中華大學

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中華大學碩士論文

題目：Mathematical modeling for battlefield -fuzzy theory approach

基於模糊理論之戰場數學模型建立

系所別：電機工程學系碩士班學號姓名：M09501031 張瑞翔指導教授：駱樂博士

中華民國九十八年七月

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基於模糊理論之戰場數學模型建立

Mathematical modeling for battlefield -fuzzy theory approach

研究生：張瑞翔 Student：Ruey-Shyang Chang 指導教授：駱樂博士 Advisor Dr. Leh Luoh

中華大學

電機工程學系碩士班

碩士論文

A Thesis

Submitted to Institute of Electrical Engineering Chung Hua University

In Partial Fulfillment of the Requirements For the Degree of

Master of Science In

Electrical Engineering July 2009

Hsin-Chu, Taiwan, Republic of China

中華民國九十八年七月

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基於模糊理論之戰場數學模型建立

研究生：張瑞翔指導教授：駱樂博士

中華大學

電機工程學系碩士班

中文摘要

本篇論文主要是使用控制系統方塊圖來建構一個戰場的模型。其中以模糊 控制理論來建立出所有戰場上的情形。並且藉由過去類似戰役的戰場情形，使用模糊關係等式作計算。最後，將計算結果讓指揮者參考，並且有效的分配至真實戰場上。

本篇論文在例子部分，使用歷史上著名的兩場戰役作為驗證。即 Normandy 與 Anzio 戰役。其中加入了 Dupuy 的 TLI 指數與各種參考指標建構成戰場資料庫。使得戰場中所有可能影響戰果的因素，皆列入參考並作計算，來更貼近真實的戰場情形。實驗結果分析，我們所求出指揮官分派部隊數量與歷史是吻合的。

此外，增派的支援數量亦能滿足 Normandy 戰場情形的變化。因此，我們相信本篇論文所建構的戰場模型可以適用於真實戰場上。

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Mathematical modeling for battlefield -fuzzy theory approach

Student : Ruey-Shyang Chang Advisor : Dr. Leh Luoh

Institute of Electrical Engineering Chung Hua University

Abstract

In this thesis, a model of battlefield is constructed by a diagram of the control system block. The fuzzy control theorem is applied to the model in order to build up all situations of battlefields. Also, the fuzzy relation equations are used to compute situations of battlefields by the situations of similar battles before. Finally, the commander references to the computed results and utilizes these results to allocate resources to real battlefields efficiently.

Two renowned battles in history, the battles of Normandy and Anzio, are used as illustrations of the thesis. The TLI index of Dupuy and all kinds of referenced indicators are also constructed into the data bases of battlefield. These parameters, which utilize all possible factors influencing the result of battle in the battlefield, are referenced and computed in order to simulate the real situation of battlefield.

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From the analysis of simulation results, the number of troops computed by this model matches the real number of troops in the historic battle. Besides, the reinforced troops estimated by this model are adaptive to the demanded number of reinforced troops when the situation of Normandy battlefield is not beneficial to our army. Thus, we believe that the model of battlefield constructed in this thesis can be adapted to the real battlefields.

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Acknowledgement

I would like to express my since gratitude for my advisor, Dr. Leh Luoh, for his helpful advice, patient guidance, encouragement, and valuable support during the course of the research. I am obliged to my classmates for their helpful discussions and all my friends for their listing to my mood. Finally I want to express my sincere gratitude for my parents for their encouragement and suggestions.

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LIST OF FIGURES

Figure 1.1 Reference of the commander orders……….2

Figure 1.2 Framework of the war...3

Figure 1.3 Game model classification..………...7

Figure 2.1 Basic configuration of the FLC system...16

Figure 2.2 Configuration of the FLC...17

Figure 2.3 Membership functions ...18

Figure 2.4 A graphical representation of the center of gravity method…...20

Figure 2.5 A graphical representation of the center of average method…………...21

Figure 2.6 A graphical representation of the mean of maximum method...22

Figure 2.7 Solution of fuzzy relation equations…...23

Figure 3.1 The Battle Field Model Block (I)...26

Figure 3.2 The Battle Field Model Block (II)...27

Figure 3.3 The landing in Normandy...45

Figure 3.4 Location of Anzio...46

Figure 3.5 Membership functionx ~₁ x ...54 ₆ Figure 3.6 Membership functiony……...55

Figure 3.7 All situations of battle field of Anzio...58

Figure 3.8 All support situations of battle field of Normandy...62

Figure 3.9 Combat results of attack of Normandy……...63

Figure 3.10 Combat results of defense of Normandy...63

Figure 3.11 Combat results of move of Normandy...64

Figure 3.12 Combat results of hit of Normandy...64

Figure 3.13 Combat results of hide of Normandy...65

Figure 3.14 Combat results of supply of Normandy...65

Figure 3.15 Attack error of combat results…………...66

Figure 3.16 Defense error of combat results…...66

Figure 3.17 Move error of combat results……...67

Figure 3.18 Hit error of combat results……...67

Figure 3.19 Hide error of combat results……...68

Figure 3.20 Supply error of combat results……...68

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Figure 4.1 Open Loop System………...71

Figure 4.2 The New Battle Field Model Block (I)..…...77

Figure 4.3 The New Battle Field Model Block (II)………..77

Figure 1 M4A3 tank……….………86

Figure 2 B-17 Fortress bomber……….………...87

Figure 3 P-51 Mustang battle plane……….………88

Figure 4 Lord Nelson class battleship………….……….………89

Figure 5 Panzerkampfwagen IV...……….………...90

Figure 6 U-submarine…………...……….………..91

Figure 7 He-177 bomber……...……….………..92

Figure 8 Bf-109 battleplane…...……….………..93

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LIST OF TABLES

Table 1.1 Matrix Game in between oppose groups………8

Table 3.1 Capability of infantry division………...30

Table 3.2 Capability of panzer division...30

Table 3.3 Capability of airborne division...……...31

Table 3.4 Capability of battleplane...32

Table 3.5 Capability of bomber…...33

Table 3.6 Capability of battleship...33

Table 3.7 Capability of fuel………...36

Table 3.8 Capability of equipment...36

Table 3.9 Capability of battle machinery...37

Table 3.10 Capability of provision……...38

Table 3.11 Capability of medical treatment...38

Table 3.12 Capability of staff…………...39

Table 3.13 Reference indicator ofR…...40

Table 3.14 Battlefield condition of Anzio and Normandy campaigns….………...53

Table 1 Anzio campaign…………..……….84

Table 2 Normandy campaign……….………..84

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CHAPTER 1 INTRODUCTION

1.1 Motivation

From ancient times, cooperation and competition exist in relationship. Human beings cooperate for common benefits but against for self benefits. There is vary competitions between human beings such as individual, business, military, politics and society, etc. The ways of competition vary according to the type of relationship.

In military relationship, war fighting is the last way of competition. Looking back to all kinds of mankind wars, it can divide into races (種族), countries (國家), areas (地區), civilest (內戰), revolutions (革命), religions (宗教) and violets (武裝暴力), and etc... People will use varieties of weapons to achieve their goal.

As the technology continuesly developing, the damage of the weapon in the war are bigger and stronger. A retired military captain T.N. Dupuy [8,38] offers an method to represent the destroy power of weapon in number, named TLI (theoretical lethality index). This theory is used to calculate the individual power for each weapon. All kinds of harmful weapon could be adjusted by the shooting rate (射擊速率), relatively inactive rate (相對失效率), efficiency of shooting (有效射程), correction (精確度), stability (可靠性), operating radius (活動半徑), and vulnerability (易損性), which is same for spears (矛), swords (劍) from old time, or the guns (槍), cannons (大砲), tanks (坦克), stray bullets (飛彈) presently. Since the harm power of the weapon is getting stronger, the effect zone of the war is also be wilder. Hence, how to use a indication to represent the result of war is apparently important.

Owing to multiple facts exist in the war, there are many uncertainties and each of them may cause results. Therefore, many scholars and militarists keep constructing

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mathematic model for the war, and hoping to trace the changing of war by mathematic model. On the other hands, the commander is the one who handles the changes of the war and provides the orders in a battle. For different commander in different situations, he could make different decision. Therefore, it’s hard to define if the order from the commander is good or not. Generally speaking, most commanders give the order according to Fig. 1.1 [43]:

Fig.1.1 Reference of the commander orders

When awareness that the war, the commander might give a wrong command and cause fatalness false, he is an outstanding leader in the urgent moment. Therefore, how to find a systematic evaluation strategy to solve this situation is the motivation of this thesis. We hope that some batter suggestions can be derived or evaluated from the proposed strategy.

1.2 Purpose

In battlefield, a commander will do his best to provide a strategy for it. The 指揮官命令的下達

歷史戰役本身經驗

主觀意識

偵查狀況

參謀建議

其他…

上級指示

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militarist, B .H. Liddell Hart, have said, “The strategies, is an art, a division, and military tool using, to achieve the political goal” [43].The Dictionary of United States Army Terms has clarified the meaning of war strategy into two definitions:

(1) The employment of units in combat.

(2) The ordered arrangement and the maneuver of units to the enemy.

It other word, if a commander can appropriately arrange all military force or resource in hand to support the war, he might increase the possibility of winning the war. This paper is try to use a mathematic way to present the real situation of the war. From which, the commander can keep the while situation of military force and backup in hand during each battle field. Also, hopefully, it can reduce the possibility of improper command to the battle. A general command dispatch in battlefield can be described as follows:

Fig.1.2 Framework of the war

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Owning to the situation of war is changing, the commander should do the best dispatch at the limit time and backup condition. If the dispatch is inappropriate, then it will affect the results of the battle directly. So, many scholars have tried many ways to solve the resource allocation problems for increasing the possibility of victory.

1.3 Review of Related Work

A success battle in ancient is based on common-sense and experience, the one with better experience has the higher chance to win. At present, however, owing to the power of weapons, the form of fighting, and the complicated development of the society, some mathematics computation have gradually be involved and adopted in battlefield, such as, Lanchester equation, Game theory, Multi-objective ooppttiimmaalliittyy and so on. These theories mostly have been used on decision and resource allocation problems. Here, we will discuss those related theory firstly.

1.3.1 Lanchester Equation

In 1914, Lanchester formulated his well-known equations of warfare [14]. It discovered concurrently and independently by Osipov [20]. Consider a Red army and a Blue army, with populations R=R t( ) and B=B t( ) varying in time. Lanchester’s equations for Area Fire is as follows.

⎪⎩

⎪⎨

⎧

−

=

−

= dt RB dB

dt BR dR

β α

B,R≥0, (1.1)

Where α and β are coefficients reflecting the relative fighting strengths of the

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Blue and Red army. One possible interpretation of α is that it is the product of the probability a blue shot kills a red opponent, and it is similarly forβ. Attrition is proportional to product of army populations. In Lanchester’s equations for Direct Fire is as follows.

⎪⎩

⎪⎨

⎧

−

=

−

= dt R dB

dt B dR

β α

B,R≥0. (1.2)

Attrition is proportional to the attacking army’s population. Lanchester himself considered the latter set of equations to be more descriptive of modern warfare [26].

The above-mentioned equations can be solved analytically. For the Area Fire equations, we obtain Lanchester’s linear law:

α

= β

−

− ) (

) (

0 0

t B B

t R

R (1.3)

for all t ≥ 0, where R₀ =R(0) and B₀ =B(0). For the Direct Fire equations, we obtain Lanchester’s quadratic law:

α

= β

−

2 2

0

2 2

0

) (

t B B

t R

R (1.4)

Lanchester didn’t caught much attention at that moment, and even be thought as paper theory without usage value. Since 1960, when electronic calculator spread using, numerous scholars started to study or using it for different battles [3,4,12,15,23,24,25].

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1.3.2 Game Theory

Game theory [2] is a branch of applied mathematics that is used in the social sciences (mostly in economics), biology, engineering, political science, international relations, computer science (artificial intelligence), and philosophy. Game theory mathematically attempts to capture behavior in strategic situations, in which an individual's success in making choices depends on the choices of each others. The field of game theory came is developed by John von Neumann and Oskar Morgenstern in 1944s. After the 1950s, game theory was broadened theoretically and applied to problems of war and politics [9,34]. In a battle, if we wish for the best result or conquer to the opposite side, we always concern about what strategy we should choose. Then, we choose the best action plan to win the greatest victory within the smallest cost. For all kinds of the game theory, it could be separated by different principles as following:

Game

Static game

Dynamic game

Alignment game

Nonalignment game

Conjoint game

Cooperative game

Finite

Two player zero-sum game Two player non-zero-

sum game Many player zero-

sum game Many player non-

zero-sum game

Infinite

Two player zero-sum game Two player non-zero-

sum game Many player zero-

sum game Many player non-

zero-sum game Differential

game

Fig.1.3 The classification of game model

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In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero. Non-zero-sum games is some outcomes have net results greater or less than zero. The two player zero-sum game occupies the prime position among most of the game models. It is also called matrix game. Here, we clarify it with some examples:

Suppose there are Blue Army and Red Army. If the Red Army try to attack Blue Army defeating territory by 10 tanks force, Blue Army anti-weapon to tanks are including single Solider Antitank Rocket in 3, Mode 60 rocket launcher in 6.

According to the experience in battlefield, the Red Army tanks would attack by the single pathway as the form, or separate into two flank in a raid. Blue Army holds anti-amour weapons with three plan, which are placing them altogether, allocate into two, and three groups. As shown in follows:

Table 1.1 Matrix Game between oppose groups

From the table, it shows the entire amount of losing from the Red Army tanks.

First of all, for making proper decision from the plan, it should pick find the worst result in comparing all the possible plans from the both sides. In mathematic, it called max-min method. Then, according to all worse results, Blue Army chooses the best one for their primary plan. To Blue Army, the disadvantage should be clarify: fewer destroyed tank amounts from the enemy side to be worst; however, to the Red Army, the worst situation would be the higher damage tank amounts to happen.

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Each plan would be able to destroy at least five tanks. On the other hand, no matter which plan the Blue Army process, the Red Army would lose at most six tanks for any pathway of attacking according to the both raid plans from the Red Army. It’s less than what cost them in seven tanks by two-way raid. As the result from the against, it’s clear to know the enemy should chose for single direction as raid plan, but us should take the two-group setting plan.

1.3.3 Multi-objective O Op pt ti im ma al li it ty y

For the problems of resource allocation, there’re numerous objects should concern, but they could also carry conflict chrematistics. The Multi-objective optimality is made to solve these problems, which carry is the conflict objects of resource allocation and decision making. That is, it is called Multi-objective opopttiimmaalliittyy prproobblleemm.. GeGenerally speaking, the three basic chrematistics to form the Multi-objective optimality problem are:

1. At least more than two objects.

2. The conflicts among the objects.

3. The different measuring units from each objects or difficulty to collect calculate for once.

The general Multi-objective ooppttiimmaalliittyy prproobblleemm could be described as follows [5,9]:

Minimize (f x)={f₁(x),f₂(x),...,f_p(x)} (1.5)

Subject to g x_i( )≤ 0, i=1,2,...,m (1.6)

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x_k^l ≤x_k ≤ x_k^u, k=1, 2,...,n (1.8)

where x denotes a vector of decision variable. The last set of constraints are called variable bound which restricts each decision variable x to take a value within lower _k x and an upper l x bound. The function (^u f x ) with p objectives is usually

called a multi-objective function. g x ) and (_i( h_j x ) are called inequality constraints and equality constraints, respectively.

The character of the Multi-objective optimality problem is among its sub- objects. It does affect each others either directly or indirectly. The conflict of the Multi-objective optimality problem is dividing into local conflict and global conflict.

The definition of the local conflict is a series variable to cause both of gradients in sub-object A and B to overlap each other into a line with the opposite direction, shown as Eq.(1.9). Therefore, For the Multi-objective optimality problem about the local conflict, we would just need to figure out the best adjustment amount for these two target functions to find the best planning spot as the solution.

∇f_i(x)=α∇f_j(x), α <0 (1.9)

According to [39] the definition of the global conflict is that a series variable to solve for sub-object A for the best solution is not necessary for the sub-object B.

Nevertheless, when the setting spot turn to be the best solution of sub-object B, it must cost some how to the sub-object A. Therefore, this series variable numbers wouldn’t be the best solution to the sub-object A. As the result, the greatest problem from the Multi-objective optimality problem of the global conflict would happen to have numerous best results. The benefits for the global conflict is not only the

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numerous best results, but also the solutions for scholars in pocket could be more flexible to seek for the most suitable one. That’s why we tend to focus our plan setting in the term of global conflict.

1.4 The Proposed Method

Noted that the two differential equations of Lanchester, either linear law or square law, unable to describe the uncertainty in battlefield. However, for its easy calculation, it’s more useful to the analysis and proven as the backup of the battle.

The advantage for using Game theory and Multi-objective optimality is for the easy calculation, which could help the commander to prove the best decision of military force allocation and delivery. However, the above-mentioned method could only fit in specific situation of the battlefield which are the weaknesses for these method. In other words, it couldn’t present the whole picture in period of battle. Thus, it would make the commander not easy to use in real battlefield. It is because the real battlefield is hard to describe due to its complicate. Thus, any single changing would affect the final result. Besides, the amount from conventional mathematic theory generally comes with the exact measurement. Therefore, the conventional in mathematic way would run to errors easily. For better describing the battle condition, and using some certain way to remain among all the changing from it, then, the more efficient analysis that commander could actual use in the term of the real battlefield.

Nevertheless, the study of the battle theory could match the brand new vision for a huge step. From the above viewpoints, we try to construct the system block diagram of battlefield. With the changing situation of battlefield, this thesis proposed a strategy that the commander add resource support battlefield to guarantee the result what he wish to. In the other hands, he could also adjust the allocation of resource, to more

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ideal arrange for different kinds of battlefields. Further more, the strategy could also use in both of economic and society with some reasonable changing.

1.5 Thesis Organization

This paper is organized as follows. In Chapter 2, the concept of command and control (C2) is dispicted first. Some basic control system block diagram will be constructed for the battlefield. Also, the basic concept the fuzzy theory is explained.

The main issues of fuzzy relation equations used in this thesis are also introduced in detail. In Chapter 3, which is the important contribution of this thesis. The method of construction battlefield model is introduced there. The famous example of Normandy war and Anzio war are also illustrated for verifying the reliability and reality of this thesis. Some discussion, analysis, and complete comparisons with other methods are described in Chapter 4. In Chapter 5, the conclusion and future works are provided. In Appendix, two notable historic wars are introduced and the reference of military strength of each troop is offered.

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CHAPTER 2 BACKGROUND

2.1 Command and Control (C2)

Command and Control (C2), can be defined as the exercise of authority and direction by a properly designated commanding officer over assigned and attached forces in the accomplishment of the mission. Command and control functions are performed through an arrangement of personnel, equipment, communications, facilities, and procedures employed by a commander in planning, directing, coordinating, and controlling forces and operations in the accomplishment of the mission.

Commanding officers are assisted in executing these tasks by specialised staff officers and enlisted personnel. These military staff are a group of officers and enlisted personnel that provides a bi-directional flow of information between a commanding officer and subordinate military units. The purpose of a military staff is mainly that of providing accurate, timely information which by category represents information on which command decisions are based. The key application is that of decisions that effectively manage unit resources. While information flow toward the commander is a priority, information that is useful or contingent in nature is communicated to lower staffs and units.

2.2 Fuzzy Theory

Fuzzy theory provides the forms for representing uncertainties. Historically,

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probability theory has been the primary tool for representing uncertainties that were assumed to follow the characteristics of random in mathematical models. However, not all uncertainties are random. Some types of uncertainties may not suitable for treating or modeling by probability theory exactly. Fuzzy theory is a marvelous tool for modeling the kinds of uncertainties associated with vagueness, imprecision, and/or a lack of information regarding a particular element of the problem at hand.

A fuzzy set is an extension of a crisp set, and it enriches the classical two-valued calculus with a deep and novel perspective. For example, we can define a crisp set A by using the membership method, which introduces a zero or one membership function described by a characteristic function μ_A( )x for A , where

( ) ¹ 0

A

if x A

x if x A

μ ^⎧⎪⎨

⎪⎩

= ∈

∉ . (2.1) However, fuzzy theory extends this concept by defining partial memberships that take value ranging from 0 to 1. A fuzzy set A defined in a universe of discourse U is characterized by a membership μ_A:U→[0,1], where μ_A( )x is denoted as the grade of membership of x in A . When U is discrete, A can be commonly represented as

_A( ) /

U

A=

∑

μ x x, (2.2) where the summation sign does not represent the arithmetic addition; it denotes the collection of all points x U∈ with the associated membership function μ_A( )x . Similarly, when U is continuous, A can be commonly represented as

_A( ) /

A=

∫

Uμ x x, (2.3) Where the integral sign does not denote the operation of mathematical integration; it just denotes the collection of all points x U∈ with the associated membership function ( )μ_A x . The complement of A is also a fuzzy set which can be denoted by

A by the following membership function

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μ_A( ) 1x = −μ_A( )x . (2.4) The intersection of A and B is still a fuzzy set in U , denoted by A B∩ , whose membership function is defined as

μ_{A B}_∩ ( ) min[x = μ_A( ),x μ_B( )]x . (2.5) The union of A and B is a fuzzy set A B∪ in U with membership function μ_{A B}_∪ ( ) max[x = μ_A( ),x μ_B( )]x . (2.6) Here we classify the commonly used operations of fuzzy sets into three parts:

Minimum operation:

( ) ( ) min[ ( ), ( )]

A x B x A x B x

μ ∧μ = μ μ (2.7) Maximum operation:

( ) ( ) max[ ( ), ( )]

A x B x A x B x

μ ∨μ = μ μ (2.8) Product operation:

( ) ( ) ( ) ( )

A x B x A x B x

μ ⋅μ =μ μ (2.9)

2.3 Fuzzy Control System

Fuzzy control was first introduced in the early 1970＇s in an attempt to design controller for systems that are structurally difficult to model due to naturally existed nonlinearities and other complexities [35]. During the past years, fuzzy control has emerged as one of the most active and fruitful areas for research in application of fuzzy set theory.

Since Mamdani and his co-worker have successfully applied the fuzzy logic controller (FLC) to steam engine control, the fuzzy control has been widely applied to many fields [19]. The characteristic of FLC is that it adopts the linguistic control strategy to control plants without realizing their mathematic models. From a set of linguistic rules that describe the operator control strategy, a control algorithm is

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constructed, where the words are defined as fuzzy sets. The main advantages of this approach seem that it is possibility implementing by “ rule of the thumb ” experience, intuition, heuristics, and the fact that it does not need a model of the process. The linguistic control strategy of FLC is constructed according to the operator experience and/or expert knowledge. As a result, the FLC can control the complex and ill-defined industrial processes as well as the skilled operators do. Experience shows that the FLC yields results superior to those obtained by traditional control algorithm in complex situation where the system model or parameters are difficult to obtain.

The basic configuration of the fuzzy logic control (FLC) system is shown in Fig. 2.1 It comprises four principle components: (1) a fuzzification interface, (2) a knowledge base, (3) an inference engine, and (4) a defuzzification interface.

In the following, we discuss the basic destination of each component that is proposed by Lee [16].

Fig. 2.1 Basic configuration of the FLC system

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Fig. 2.2 Configuration of the FLC

2.3.1 Fuzzification Interface

The fuzzification interface consists of a predefined set of linguistic values. Its main function is converted non-fuzzy inputs to fuzzy variable. For example, the linguistic sets can be assigned as

PB: Positive Big NB: Negative Big PM: Positive Medium NM: Negative Medium

PS: Positive Small NS: Negative Small ZO: Zero

According to the expert experience/knowledge determined the shapes of the membership functions of the linguistic sets and the types of the membership functions are always classified into three categories: bell-shaped, triangle shaped, and trapezoid-shaped. Abovementioned membership functions are shown in Fig. 2.3 (a)(b)(c) [27].

Fuzzification Interface

Defuzzification Interface

Inference Engine Knowledge

Base Fuzzy Controller

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(a) bell-shaped (b) triangle shaped (c) trapezoid-shaped Fig. 2.3 Membership functions

2.3.2 Knowledge Base

The knowledge base comprises knowledge of application domain and the control goals to meet. The knowledge base consists of two sections: a database and a rule base. The database provides the fuzzy set definitions of the linguistic values for the linguistic variables of the fuzzy system. It contains information about the boundaries, possible transformations of the domains, and the fuzzy sets with their corresponding linguistic terms. The rule base collects a group of fuzzy IF-THEN rules to represent a mapping of the system from a set of fuzzy inputs to a set of fuzzy outputs. The rules are fuzzy conditional statements in the form of

IF premise, THEN conclusion. (2.10) Fuzzy rules typically express an inference such that if we know a fact called a premise, we can infer, or derive, another fact called a conclusion. The canonical fuzzy IF-THEN rules in the form of Eq.(2.10) is

Rule :^{( )}^l IF x₁ is A₁^l and … and x_n is A_n^l , THEN y is B^l, (2.11) where A_i^l and B^l are fuzzy sets , and x=( , ,..., )x x₁ ₂ x_n ^T∈U and y∈V are the input and output (linguistic) variables of the fuzzy system, respectively. Let M be the number of rules in the fuzzy rule base; that is, l=1, 2,...,M in Eq. (2.11).

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The other fuzzy IF-THEN rules are stated as follows [31]:

(a) “Partial rules”:

Rule :( )^l IF x₁ is A₁^l and … and x_m is A_m^l , THEN y is B^l, where m<n.(2.12) (b) “Or rules”:

Rule :( )^l IF x₁ is A₁^l and … and x_m is A_m^l

or x_m₊₁ is A_m^l₊₁ and … and x_n is A_n^l, THEN y is B^l (2.13) (c) single fuzzy statement:

y is B^l (2.14) (d) “Gradual rules,” for example:

The smaller the x , the bigger the y (2.15) (e) Non-fuzzy rules (that is, conventional production rules).

2.3.3 Inference Engine

There are many compositional methods in fuzzy inference engine such as product inference engine and minimum inference engine etc [27,31,37]. Let U and

V denote the universes of discourse of input and output, respectively. Suppose R

be a fuzzy relation on U V× , and R A: →B be the implication representation where

A U∈ and B∈V . Now given the fuzzy relation R and the input fuzzy set A', then the output fuzzy set B' can be inferred by

Product inference engine:

_'( ) [ _'( ) _'( , )]

i

B x x X A xi R x yi i

μ μ μ

= ∨∈ ⋅ ∀ ∈y_i V (2.16) Minimum inference engine:

'( ) [ '( ) '( , )]

i

B A i R i i

x X

x x x y

μ μ μ

= ∨∈ ∧ ∀ ∈y_i V , (2.17) where the operations ⋅ , ∧ and ∨ are denoted the product, minimum operations as

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2.3.4 Defuzzification Interface

The defuzzification Interface converts the fuzzy output of the rule-base in to a non-fuzzy value. It is defined as a mapping from fuzzy sets B' in V to crisp point

*

y ∈V. Three often-used methods of defuzzification are the center of gravity method, the center average method, and the mean of maximum method. The center of gravity method specifies the y* as the center of the area covered by the membership function of B', that is,

^'

'

* ( )

( )

V B

y y dy y

y dy μ μ

=

∫

⋅

∫

, (2.18) where

∫

Y is the conventional integral. Fig.2.4 shows this operation graphically.

Fig. 2.4 A graphical representation of the center of gravity method

Because the fuzzy sets B' is the union or intersection of ^M fuzzy sets, a good approximation of Eq. (2.18) is the weighted average of the centers of the M

fuzzy sets, with the weights equal the heights of the corresponding fuzzy sets.

Specially, let y^l be the center of the l'th fuzzy set and w_l be its height, the center average method determines y* as

1

*

M l

l l

M

l l

y w y

w

=

∑

(2.19)

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Fig. 2.5 illustrates this operation graphically for a simple example with ^M = 2.

Fig. 2.5 A graphical representation of the center of average method

Conceptually, the maximum method chooses the y* as the point in V at which μ_B_'( )y achieves its maximum value. Define the set

' '

( ') { _B( ) sup _B( )}

y V

hgt B y V μ y μ y

∈

= ∈ = (2.20) that is, hgt B( ') is the set of all points in V at which μ_B_'( )y achieves its maximum value. The maximum method defines y* as an arbitrary element in hgt B( '), that is,

*

y =any point in hgt B( '). (2.21) If hgt B( ') contains a single point, then y* is uniquely defined. If hgt B( ') contains more then one point, then we may still use Eq.(2.21) or use the mean of maximum method.

The mean of maximum method is defined as

( ') ( ')

* ^{hgt B}

hgt B

ydy y

dy

=

∫

(2.22) where

( ') hgt B

∫

is the usual integration for the continuous part of hgt B( ') and is summation for the discrete part of hgt B( '). Fig. 2.6 shows this operation graphically.

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Fig. 2.6 A graphical representation of the mean of maximum method

2.4 Fuzzy Relation Equations

Fuzzy relation equations play an important role in both theoretical and applied research in fuzzy set theory [6]. The notion of fuzzy relation equations is associated with the concept of composition of binary relations. In general, a fuzzy relation equations problem is described as follows. A matrix Q=[q_{jk m n}] _× called the state-matrix and a n-dimensional vector r=[ ]r_k ₁_×_n called the output-vector with

[0,1]

qjk∈ ,r_k∈[0,1] for all j J∈ and k∈K are given, where J ={1,2,..., }m and {1,2,..., }

K = n , ,m n∈ . The problem is to determine all N m-dimensional vectors [ ]_j 1_m

p= p _× , where p_j∈[0,1] satisfying

p Qo =r (2.23) where o denotes max-min composition with ⁽ ^{( ,}j jk⁾⁾ k

j J

max min p q r

∈ = or max-product

composition with ( ( _j _jk)) _k

j J

max min p q r

∈ × = for all k∈K in general [7].

The problem of fuzzy relation equations was first recognized and studied by Sanchez [22]. Although the fuzzy relation equations can be applied to many categories [6], however, the solving procedure of fuzzy relation equations may be tedious [6,10,13]. [10] extendedEq. (2.23) to the case

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P Qo =R, (2.24) where P is an unknown s m× matrix, Q and R are two known m n× and

s n× matrices, respectively. The entries of P , Q , and R belong to the set [0,1]. It is easy to see that this problem can be decomposed into a set of Eq. (2.23). That is, Eq.

(2.24) can be solved s times, repeatedly, by using the method of solving Eq. (2.23).

Above P the solution not only one, possibly to have many P to satisfied.

If p Q R denotes the set of all solutions of p Q( , ) o =r, we call p∈p Q r( , )the maximum solution of ( , )p Q r , if p≤^__P for all p∈p Q r( , ). Meanwhile, p∈p Q r( , ) is called a minimal solution of ( , )p Q r , if p≤ implies p PP = for ( , )p∈p Q r . The set of all minimal solutions of ( , )p Q r is denoted by ( , )p Q r .Let’s explain this conceptual design framework more detail in Fig.2.7.

Fig. 2.7 Solution of fuzzy relation equations

Basically, the procedure for solving Eq. (2.23) has three steps: (a) Check the existence of the solution [18], (b) Find the maximum solution [10], and (c) Find all minimal solutions.

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2.4.1 Check the Existence of the Solution

For the max-min or max-product composition case, if there is some k satisfying _k _jk

j J

maxr max q

> ∈ , (2.25) for all j , ( , )p Q r = . This proposition allows us, in certain cases, to determine quickly φ that Eq. (2.23) has no solution; its negation, however, is only a sufficient, not necessary condition. If Eq. (2.25) does not hold, it does not mean that some solution exists.

On the other hand, if the solution set is empty, [17,21] have provided an approximate solution.

2.4.2 Find the Maximum Solution

For the max-min composition case, the maximum solutionp=[ ,...,p₁ p_j,...,p_m] of Eq. (2.23) is

( _jk, ),_k

j k K

p minσ q r

= ∈ (2.26) where ,

( , ) 1,

k jk k

jk k

r if q r

q r

otherwise

σ _{≡ ⎨}^⎧ ^>

⎩ .

For the max-product composition case, the maximum solution p =[ ,...,p₁ p_j,...,p_m] of Eq. (2.23) is given by Eq. (2.26), where

( , ) ,

1,

k

jk k

jk k jk

r if q r q r q

otherwise σ

⎧ >

≡ ⎨⎪

⎪⎩

.

2.4.3 Find All Minimal Solutions

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The minimal solutions can be found in the previous papers [7,13,18]. Assume that we have a method for solving Eq. (2.23) only for the first decomposition problem (given Q and R).

Then, we can indirectly utilize this method for solving the second decomposition problem as well. We simply rewrite Eq. (2.23) in the form

T T T

P Qo = ⇔R Q oP =R (2.27) Employ transposed matrices. We can now solve Eq. (2.27) Q^Tby our method and, then, obtain the solution of Eq. (2.23) by

(Q^{T T}) = (2.28) Q

2.5 Fuzzy Composition

It is feasible to use the method of fuzzy relation equation (F.R.E.) in control problems. F.R.E. is described detailedly in [28] and the related applications of F.R.E.

are introduced in [33]. While fuzzy controllers have been successful in a variety of practical situations (see Tong [29] for a review), their design has relied on adhoc techniques. That is, the controllers were not synthesized using an underlying theory but were arrived at by trial and error. The results presented here were motivated by, and provide a partial resolution of, this situation. The analysis is not exhaustive, but it does provide some insight into the basic operation of fuzzy feedback systems and does indicate how a wider range of systems problems might be addressed.

The correspondence is concerned only with finite discrete relational fuzzy systems in that the state, input, and output variables are treated as finite discrete multidimensional fuzzy sets. There are two main reasons for this. First, in all practical situations the power of the fuzzy approach comes from the ability to express process behavior, design goals, and other important systems features in a linguistic form. The

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most natural and simplest representation of such information is in relational terms.

Second, any implementation of the ideas must involve a digital computer, which implies both finiteness and discreteness. Rather than consider continuous relational systems and then have to approximate them, it is better to make the practical issues explicit.

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CHAPTER 3 COMMAND AND CONTROL (C2), FUZZY RELATION EQUATION APPROACH

This chapter will describe the battlefield construct process and patterns carefully in this paper. This paper will be more truly and correctly, because the Chapter 2.3 and Chapter 2.4 to describe fuzzy control system and fuzzy relation equation (F.R.E.) methods. Then it can go with feedback control theorem to build battlefields block diagram and take the real battlefields for example to prove.

3.1 Notation

This paper is using control system block to built the battlefield model as follows Fig. 3.1：

Fig. 3.1 The Battle Field Model Block (I)

Above the battlefield model as follow:

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Fig. 3.2 The Battle Field Model Block (II)

Where P denote the amount of the military force that commander has to send. M denote the average value of all the battlefield conditions in past battlefields. Q is made from Fig. 3.2 between C and N fuzzy relation, it denotes the average value of all the battlefield conditions in actual battlefields. C is the amount of the support that commander has to send. N is the average value of all the battlefield conditions of support in actual battlefields. R is the battle result as commander except. 'R is the actual battle result. E is error between R and 'R . They are matrixes that are symbols at above in the model, called MIMO system.

In every matrix, the elements are 0 ~ 1 degree. The lowest degree is 0 but the highest degree is 1. The R 、 M and N have known already in this matrix then C and N are square matrixes.

3.2 Requirement & Practical Method of the Block Diagram

When a commander allocates the military force for a battlefield, he would concern according to the past similar fight from his experience or history reference according to the average condition ( M ) and the exceptive result ( R ). For the past conditions (M₁,M₂,…,M_i) , he could decide the amount of the military force to send ( P ) due to which fighting result he wished, to send forward to the battlefield. When the sending military force ( P ) is changing due to the condition varying, the real and

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the expected result of the commander could happen to come out error ( E ) To reduce the gap of error for the failure of the battle, the commander should continue to increase the support (C) the error could remain in the acceptance range (ε ) by conclude the changing of the real battlefield. However, the two battlefields require the following limits:

(1) They have to be the same type of battlefield.

(2) The resource and the weapon from them among all the military groups should not be too big in gap.

(3) The weather and the landforms in these two battlefields should not be too much different.

3.3 The Formation Ways of the Battle Condition

This section is for further description for a battlefield; from its battle condition since begin till the end, which is among all the attending military in the term of ability value changing. This ability value would be vary due to the amount of the military force, the cooperation military force, the enemy military force, and the cooperation military force for our enemy side. Therefore, we could create the battlefield conditions and its changing according to the above reference values, and its support for each military abilities in the battlefield.

The conditions of each battle is the sequence changing from begin till the end, and we could choose some prime issues that could affect the battle result as our major battle conditions, such as,M₁,M₂,…,M_i. When the selections of the battle event more, the descriptions for it could be more in detail; however, the calculation would be more complicate and time would needed longer. Therefore, the amount of battle condition select for a commander could also be another attention issue.

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3.3.1 Reference Ability value Among All Military in the Battlefield

Due to the different abilities among each military, the vary types of weapon and the bombs in the battlefield, the concern requirement would be much different. As the result, we have to make the ability into numerical value way for using for the calculation in the math model in this section. According to the performance in the battlefield, we divide the ability into six, Attack (Ak), Defense (De), Move (Me), Hit (Ht), Hide (He), and Supply (Sy). Attack means the ability for us to damage the enemy force during the fighting in the battlefield. Defense means the ability for the enemy to cause our lost during the fighting in the battlefield. Move means the ability for us to do the movement and evade from the enemy gunfire in the battlefield. Hit means the ability for us to hit the enemy military in the battlefield. Hide means the ability for the enemy to hit our military in the battlefield. Supply means the ability for us to support from our backup teams while we are pushing forward in the battlefield.

In the other hands, the differentiation for the military kinds is also according to the attending armies in each battlefield. According to our using samples of the battle, the types of army could divide into six: Infantry division (ID), Panzer division (PD), Airborne division (AD), Battle plane (BP), Bomber (Bb), and Battleship (BS). Thus, we would take these six army types as our reference index. The reference index shown as Table 3.1~Table 3.6:

s1 s₂ s ₃ s₄

Attack Crack degree The enemy disperses degree

Defense of the enemy

TLI

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Defense Crack degree Intensity of the fortification

Attack of the enemy

Equipment

Move Topography Weight of equipment

Move the pace

Intensity of enemy's fire power strength

Hit Climate Topography Distance of

goals

Hide of enemy

Hide Topography Climate Barrier Hit of enemy Supply Food and

water

Ammunition quantity

Deepen the degree

Capability of supply Weight w₁ =0.2 w₂ =0.2 w₃ =0.3 w₄ =0.3

Table 3.1 Capability of infantry division

s1 s₂ s ₃ s₄

Attack Tank type Infantry's ability

TLI

Defense Armoured thickness

Intensity of the fortification

Attack of the enemy

Tank intensity of enemy Move Topography Weight of the

tank

Move the pace

Intensity of enemy's fire power strength

goals

Hide of enemy

Hide Topography Climate Tank volume Hit of enemy Supply The fuel

losses

Ammunition quantity

Deepen the degree

Capability of supply

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Weight w₁ =0.2 w₂ =0.2 w₃ =0.3 w₄ =0.3 Table 3.2 Capability of panzer division

s1 s₂ s ₃ s₄

Attack Crack degree The enemy disperses degree

TLI

Defense Crack degree Place of landing Attack of the enemy

Equipment

Move Topography Weight of equipment

Move the pace Intensity of enemy's fire power

strength

goals

Hide of enemy

Hide Topography Climate Barrier Hit of enemy Supply Food and

water

Ammunition quantity

Deepen the degree

Capability of supply Weight w₁ =0.2 w₂ =0.2 w₃ =0.3 w₄ =0.3

Table 3.3 Capability of airborne division

s1 s₂ s ₃ s₄

Attack Battleplane type Ammunition type

TLI

Defense The largest height

Fight ability Attack of the enemy

Intensity of enemy

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Move Climate Weight of the battleplane

Flying speed Antiaircraft intensity of

enemy Hit Wind direction Fight radius Distance of

goals

Hide of enemy

Hide Radar sensitivity of enemy

Climate Volume of

battle plane

Hit of enemy

Supply The fuel losses Ammunition Deepen the degree

Distance of the airport Weight w₁ =0.2 w₂ =0.2 w₃ =0.3 w₄ =0.3

Table 3.4 Capability of battleplane

s1 s₂ s ₃ s₄

Attack Bomber type Ammunition type

TLI

Defense The largest height

The battle plane convoys ability

Attack of the enemy

Intensity of enemy Move Climate Weight of the

bomber

Flying speed Antiaircraft intensity of

enemy Hit Wind speed Release altitude Goal range Hide of enemy

Hide Radar

sensitivity of enemy

Climate Volume of

bomber

Hit of enemy

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Supply The fuel losses Ammunition quantity

Deepen the degree

Distance of the airport Weight w₁ =0.2 w₂ =0.2 w₃ =0.3 w₄ =0.3

Table 3.5 Capability of bomber

s1 s₂ s ₃ s₄

Attack Battle ship type Cannon quantity Defense of the enemy

TLI

Defense Armoured thickness

Convoy warship ability

Attack of the enemy

Intensity of enemy's fire power strength Move Topography of

the bank

Displacement Sail speed Intensity of artillery fire of

enemy Hit Wind direction Distance of goal Goal range Hide of enemy Hide Radar

sensitivity of enemy

Climate Volume of

battle ship

Hit of enemy

Supply The fuel losses Ammunition quantity

Distance of the bank

Distance of port

Weight w₁ =0.2 w₂ =0.2 w₃ =0.3 w₄ =0.3 Table 3.6 Capability of battleship

According to military reference index, the calculation formula of ability value as follows:

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1 n

k k k

A s w

=

= ∑

(3.1)

where A denote capability value of the army. s is reference index of every _k abilitys. w is weighting. If we want to know the armies attack power of allied _k forces at battlefield, the calculation formula as follows：

Attack of infantry division = (Crack degree×0.2)+( The enemy disperses degree ×0.2)+( Defense of enemy ×0.3)+(TLI×0.3).

For the attacking ability for this paper, please use TLI value [8] as the reference.

However, due to destroy force in this book is only be useful in the term of the attacking but not for other terms; thus, we would adjust other ability from the author’s own reference. For further, if it could hold any better description database for reference, we would provide for more specify as the information.

3.3.2 The setting methods of the vary ability value among armies

After the value each ability from the individual army is present, we would make it as the first index from begin of the war, called Initial Battlefield Condition (M₁).

The later battlefield condition would change by following the ability from the individual army in varying. Let’s using the method of the fuzzy control system, from Chapter 2.2, to calculate Fig. 2.2; first, we have to define the word term of variable in both of the front and the rear part; also the shape types of the membership function, from the fuzzy rule base, as shown in Chapter 2.2.1. Second, we would use IF-THEN

中 華 大 學

中 華 大 學 碩 士 論 文

題目：Mathematical modeling for battlefield -fuzzy theory approach

基於模糊理論之戰場數學模型建立

系 所 別：電機工程學系碩士班 學號姓名：M09501031 張瑞翔 指導教授：駱 樂 博士

中華民國 九十八 年 七 月

基於模糊理論之戰場數學模型建立

Mathematical modeling for battlefield -fuzzy theory approach

中 華 民 國 九 十 八年 七 月

基於模糊理論之戰場數學模型建立

Mathematical modeling for battlefield -fuzzy theory approach

Abstract

Acknowledgement

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF TABLES

CHAPTER 1 INTRODUCTION

1.1 Motivation

1.2 Purpose

1.3 Review of Related Work

1.3.1 Lanchester Equation

1.3.2 Game Theory

1.3.3 Multi-objective O Op pt ti im ma al li it ty y

1.4 The Proposed Method

1.5 Thesis Organization

CHAPTER 2 BACKGROUND

2.1 Command and Control (C2)

2.2 Fuzzy Theory

∑

∫

2.3 Fuzzy Control System

2.3.1 Fuzzification Interface

2.3.2 Knowledge Base

2.3.3 Inference Engine

2.3.4 Defuzzification Interface

∫

∫

∫

∑

∑

∫

∫

∫

2.4 Fuzzy Relation Equations

2.4.1 Check the Existence of the Solution

2.4.2 Find the Maximum Solution

2.4.3 Find All Minimal Solutions

2.5 Fuzzy Composition

CHAPTER 3

COMMAND AND CONTROL (C2), FUZZY RELATION EQUATION APPROACH

3.1 Notation

3.2 Requirement & Practical Method of the Block Diagram

3.3 The Formation Ways of the Battle Condition

3.3.1 Reference Ability value Among All Military in the Battlefield

A s w

= ∑

3.3.2 The setting methods of the vary ability value among armies

中華大學

中華大學碩士論文

系所別：電機工程學系碩士班學號姓名：M09501031 張瑞翔指導教授：駱樂博士

中華民國九十八年七月

中華民國九十八年七月