Weapon choice job is a strategic issue and has important impacts on the efficiency of a defence system. Several options must be considered and evaluated in footings of many different conflicting standards and bomber standards and hence an effectual rating attack is indispensable to better determination quality. Analytic hierarchy procedure ( AHP ) is one such technique used by the research workers over the old ages in set uping the comparative values of military arm system.

Despite its popularity, some defects of AHP have been reported in the literature, which have limited its pertinence. For illustration, the figure of judgements to be elicited in AHP additions as the figure of options and standards addition. The issue of rank reversal is another outstanding restriction of traditional AHP. This survey, showing a scientific model to measure arm systems, uses a intercrossed attack of Data Envelopment Analytic Hierarchy Process ( DEAHP ). The freshness of this research lies in the application of this intercrossed attack to a arm choice job and concludes that the DEAHP method outperforms the AHP method and besides does non endure from the rank reversal phenomenon.

Keywords: AHP, DEA, Multi standards determination devising.

## Introduction

Because arm systems are regarded as important to the result of war, the choice of arm systems is a critical national determination. It is an of import issue as an improper arm choice can negatively impact the overall public presentation and productiveness of a defense mechanism system. Choosing the new arm is a time-consuming and hard procedure, necessitating advanced cognition and deep experience. The rapid development of military engineerings makes arm systems of all time more sophisticated, expensive, and rapidly accelerates research on methods for choice of these systems. The Republic of Korea ( ROK ) Ministry of National Defense [ 1 ] ( MND ) has been raising its force investing budget to more than 30 % of its defence budget, most of which is for arm systems procurance. Finance curate Pranab Mukherjee has announced more than 17 % hiking in India ‘s defence outgo for the fiscal twelvemonth 2012-13, as the state looks to off-set turning Chinese laterality in Asia. All this reflects the current system demands from the analysts to develop concrete and touchable methods for the choice of arm systems.

Like most real-world determination devising jobs, the choice of a arm systems requires a multiple standard determination analysis ( MCDA ). Ho [ 2 ] classified MCDAs into two proficient classs, multiple nonsubjective determination devising ( MODM ) and multiple attribute determination devising ( MADM ). MADM selects the best option among the assorted properties that are to be considered. One of the most popular MADM techniques includes AHP [ 3 ]. AHP structurally combines touchable and intangible standards with options in determination devising and logically integrates the judgement, experience, and intuition of determination shapers. Because of its serviceability and flexibleness, AHP has been widely applied to complex and unstructured determination doing jobs such as military determination doing. Several literatures are available which makes usage of AHP for arm system choice. Some of the prominent are by Cheng [ 4 ] which proposes public presentation rating and optimum choice of arm systems holding multi-level and multi-factor characteristics. On arm system undertakings, some research workers applied combined attacks such as a intercrossed AHP-integer scheduling attack to test weapon systems undertakings ( Greiner et.al [ 5 ] ) and an AHP attack based on lingual variable weights ( [ 4 ], [ 6 ] ).

Associating AHP with DEA

Despite the utility of AHP, some defects of AHP have been reported in the literature, which have limited its pertinence. The figure of judgements to be elicited in AHP additions as the figure of options and standards addition. This is frequently a boring and exercising exercising for the determination shaper. For illustration, the figure of judgements to be elicited in AHP additions as the figure of options and standards addition. The issue of rank reversal [ 7 ] is another outstanding restriction of traditional AHP. The ranking of options determined by the traditional AHP may be altered by the add-on or omission of another option for consideration. Hence linking of AHP with other techniques such as DEA [ 8 ] has been suggested to get the better of its restrictions. One of the earliest efforts to incorporate DEA with multi-objective additive scheduling was provided by Golany [ 9 ].

Since so, there have been several efforts to utilize the rules of DEA and AHP in the MADM literature. For illustration, some articles [ 10-12 ] have used AHP for managing subjective factors and so used DEA to place efficiency mark based on the full informations, including those generated by the AHP. In add-on, DEA and AHP have been linked with other techniques for specific applications [ 13-14 ]. In this paper, the constructs of efficiency measuring in DEA are integrated with the constructs of weight measuring in AHP and intercrossed attack of Data Envelopment Analytic Hierarchy Process [ 15 ] has been used for arm system choice as an option to the conventional and remarkable methods of weight derivation in analytic hierarchy procedure ( AHP ). To the best of cognition, it is by far a first effort of its sort in the field of defence.

The balance of the paper is organized as follows. The following subdivision briefly discusses the methodological analysiss of DEA, AHP and explains the DEAHP methodological analysis. This is so followed by the application of the DEAHP method to a conjectural missile system choice instance. Decisions are presented in the concluding subdivision.

## Methodology

## 2.1 Analytic Hierarchy Procedure

The AHP consists of three chief operations, including hierarchy building, precedence analysis and consistence confirmation. First of wholly, the determination shapers need to interrupt down complex multiple standards determination jobs into its constituent parts of which every possible properties are arranged into multiple hierarchal degrees. After that, they have to compare each bunch in the same degree in a brace wise manner based on their ain experience and cognition. Since the comparings are carried out through personal or subjective judgements, some grade of incompatibility may be occurred. To vouch the judgements are consistent, the concluding operation called consistence confirmation is incorporated in order to mensurate the grade of consistence among the brace wise comparings by calculating the consistence ratio. If it is found that the consistence ratio exceeds the bound ( i.e. if CR & gt ; 0.1 ), the determination shapers should reexamine and revise the brace wise comparings. Once all brace wise comparings are carried out at every degree, and are proved to be consistent, the judgements can so be synthesized to happen out the precedence ranking of each standard and its properties.

## 2.2 Data enclosure analysis ( DEA )

Data Envelopment Analysis ( DEA ) is a widely applied non-parametric mathematical scheduling attack for analysing the productive efficiency and public presentation rating of Decision Making Unit of measurements ( DMUs ) or with the same multiple input and multiple end products. For each option ( DMU ), the efficiency is measured and an optimisation will be proposed harmonizing to the below indicated additive plan. The DMUs are denoted by. Each DMU I employs m inputs to bring forth s different end products. Specifically, DMU J consumes sum of input I and produced sum of end product r. It is assumed that and and that each DMU has at least one positive input and one positive end product value. Let and denote the input and end product weights severally. Then the inputs and end product weights are calculated for the ascertained DMU0 utilizing the undermentioned additive programming job [ 16 ].

Subject to

( P1 )

The optimum nonsubjective map values of job ( P1 ), when solved, stand for the efficiency mark of the ascertained DMU. This DMU is comparatively efficient if their optimum nonsubjective map value peers integrity. For inefficient units, DEA besides provides those efficient units ( namely equals or benchmarks ), which the inefficient units can emulate to register public presentations that could better their efficiency tonss.

## 2.3 Data Envelopment Analytic Hierarchy Process ( DEAHP )

Here, it is proposed that DEA constructs can be used in the last two stairss of using AHP to a determination problem-namely, deducing local weights from a given judgement matrix and aggregating local weights to acquire concluding weights. Efficiency computations utilizing DEA require end products and inputs. Each row of the judgement matrix is viewed as a DMU and each column of the judgement matrix is viewed as an end product. Thus a judgement matrix of size NA-J will hold N DMUs and J outputs [ 15, 17 ]. Since DEA theoretical accounts can non be used wholly with end products and require at least one input, a silent person input column has been added holding input value as 1 for all DMUs. It is proposed that the efficiency tonss calculated utilizing DEA theoretical accounts could be interpreted as the local weights of the DMUs.

## Traditional AHP View Proposed DEA View

O1

O2

## aˆ¦

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 comparing of the traditional AHP position and the proposed DEA position

( here ALT: option ; Cr: standards O: end product )

While calculating the concluding weights of the chief factors every bit good as the public presentation evaluation of employees, following theorem has been used.

Theorem: Let the local weights of options with regard to different standards be given by the matrix below

Where is the local weight of alternate I ( i=1,2, N ) with regard to standard J. If the importance of standards is incorporated in the signifier of multipliers so concluding weights aggregated utilizing DEA is relative to the leaden amount for the ith option.

## 3. EMPIRICAL Illustration

To exemplify the process involved in the proposed intercrossed attack and to show its effectivity, a conjectural instance on surface-to-air missile system choice is presented.

## 3.1 Problem apparatus and informations

The job is designed as a hierarchal construction of four degrees: First the end of the determination job, followed by the standards, bomber standards, and alternate degrees. As shown in FIGURE 2, to choose an optimum option, six campaigner missile systems are considered and evaluated based on three standards and 18 bomber standards. Each bomber standard, identified and structured in the old phase, has its ain characteristic informations about the campaigner missile system. The standards and characteristic informations were identified based on Ahn ‘s survey [ 18 ]. Data has been arbitrary but meaningfully generated to suits the demands.

Range ( R )

Altitude ( AL )

Hit chance ( HP )

Chemical reaction clip ( RT )

Put up clip ( ST )

Detection marks ( DT )

Engagement marks ( ET )

Inter operability ( IO )

Electronic countermeasures

Anti ARM ( AA )

Mobility ( MO )

Trainability ( TY )

Acquisition cost ( AC )

Care cost ( MC )

Offset trade ( OT )

Technological effects ( TE )

Industrial effects ( IE )

Corporation growing ( CG )

M -1

M-2

M-3

M-4

M-5

Basic capablenesss ( BC )

Costss & A ; proficient effects ( C & A ; T )

Operational capablenesss ( OC )

LEVEL-1

LEVEL-2

LEVEL-3

Missile system choice FIGURE 2: Hierarchical construction for missile system choice

After structuring the hierarchy of factors impacting the missile public presentation, sentiment of the experts can be obtained on the undermentioned issues:

a ) Comparative effects of chief standards on the public presentation of the Missile.

B ) Comparative part of assorted sub factors on the factors mentioned above, e.g

consequence of scope, hit chance, altitude etc. on the basic capableness of the missile.

degree Celsius ) Relative ranking of each alternate missile with regard to each bomber factor.

The qualitative information is obtained in a appropriately designed format enabling pair wise comparing of factors, sub factors and missiles ( FIGURE 3 ). This is communicated in the format by taging ‘X ‘ suitably in one of the columns depending on strength of comparings, i.e equal, moderate, strong, really strong and highly strong. It may be mentioned that in multiple standards determination devising jobs, the sentiment though consistent may be prejudiced or biased towards a specific facet of system. It is hence suggested that to extinguish such prejudice, the sentiment of several experts from different subjects may be elicited. To unite their sentiment geometric mean of the corresponding values of the mated comparing at each phase in the hierarchy may be used for the concluding analysis.

Missiles

Einsteinium

Volt

Second

Multiple sclerosis

Equivalent

Multiple sclerosis

Second

Volt

Einsteinium

Missiles

Garand rifle

M-2

Garand rifle

Ten

M-3

Garand rifle

Ten

M-5

M-2

M-3

M-2

Ten

M-4

M-2

Ten

M-5

M-3

Ten

M-5

M-4

Ten

M-5

FIGURE 3: Format used for brace wise comparings

## 3.2 Calculation of weights of standards

After obtaining the comparing matrices, following measure involves the weight computation of each degree to obtain the overall mark of each missile with regard to all 18 sub-criteria and pair-wise comparings of the chief choice standards.

## 3.2.1 Evaluation of the 3rd degree determination options

The 3rd degree of the hierarchy, as antecedently described, has been analyzed utilizing the DEAHP and AHP methodological analysiss. Decision-makers were asked to stipulate the comparative importance of missile choice standards. In table 1 panel A, seven missile choice standards related to basic capablenesss of the missile choice system which include scope, height, hit chance etc. are compared with each other in pair-wise signifier. In panel B inter comparing of missiles with regard to one of the bomber standards i.e. scope has been shown. In the similar mode inter comparing of missiles with regard to other sub standards of the basic capablenesss has been computed. They have non been shown here due to limitation of research paper length. ( Here R: Scope ; AL: height ; HP: hit chance ; RT: reaction clip ; ST: apparatus clip ; DT: Detection marks ; ET: Engagement marks ).

The computations of the DEAHP attack are besides illustrated utilizing the entries of Table 1. The local weights of options utilizing chip AHP and DEAHP methods are shown in the last two columns of table 1. In the DEAHP method, in order to determine how to deduce local weight from the pair-wise matrix, an case of shipment standard is illustrated in the undermentioned theoretical account.

## Comparison of sub standards with regard to basic capableness standards

## Roentgen

## Aluminum

## Horsepower

## RT

## ST

## DT

## ET

## Input signal

## AHP

## DEAHP

## Roentgen

## 1

## 5

## 4

## 5

## 1

## 4

## 4

## 1

## 0.31629

## 1.000

## Aluminum

## 1/5

## 1

## 1

## 5

## 2

## 1/2

## 4

## 1

## 0.1387

## 1.000

## Horsepower

## 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

## A?

## 2

## 1/2

## 5

## 1/2

## 1

## 4

## 1

## 0.133025

## 1.000

## ET

## A?

## 1/3

## 1/4

## 2

## 1/5

## 1/3

## 1

## 1

## 0.043844

## 0.4

## Comparison of missiles with regard to run bomber standards

## Garand rifle

## M-2

## M-3

## M-4

## M-5

## Input signal

## AHP

## DEAHP

## Garand rifle

## 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 capablenesss

For panel B of table 1, the undermentioned theoretical account is generated.

## ( P2 )

The above theoretical account ( P2 ) can be easy solved utilizing Excel-Solver or Lingo package. The optimum nonsubjective map value of this theoretical account will give the local weight of alternate M-1 i.e. 0.4445. To obtain the local weight of other missiles classs, similar theoretical accounts are used by altering the nonsubjective map can be formulated and solved. The DEA efficiency tonss ( i.e. optimum values of the nonsubjective map in these jobs ) stand foring the local weights of DMUs ( i.e., the Missiles here ) are presented in the 2nd last column of Table 1 panel B.

## Evaluation of the 2nd degree determination options

Once local weights of providers are obtained in the 3rd degree, so they are aggregated to obtain 2nd degree of weights of the determination options. We can now use the DEA attack once more to aggregate the local weights obtained from old informations to bring forth concluding weights which will give the basic capableness of the missile as follows:

Roentgen

Aluminum

Horsepower

RT

ST

DT

ET

Basic capableness ( BC )

Relative weightage

1

1

0.9231

0.2

1

1

0.4

Garand rifle

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: Calculation of concluding weights for the missiles with regard to basic capableness standards

Collection utilizing DEA utilizing the local weights of sub factors: In this instance, once more the DEA theoretical account has been applied for the local weights in Table 2. The local weights are considered as end products of options ( missiles ), and a dummy input is introduced. For illustration, to acquire the concluding weight of missile M-1, the undermentioned theoretical account can be used:

## ( P3 )

The optimum nonsubjective map value of ( P3 ), when solved, will give the concluding weight of alternate missile M-1. To acquire the concluding weight of other missiles, theoretical accounts similar to the above theoretical account should be solved by altering the nonsubjective map. Using the values of local weights of bomber standards, the extra restraints is introduced in the DEA theoretical account ( P3 ) that calculates the concluding weight ( mobility ) of DMU ( missile M-1 ). The ensuing concluding weights of missiles which gives the basic capablenesss are shown in the last column of Table 2, are 0.5410, 1.000, 0.5842, 0.4671and 0.7516.

The concluding weights are relative to the leaden amount of local weights. For illustration, for alternate missile – T1, the leaden amount 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 leaden amount for the five missiles are 2.8630, 5.3284, 3.8952, 3.2677 and 4.0211 which are relative to 0.5410, 1.000, 0.5842, 0.4671and 0.7516. Similarly, the concluding 8s of missiles with regard to other standards such as operational capablenesss and costs can be obtained.

3.2.3 Level 1 analysis

At degree 1 three chief standards viz. basic capableness, operational capableness and costs and proficient effects has been identified and so once more the rules of AHP and DEA has been used as in the above mode to calculate the local weights of the chief factors besides.

## BC

## OC

## C & A ; T

## Input signal

## AHP

## DEAHP

## BC

## 1

## 1/3

## 1/2

## 1

## .334

## OC

## 3

## 1

## 2

## 1

## 1

## C & A ; T

## 2

## 1/2

## 1

## 1

## .67Table 3: Comparison of chief standards with regard to objective

Performance evaluation of missiles: Finally the public presentation evaluation of missiles can be computed by once more utilizing the same DEAHP attack as discussed above. In this instance local precedences or the comparative weights of three chief factors i.e. basic capablenesss, operational capablenesss, costs and proficient effects are taken into consideration. Performance index of assorted missile s comes out as the concluding weights of different missiles.

Standards

BC

OC

C & A ; T

Missile public presentation index

Performance evaluation

Relative weightage

0.334

1

0.67

Garand rifle

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: Options ( missiles M-1, M-2, M-3, M-4, M-5 )

For illustration, to acquire the concluding weight of missile M-1, the following theoretical account has been used:

We can deduce from the consequences of the above analysis that the public presentation evaluation of missile M-5 is 1.54 times the public presentation evaluation of missile M-1. For a comparing of cost-effectiveness of two missile s, their comparative evaluations can be used as effectiveness index. For illustration here if we consider missile M-2, it is about twice every bit efficient as compared to missile M-1.

## Decision

An effort here has been made to supply an alternate attack to the traditional AHP method for the calculation of local weights or precedences and the concluding weights. For a absolutely consistent opinion matrix, this attack gives the similar weights as given by AHP attack where as for inconsistent matrices it tries to take in consistence. Further when concluding weights of the options are computed with regard to each of the chief standards, it was found that they are the leaden amount of local weights.

The comparative weights of each standards factor, viz, basic capablenesss, operational capablenesss, cost and proficient effects is appropriately aggregated along with the comparative weightage of each bomber factor and the evaluations of each bomber factor and the evaluations of each missile with regard to each bomber factor, to give an overall public presentation index of each missile. From the missiles public presentation index, public presentation evaluation of each missile can be calculated.