missile systems are considered and evaluated based on three criteria and 18 sub criteria. Each sub criterion, identified and structured in the previous stage, has its own characteristic data about the candidate missile system. The criteria and characteristic data were identified based on Ahn's study [18]. Data has been arbitrary but meaningfully generated to suits the requirements.
Range (R )
Altitude (AL)
Hit probability (HP)
Reaction time (RT)
Set up time (ST)
Detection targets (DT)
Engagement targets (ET)
Inter operability (IO)
ECM
Anti ARM (AA)
Mobility (MO)
Trainability (TY)
Acquisition cost (AC)
Maintenance cost (MC)
Offset trade (OT)
Technological effects (TE)
Industrial effects (IE)
Corporation growth (CG)
M -1
M-2
M-3
M-4
M-5
Basic capabilities (BC)
Costs & technical effects (C&T)
Operational capabilities (OC)
LEVEL-1
LEVEL-2
LEVEL-3
Missile system selection FIGURE 2 : Hierarchical structure for missile system selection
After structuring the hierarchy of factors affecting the missile performance, opinion of the experts can be obtained on the following issues:
a) Comparative effects of main criteria on the performance of the Missile.
b) Comparative contribution of various sub factors on the factors mentioned above, e.g
effect of range, hit probability, altitude etc. on the basic capability of the missile.
c) Relative ranking of each alternative missile with respect to each sub factor.
The qualitative information is obtained in a suitably designed format enabling pair wise comparison of factors, sub factors and missiles (FIGURE 3). This is communicated in the format by marking 'X' appropriately in one of the columns depending on intensity of comparisons, i.e equal, moderate, strong, very strong and extremely strong. It may be mentioned that in multiple criteria decision making problems, the opinion though consistent may be prejudiced or biased towards a specific aspect of system. It is therefore suggested that to eliminate such bias, the opinion of several experts from different disciplines may be elicited. To combine their opinion geometric mean of the corresponding values of the paired comparison at each stage in the hierarchy may be used for the final analysis.
Missiles
ES
VS
S
MS
EQ
MS
S
VS
ES
Missiles
M-1
M-2
M-1
X
M-3
M-1
X
M-5
M-2
M-3
M-2
X
M-4
M-2
X
M-5
M-3
X
M-5
M-4
X
M-5
FIGURE 3: Format used for pair wise comparisons
3.2 Calculation of weights of criteria
After obtaining the comparison matrices, next step involves the weight calculation of each level to obtain the overall score of each missile with respect to all 18 sub-criteria and pair-wise comparisons of the main selection criteria.
3.2.1 Evaluation of the third level decision alternatives
The third level of the hierarchy, as previously described, has been analyzed using the DEAHP and AHP methodologies. Decision-makers were asked to specify the relative importance of missile selection criteria. In table 1 panel A, seven missile selection criteria related to basic capabilities of the missile selection system which include range, altitude, hit probability etc. are compared with each other in pair-wise form. In panel B inter comparison of missiles with respect to one of the sub criteria i.e. range has been shown. In the similar manner inter comparison of missiles with respect to other sub criteria of the basic capabilities has been computed. They have not been shown here due to restriction of research paper length. (Here R: Range; AL: altitude; HP: hit probability; RT: reaction time; ST: setup time; DT: Detection targets; ET: Engagement targets).
The calculations of the DEAHP approach are also illustrated using the entries of Table 1. The local weights of alternatives using crisp AHP and DEAHP methods are shown in the last two columns of table 1. In the DEAHP method, in order to ascertain how to derive local weight from the pair-wise matrix, an instance of shipment criterion is illustrated in the following model.
Comparison of sub criteria with respect to basic capability criteria
R
AL
HP
RT
ST
DT
ET
Input
AHP
DEAHP
R
1
5
4
5
1
4
4
1
0.31629
1.000
AL
1/5
1
1
5
2
1/2
4
1
0.1387
1.000
HP
1/4
1
1
4
1
2
4
1
0.143133
0.9231
RT
1/5
1/5
1/4
1
1/5
1/5
1/2
1
0.031887
0.2
ST
1
1/2
1
5
1
2
4
1
0.046448
1.000
DT
¼
2
1/2
5
1/2
1
4
1
0.133025
1.000
ET
¼
1/3
1/4
2
1/5
1/3
1
1
0.043844
0.4
Comparison of missiles with respect to range sub criteria
M-1
M-2
M-3
M-4
M-5
Input
AHP
DEAHP
M-1
1
1/6
1
4
1/7
1
0.084729
0.4445
M-2
6
1
6
7
1
1
0.40800
0.99999
M-3
1
1/6
1
4
1/7
1
0.080868
0.4445
M-4
1/4
1/7
1/4
1
1/8
1
0.036765
0.143
M-5
7
1
7
9
1
1
0.402633
1
TABLE1: Basic capabilities
For panel B of table 1, the following model is generated.
(P2)
The above model (P2) can be easily solved using Excel-Solver or Lingo software. The optimal objective function value of this model will give the local weight of alternative M-1 i.e. 0.4445. To obtain the local weight of other missiles categories, similar models are used by changing the objective function can be formulated and solved. The DEA efficiency scores (i.e. optimal values of the objective function in these problems) representing the local weights of DMUs (i.e., the Missiles here) are presented in the second last column of Table 1 panel B.
Evaluation of the second level decision alternatives
Once local weights of suppliers are obtained in the third level, then they are aggregated to obtain second level of weights of the decision alternatives. We can now apply the DEA approach again to aggregate the local weights obtained from previous data to generate final weights which will give the basic capability of the missile as follows:
R
AL
HP
RT
ST
DT
ET
Basic capability (BC)
Relative weightage
1
1
0.9231
0.2
1
1
0.4
M-1
0.45
0.75
0.78
0.5625
0.395
0.275
0.26
0.5410
M-2
1
0.821
0.63
1
1
1
1
1.000
M-3
0.45
1
1
0.4375
0.184
0.275
0.209
0.5842
M-4
0.14
0.714
1
0.4375
0.184
0.275
0.209
0.4671
M-5
1
0.392
0.23
0.157
0.95
0.775
0.651
0.7516TABLE 2: Computation of final weights for the missiles with respect to basic capability criteria
Aggregation using DEA using the local weights of sub factors : In this case, again the DEA model has been applied for the local weights in Table 2. The local weights are considered as outputs of alternatives (missiles), and a dummy input is introduced. For example, to get the final weight of missile M-1, the following model can be used:
(P3)
The optimal objective function value of (P3), when solved, will give the final weight of alternative missile M-1. To get the final weight of other missiles, models similar to the above model should be solved by changing the objective function. Using the values of local weights of sub criteria, the additional constraints is introduced in the DEA model (P3) that calculates the final weight (mobility) of DMU(missile M-1). The resulting final weights of missiles which gives the basic capabilities are shown in the last column of Table 2, are 0.5410, 1.000, 0.5842, 0.4671and 0.7516.
The final weights are proportional to the weighted sum of local weights. For example, for alternative missile - T1, the weighted sum can be calculated as [(0.4445 * 1) + (0.75 * 1) + (.78 * 0.9231) + (.5625* 0.2) + (.395*1) + (.275*1)+ (.26*.4)+(.2*.4)] = 2.8630. The weighted sum for the five missiles are 2.8630, 5.3284, 3.8952, 3.2677 and 4.0211 which are proportional to 0.5410, 1.000, 0.5842, 0.4671and 0.7516. Similarly, the final eights of missiles with respect to other criteria such as operational capabilities and costs can be obtained.
3.2.3 Level 1 analysis
At level 1 three main criteria namely basic capability, operational capability and costs and technical effects has been identified and then again the principles of AHP and DEA has been used as in the above manner to compute the local weights of the main factors also .
BC
OC
C&T
Input
AHP
DEAHP
BC
1
1/3
1/2
1
.334
OC
3
1
2
1
1
C&T
2
1/2
1
1
.67Table 3: Comparison of main criteria with respect to objective
Performance rating of missiles: Finally the performance rating of missiles can be computed by again using the same DEAHP approach as discussed above. In this case local priorities or the relative weights of three main factors i.e. basic capabilities, operational capabilities, costs and technical effects are taken into consideration. Performance index of various missile s comes out as the final weights of different missiles.
Criteria
BC
OC
C&T
Missile performance index
Performance rating
Relative weightage
0.334
1
0.67
M-1
0.5410
0.5
0.578
0.5469
1.0
M-2
1
1
1
1.000
1.83
M-3
0.5842
0.42
0.367
0.4951
0.9053
M-4
0.4671
.212
0.182
0.3430
0.6272
M-5
0.7516
0.94
0.94
0.8372
1.531
Table 4: Alternatives (missiles M-1, M-2, M-3, M-4,M-5)
For example, to get the final weight of missile M-1, the following model has been used:
We can infer from the results of the above analysis that the performance rating of missile M-5 is 1.54 times the performance rating of missile M-1. For a comparison of cost-effectiveness of two missile s, their relative ratings can be used as effectiveness index. For example here if we consider missile M-2, it is approximately twice as efficient as compared to missile M-1.
CONCLUSIONS
An attempt here has been made to provide an alternative approach to the traditional AHP method for the computation of local weights or priorities and the final weights. For a perfectly consistent judgement matrix, this approach gives the similar weights as given by AHP approach where as for inconsistent matrices it tries to remove in consistency. Further when final weights of the alternatives are computed with respect to each of the main criteria, it was found that they are the weighted sum of local weights. The relative weights of each criteria factor, viz , basic capabilities , operational capabilities , cost and technical effects is suitably aggregated along with the relative weightage of each sub factor and the ratings of each sub factor and the ratings of each missile with respect to each sub factor , to give an overall performance index of each missile. From the missiles performance index, performance rating of each missile can be calculated.
