THE EQUIVALENT FORMS OF MIXED INTEGER LINEAR …€¦ · The Fourier-Motzkin elimination method...

NEW ZEALAND JOURNAL OF MATHEMATICS Volume 27 (1998), 301-315

THE EQUIVALENT FORMS OF MIXED INTEGER LINEAR PROGRAMMING PROBLEMS

N a n Z h u a n d K e v i n B r o u g h a n

(Received February 1997)

Abstract. Using the dual of Fourier-Motzkin elimination method, a Mixed Integer Linear Programming problem can, in general, be transformed into an equivalent continuous Knapsack problem together with a series of linear homogeneous congruences. The number of the congruences is the number of integer variables in the original problem. By ignoring the congruences, a procedure for the Linear Programming relaxation can be obtained directly If all the variables in the mixed problem are integral, this problem can, in the same manner, be transformed into another form that is an integer Knapsack problem together with a series of linear homogeneous congruences in the standard form. Two examples are included.

1. Introduction

The Fourier-Motzkin elimination method (F -M E ) eliminates variables in a linear system, in step by step manner, by constructing a series of inequalities, Dantzig [7]. This method may be regarded as an extension of Gaussian elimination method for a system of linear equations. In [7], F-ME is used to discuss Linear Programming (LP ) problems, and a LP example is investigated. There are many applications of F-ME. For example see the works of (arranged alphabetically): Bradley and Wahi[3], Cabot [4], Chandru [5], Cook and Cooper [6], Duffin [10], Eaves and Rothblum[11], Kohler [19], Kuhn [20], Lee [21], and Williams [23]. ‘ Cook and Cooper are the first to use F-ME to solve Integer Linear Programming (ILP ) problems (see [3, 6]). In [4], F-ME is used to solve an inequality constrained Knapsack problem, and computational experiments are reported.

The dual of Fourier-Motzkin elimination (D F -M E ) is provided by Dantzig and Eaves [8]. This method eliminates homogeneous equations of a LP problem step by step using a series of transformations of variables. In [8], it is also suggested DF-ME might be applied to some particular types of ILP problems after, where possible, taking suitable linear combinations of constraints to produce homogeneous equations for which all coefficients are 0, or ±1. DF-ME is described by Beale [2], and discussed by Knolmayer [16] where effects of the elimination of constraint equations on problem size and computational experiments with different model formulations are reported. This method is further studied for reformulation of production decision models by Knolmayer [18].

1991 A M S Mathematics Subject Classification: 90C10.K ey words and phrases: Dual of Fourier-Motzkin elimination; Mixed integer linear programming;Knapsack problem; Linear Diophantine equation.

302 NAN ZHU AND KEVIN BROUGHAN

Williams [25] shows that DF-ME can be used to eliminate constraint equations of an ILP problem. The result will, in general, be to reduce the constraints of the ILP to a single Diophantine equation, together with a series of linear homogeneous congruences. Extreme continuous solutions to the Diophantine equation give extreme solutions to the LP relaxation. Integral solutions to the Diophantine equation and the congruences give all the solutions to the ILP problem.

Following the work of [8, 25], Zhu [29] uses DF-ME to further discuss the structure of all feasible solutions to a Mixed Integer Linear Programming (M ILP) problem, where only some nonnegative variables are restricted to integer values and the corresponding coefficients may be 1 in the absolute value. Using DF-ME, further investigation is done by Zhu (see Section 2 below).

This paper is written based on the work of [8, 25, 29]. In Section 2, using DF-ME, we prove that a MILP problem, in general, can be transformed into an equivalent continuous Knapsack problem (C K P ) together with a series of linear homogeneous congruences. By ignoring the congruences, a procedure for the LP problem, that is a relaxation of the original mixed problem, can be obtained directly. As another special case, if a MILP is a Pure Integer Linear Programming (.PILP) problem, then this problem can also be equivalently transformed into an integer KP, that is an equality constrained form, together with a series of linear homogeneous congruences in the standard form. In Section 3, two numerical examples are presented to illustrate these results. Conclusions with potential research directions are given in Section 4.

2. M ain Results

A MILP problem can be expressed in the standard form:max C\x + C2y

subject to A\x + A 2y = B, (2.1)x > 0; y > 0 integer vector,

where A\ and A 2 are, respectively, m x n i and m x n 2 integer matrices, C\ and C2 are, respectively, 1 x n i and 1 x n2 rational matrices, B is a m x 1 integer matrix, and x and y are, respectively, the real and integral solution vectors.

If integer restrictions on y are ignored (or n2 = 0), we have a LP problem:max; C X

subject to A X = B , (2.2)X > O,

where A = (Ai, A 2), C = (C\,C 2), O is a zero vector, X = (x ,y )T . If m = 1 and all the coefficients in the left-hand side of the equation are nonnegative, this type of LP(2.2) can be considered to be a CKP.

If integer restrictions on x are added to problem (2.1) (or n\ = 0 ) , we have a PILP problem:

max C X

subject to AX = B, (2.3)

> O integer vector.

MIXED INTEGER LINEAR PROGRAMMING PROBLEMS 303

The linear equations of problem (2.3) are, in fact, a series of linear Diophantine equations in Number Theory (Dickson [9]). If m = 1 and all the coefficients in the left-hand side of the above equation are nonnegative, problem (2.3) can be considered to be an integer KP.

In the following, we provide three main results for the above problems using the sequential elimination procedure of DF-ME. The concept of equivalence, in this paper, means that the feasible sets of the two problems are corresponding, their optimal solutions are corresponding, and their optimal solution values are the same.

Using DF-ME in MILP(2.1), the following result is obtained:

Theorem 2.1. In general, a MILP(2.1) is equivalent to an equality constrained CKP, that is obtained using DF-ME, together with a series of linear homogeneous congruences. The number of the congruences is the number of integer variables in the MILP(2.1).

P roof. As in [25, 29], the elimination procedure can be divided into three main steps.Step 1. Assume y — ( j/i ,. . . ,y n2)T and reformulate problem (2.1) in the form:

max C\x + C2y

B y 0 = 0 , (2.4)

yo = 1, (2.5)

Vi = 0 (mod 1), (2.6)

i e ■{!, ,n 2}.x, yi and y0 > 0,

Step 2. Repeatedly (at most m times) eliminate each of the homogeneous equations in (2.4) using DF-ME, and at the same time, use congruences (2.6) to guarantee that original variables yi are integral. After eliminating a homogeneous equation of (2.4) by a transformation of variables, we obtain a new problem where the number of homogeneous equations is one less than the previous problem’s.

Taking the first equation in (2.4) as an example, the use of DF-ME and the congruence are detailed here. In general, the first equation can be rewritten as:

p q

^ 2 ®iui - Y P m = (2-7)j =i

where all the coefficients ai and (3j are positive integers. Introduce p x q new real variables Zij, and perform following substitutions (2.8) to eliminate equation (2.7):

q p

OtiUi — ^ ̂Z jj , PjVj = ^ ̂%ij5 (2 -8 )j=1 i= 1

Zij > 0 , i e { i , . . . , p } , j e { l Equations (2.7) and (2.8) can be considered to be the constraints of a classical

balanced transportation problem ([7], Guan and Zheng [15]). The amount of supply at origin i is a:*it*, the amount of demand at destination j is fijVj, and (2.7) means the total supply equals the total demand, and represents the amount transported


from origin i to destination j . An interesting property of the constraints is that if all the Ui and v3 take nonnegative (integer) values, then all the nonnegative (integer) values, Zij, are guaranteed to exist.

After using substitution (2.8), equation (2.7) is automatically eliminated. At the same time, the old variables also need to be substituted from other equations, the congruences, and the objective function. To produce integral coefficients in the constraints of the new problem, these constraints may need to be multiplied by appropriate factors.

For example, assuming u\ = yi, from (2.6) and (2.8) we have a congruence:Q

y : Z\j = 0 (mod Qi), (2.9)j = i

where a\ may be 1. Congruence (2.9) guarantees that y\ is an integer. Note that for non-integral variables x of MILP(2.1), congruence relations do not need to be specified at all.Step 3. After all equations (2.4) are eliminated, in general, we obtain a problem (called problem (II) below), that is a CKP together with n2 linear homogeneous congruences:

imax djWj

i=ii

subject to Y^ajW j = n, (2.10)3 = 1

IY^bijWj = 0 (mod rn*), (2-11)3 = i

Wj > 0, i e {1 , . . . , n 2}, j € {1, where all integral coefficients aj, bij > 0. Linear equation (2.10) is obtained from (2.5).

Note that since congruence (2.6) can be written as yi = 1-yi, where yi is integral, congruence (2.11), in fact, represents the linear equation (2.12):

i^ > bjj^j — ' Vi, (2-12)j =i

where i G {1 , . . . , h2}.Now we prove a MILP(2.1) (called problem (I)) is equivalent to a problem (II).

First of all, the solution sets of problems (I) and (II) are corresponding. If (x, y)T is a feasible solution to (I), through DF-ME we can always obtain a corresponding feasible solution w to (II). The converse also holds. If the solution set of (I) is empty, so is (II)’s. Therefore, all the nonnegative solutions to linear equation (2.10) and congruences (2.11) generate all the nonnegative solutions to problem (I).

We prove max f ( x ,y ) = maxg(u>), where f ( x ,y ), g(w) are the objective functions of (I) and (II) respectively. Assuming that the solution set of (I) is not empty and (x * ,y * )T is an optimal solution to (I) with the optimal solution value


a corresponding feasible solution w' to (II) with the objective value g(w') = f(x * ,y * ) can always be obtained. Now assuming w* is an optimal solution to (II) with the optimal solution value g{w*), then it is true g(w') < g(w*). Supposing g(w') < g(w*), by a series of back substitution, a feasible solution (x", y ")T that is corresponding to w* can always be obtained and f { x " ,y " ) = g(w*). Hence, f(x * ,y * ) < f ( x " ,y " ) , and this is a contradiction. Therefore, g{w') = g(w*), and w ' , in fact, is an optimal solution to (II).

If (I) is infeasible, then either (II) is infeasible or the form of (II) can not be obtained from the above steps. □

Applying Theorem 2.1 to LP(2.2), which is a special case of MILP(2.1), we give:

Theorem 2.2. In general, a LP(2.2) is equivalent to an equality constrained CKP obtained using DF-ME.

Proof. From the Proof of Theorem 2.1, we have that a LP(2.2) is, in general, equivalent to a CKP after using DF-ME:

imax E djWj

3=1

Isubject to Y^ajW j = n, (2.13)

j =i

Wj > 0 , j e { 1 ,... , /} ,

where all aj > o- This problem can be considered as the problem formed when congruences (2.11) are removed from problem (II). □

In [25], it is indicated that all solutions to LP(2.2) can be generated by all solutions to equation (2.13), and all extreme solutions to LP(2.2) can be obtained by allowing only one variable to be non-zero in equation (2.13). From the optimization point of view, using Theorem 2.2, the optimal solution to LP(2.2) with the optimal solution value can be obtained by solving CKP(2.13) directly.

If n < 0 in (2.13), LP(2.2) is infeasible, and the optimal solution value does not exist. If n > 0, CKP(2.13) can be solved by two cases: one case is all aj > 0; and the other is some aj > 0, and aj — 0 otherwise.

Case 1. All aj > 0. It is easy to prove that CKP(2.13) has an optimal solution: w* ̂ — n /a j1, and w* = 0 otherwise, with the optimal solution value dj1n /a j1, where d ji /aj i ~ maxi< j< i{d j/a j}. By a series of back substitution, a corresponding optimal solution X * to LP(2.2) will be obtained. If j\ is not unique, more optimal solutions to the original LP may be found using the convex combination of optimal solutions {X * } obtained through back substitutions.

Case 2. Some aj > o and aj = 0 otherwise. To those aj = 0 , if at least one dj > 0 , the original LP(2.2) has m a x /(X ) = +oo; if all dj < 0, the corresponding — 0 in the optimal solution to CKP(2.13), and Case 2 is reduced to Case 1.


The above procedure for LP(2.2), which is a subsidiary result of Theorem 2.1, is reported by Zhu [29, 30] based on the work of [8, 25]. Because of the large size of the transformed variables and resulting data storage requirements, DF-ME, as F-ME, is impractical as a computational tool for general LP problems. However, an equivalent characterisation between LP(2.2) and CKP(2.13) can be directly found through using DF-ME. In [30], this procedure is illustrated by a LP example taken from Beale [1]. Using DF-ME, Zhu and Hu [31] extend the work of [30] to study a LP problem with a parametric objective function, and discuss a numerical example of [15], that example comes originally from Gass ([13], pp. 139-141).

Williams [26] uses DF-ME to study LP problems. Based on the work of [19] and Knolmayer [17], through a numerical example, in [26] it is also shown that some transformed variables can be identified as redundant and assigned the value of zero. It is suggested that some strategies might be adapted for improving computational efficiency. F-ME and DF-ME are discussed further by Williams [27] for two cases of a general LP problem: one case is with parametric right-hand side values of the constraints, the other case is with a parametric objective function. The theoretical insights given by the two methods are demonstrated, as well as their clear geometrical interpretation, in [27].

The procedure provided in Theorem 2.1 can also be applied to PILP(2.3) directly. That is to say, in general, using DF-ME, a PILP(2.3) is equivalent to a CKP together with a series of linear homogeneous congruences, where the number of the congruences is the number of variables in problem (2.3). Because all variables in the problem (2.3) are integral, another equivalent form of PILP(2.3), using the integer characterisation of DF-ME (e.g. see (2.7) and (2.8)), is provided in Theorem 2.3.

Theorem 2.3. In general, a PILP(2.3) is equivalent to an equality constrained integer KP, that is obtained using DF-ME, together with a series of linear homogeneous congruences in the standard form.

Proof. The proof is similar to the proof of Theorem 2.1. The main difference is in specifying congruence relations. As [25], in Step 1, we reformulate (2.3) into the form:

From the proof of Theorem 2.1, we know that, if all variables in (2.3) are nonnegative integers, when using substitution (2.8) to eliminate the homogeneous Dio- phantine equation (2.7), all the transformed variables, Zij, can be guaranteed to be nonnegative integers. Hence, using DF-ME for a PILP problem, we can always choose transformed variables as nonnegative integers so that the new problem is still an integer optimization problem. Thus, in Step 2, when using (2.8) to eliminate (2.7) we should specify p x q congruence relations:

max C X

subject to A X — Byo = O,

Vo = 1,X > O integer vector.

(2.14)

(2.15)

Q P

(2.16)


If some moduli are 1, the corresponding congruences of (2.16) become redundant due to all Zij being integral. For example, if a\ — 1, congruence (2.9) is redundant and should be deleted immediately. Then use the same strategy to eliminate the remaining homogeneous Diophantine equations in the whole of Step 2.

In Step 3, in general, we havei

max E djWj3 = 1

Isubject to ^ a j W j = n, (2-17)

3= i

i

5 2 b'ijwi = 0 (m odm i)> (2-18)3=1

Wj > 0 integer, i E {1 , . . . , A:}, j E { 1 , . . . , / } ,

where all integral coefficients aj, b'̂ > 0, and the modulus m\ > 2.Congruences (2.18) can be expressed in a number of standard forms. Two pos

sibilities are described in [25]. Here, we use the idea of [25] and the Smith normal form of the integer matrix (Garfinkel and Nemhauser [12]) to give a general procedure to derive a new standard form, in which some redundant congruences are eliminated, and the modulus of each congruence divides the modulus of the next congruence. The procedure consists of two steps:Step A (Modulus Decomposition). Assume m!i = flt= i Pt* ■> where p i , . . . ,ps are the distinct prime factors of the least common multiple of the integers {m!l : . . . , m'k}, integer ati > 0, t E {1 , . . . , s} and i E {1 , . . . ,&}• Then congruences (2.18) are equivalent to the following congruences:

iY^b"jW j = 0 (mod Ptu ), (2.19)j - 1

where b"• = b' ̂ (mod p fu), i E {1 , . . . , k}, j E {1 , . . . , /} , and t E {1 , . . . , s}.Some congruences in (2.19) may be redundant in the problem. The right-hand

side value, n, in Diophantine equation (2.17) should also be decomposed in order to identify and eliminate redundant congruences.

Step B (Use the Smith Normal Form). The remaining congruences (assuming the number of congruences is v) can be expressed as a matrix equation: Dw = Bt, where D = (6^)vx«, w = (wi, . . . , wi)T, B is a (v x v) diagonal matrix in which the diagonal elements are the moduli of the congruences, and t = ( i i , . . . ,tv)T is an integer vector. For the given matrix B , we have a unique matrix A = R B C , where A is the Smith normal form of 5 , and R and C are unimodular integer matrices. An algorithm for the Smith normal form can be found in [12]. Letting t' = C _ 1t, we obtain RDw = A t', where t' is guaranteed to be a ii-dimensional integer vector. Transform the matrix equation into a set of linear homogeneous congruences in the


standard form:i

= 0 (mod 8i), (2 .20)j =i

where the integers, fc j, are the elements of RD, the modulus, Si, is the diagonal element of A, in which <5i = . . . = <5r_i = 1, Si > 1, <5j_i is a divisor of Si,i G { r , . . . , t;}, and j G {1 , . . . , /}. Obviously, the first r — 1 congruences of (2.20) should be eliminated since the moduli are 1.

Therefore, in general, using DF-ME and the procedure for the congruence standard form, a PILP(2.3) is equivalent to the following problem:

imax djWj

j =i

isubject to Y^ajW j = n , (2-21)

j =i i

= 0 (mod Si), (2 .22)3=1

Wj > 0 integer, i £ { r , . . . ,v } , j G { ! , . . . , / } .

□As in [12, 25], congruences (2.22) can be expressed as a group equation over a

direct sum group G(Sr, . .. ,SV) of order n iW

3. Tw o Numerical Examples

First, consider a MILP example, where the corresponding PILP problem appears in [12]. A characterisation of the feasible solutions to this PILP example is discussed in [25]. A further discussion on the MILP example is presented in [29], where the MILP example is transformed into a PILP problem using a result of Wolsey (Theorem 1 in [28]) before applying DF-ME.

Example 3.1. Use the results of Section 2 to discuss the following MILP example:max 2x\ + yi

subject to x i+ y\ < 5, (3.1)

- x i + yi < 0, (3.2)

6xi+2yi < 21, (3.3)

X\ > 0 ; y\ > 0 integer.Introducing slack variables X2, £3 and X4, and a new variable yo, we have the

equivalent form:max 2xi + y\


subject to x \ + x 2 + y i~ 5yo = 0, (3.4)

- x i + x 3 + yi = 0 , (3.5)

6x1 + x 4+2yi-21y0 = 0, (3.6)

Vo = 1, (3.7)

V\ = 0 (mod 1), (3.8)

xi, x 2, £3, x4, yi, y0 > 0 .

Using Theorem 2.1, the substitutions required to eliminate the homogeneous equations (3.4), (3.5) and (3.6) are, respectively:

• x 1 = Z\, yi = z2, x2 = and 5y0 = Z\ + z2 + z3, where z 1, z2, and 23 > 0;

• z 1 = Vi + v2, z2 = ui, and x 3 = v2, where Vi, and v2 > 0 ;

• 2v\ = wi + w2, 9v2 — w\ + w3, 2 1z3 — w3 + w4, and 5x4 = w2 + where w 1, ^ 2, ^ 3, and w4 > 0 .

Hence, we obtain an equivalent form:

max (1/18) • (31u?i + 27w2 + 4w3)

subject to 70w i +63w 2+ 10 w3+ 3 w4 — 315, (3.9)

w i+ w2 = 0 (mod 2), (3.10)

wi, W2, W3, w4 > 0, where congruence (3.10) is obtained from congruence (3.8).

Congruence (3.10) is equivalent to wji + W2 = 2 • yi, where yi is the integral variable of the original problem. All feasible solutions that satisfy (3.9) and (3.10) generate all the feasible solutions, corresponding to the segments O C , 0\C\ and 0 2C2 in Figure 1, of Example 3.1 by a series of back substitution. Using the branch-and-bound algorithm for solving the above equivalent problem, we obtain an optimal solution: (u;£, w2, w3, w4)T — (4,0, 7 /2 ,0)T, with the optimal solution value 23/3. Hence, an optimal solution to the example is: (x\ ,yl)T == (17/6, 2)T, corresponding to the vertex C2 in the Figure.

Ignoring congruence (3.10), we have a CKP that is equivalent to the original LP relaxation. All feasible solutions to the CKP generate all the feasible solutions, corresponding to the convex set O A B C in Figure 1, to the LP. In [25], it is showed that vertex solutions A, B , C and O to the original LP can be generated by the extreme solutions to CKP through making back substitutions. Since all the data in (3.9) are positive, and max{31/70,27/63,4/10,0/3} = 31/70, an optimal solution to the CKP is found directly: {w\,w2 ,w 3 ,wX)T = (9 /2 ,0 ,0 ,0)T, with the optimal solution value 31/4. Thus, using Theorem 2.2 an optimal solution to the original L P is: (x \ ,y {)T = (11/4,9/4)T, corresponding to the vertex B in Figure 1.


F i g u r e 1. The solution to Example 3.1

If an integer restriction on x\ is added into the example, using Theorem 2.1, we obtain an equivalent form of the PILP:

max (1/18) ■ (31i(/i + + 4W3)

subject to 70w;i+63w2+10u;3+3w;4 = 315, (3-11)

w i+ u>2 = 0 (mod 2), (3-12)

ll^i-t- 9w 2 + 2w 3 = 0 (mod 18), (3.13)

wu w2, w3, w4 > 0, where congruence (3.13), which is obtained from Xi = 0 (mod 1), is the equivalent form of llw i + 9w2 + 2w^ = 18 • x\.

Using Theorem 2.3 in the corresponding PILP of Example 3.1, i.e. using the same substitutions given above and letting all transformed variables be integral at the same time, we can obtain another equivalent form of the PILP:

. max (1/18) • (31w;i + 27^2 + 4w;3)

subject to 70^1+63w;2+10«;3+3u!4 = 315, (3-14)

w i+ w2 = 0 (mod 2), (3.15)

W3+ W4 = 0 (mod 21), (3.16)

w\ + w3 = 0 (mod 9), (3-17)

w2 + W4 = 0 (mod 5), (3.18)

wi, w2, W3, W4 > 0 integers.


Now use modulus decomposition in congruences (3.15) - (3.18). Since 21 = 3 • 7 and 315 = 32 • 5 • 7, w3 + w4 = 0 (mod 7) in (3.16) and congruence (3.18) are identified as being redundant and eliminated. Thus, the remaining congruences are equivalent to:

W1+W 2 — 0 (mod 2), (3.19)

W3+W 4 = 0 (mod 3), (3.20)

w\ + w 3 = 0 (mod 9). (3-21)Now use the Smith normal form. Express the congruences as a matrix equation:

(3.22)0 0 > 1 1 1 0 y

/ W i \1 w2! w3

\ w 4 J

' 1 1 0 ° \0 0 1 11 0 1 0 J

-1 1 0 \1 CO 2 1

9 - -6 - 2 J

- 2 CO 0 >

1 to 2

CO

1 -1 -1 J

D =

Hence, we have:

R =

C ~ l =

Letting t' = C ~ 1t , (3.22) is equivalent to( w 1 \

w2 w3

\ w4 )

Finally, expressing (3.23) as congruences, we obtain a KP together with a series of linear homogeneous congruences in the standard form:

max (1/18) ■ (31u>i + 27^2 + 4w3)

(3.23)

subject to 70u>i+63,u;2+10w;3+ 3^4 = 315, (3.24)

wi + 2w4 = 0 (mod 3), (3.25)

7u;i+ 9u>2+10it> 3+ 12^4 = 0 (mod 18), (3.26)

wi, w2, w3i W4 > 0 integers..It is easy to verify that the above problem has all twelve nonnegative integer

solutions (see also [25]). These solutions generate all the eight nonnegative integer solutions, corresponding to eight integral points in the convex set O A B C in Figure 1, to the original PILP. .There are three optimal integer solutions:


(wi, W2 , w%, w\)T = (0,2,18, 3)t , (1,1,17, 4)t and (2,0,16,5)T, in the equivalent problem with the optimal solution value 7. Hence, through making back substitutions, the original PILP has the optimal integer solution: (x*,t/i)t = (3 ,1)T, corresponding to the integral point D in Figure 1.

In the following we investigate a Set Partitioning example. The constraint set of the example is taken from Williams [24]. Set Partitioning problems appear frequently in applications, e.g. in the airline industry, Ryan [22].

Exam ple 3.2. Use DF-ME to discuss the following problem:max 3yi + 2y2 + 3y3 + 52/4 + 4y5 + 2y6

subject to 2/1+ 2/2 + 2/5 = 1,

2 /i + 2 / 3 = 1 ,

2/2 + 2 / 4 = 1 ,

2/3 + 2 / 6 = 1 ,

2 /2 + 2 /3 + 2 / 6 = 1 ,

all yi e {0 ,1}.

First we transform the example to an equivalent representation: max 3yi + 2y2 + 3y3 + 52/4 + 4y5 + 2y6

subject to 2/1+ 2/2 +J/5 - 2/0 = 0, (3.27)

2 /i + 2 / 3 - 2 / 0 = 0, (3.28)

2/2 + 2 / 4 - 2 / 0 = 0, (3.29)

2/3 +2/6-2/0 = 0, (3.30)

2/2+ 2/3 + 2/6- 2/0 = 0 , (3.31)

2/o = 1, (3.32)

all yi € {0 ,1}.Applying Theorem 2.3 to this PILP problem, the substitutions required to elim

inate the homogeneous equations (3.27), (3.28) and (3.29) are, respectively:

• 2/1 = wi> 2/2 = U2, 2/5 = U3, and 2/0 = « i + «2 + u3, where all Uj e {0 ,1};• 2/3 = vi + V2, w2 = Vi, and U3 = v2, where all Vj € {0 ,1 };• 2/4 = wi + ^ 2, wi = iui, and ^2 = ^ 2, where all Wj E {0 ,1}.

We have y§ — w\, v\ = 0, and an equivalent 0 — 1 Knapsack problem is:max 10u>i + U w 2

subject to wi + w2 = 1,

wu w2 E {0 ,1}.

(3.33)


Constraint (3.33) is obtained from constraint (3.32).It is easy to obtain an optimal solution to the KP: (tuj, = (0 ,1)T, with the

optimal solution value 12. Hence, through making back substitutions, an optimal solution to the original problem is:

(vlvlvbvlvt^f = ( 0 , 0 , 1 , 1 , 1 , O f .We also find a feasible solution to the KP: (u?i,«;2)t = (1,0)T, with the solution value 10. Hence, a feasible solution to the original problem is:

(2/1 , 2/2, 2/3, 2/4, 2/5, 2/e)r = ( 1 , 0 , 0 ,1 , 0 ,1)T.

Note that when applying Theorem 2.3 to Example 3.2, congruences are not needed. We can also use techniques described in [8] to deal with this example.

4. Conclusions

We have discussed equivalent forms of MILP (including LP and PILP) problems using the sequential elimination procedure of DF-ME. Some results have been provided, and two examples have been given. Studying an equivalent form is of interest because this formulation may be easier to solve than the original one. As an elimination procedure, DF-ME is different from another reduction method, called aggregation method, in Integer Programming (see, for example, Glover and Babayev [14], and Zhu and Broughan [32]). Using DF-ME, the optimality of the LP relaxation is always retained. It is not generally true when the aggregation method is used.

Applying DF-ME to a MILP problem, a large quantity of transformed (integer) variables are produced. To a LP problem, some redundant transformed variables can be identified and eliminated [26]. It might be worthwhile to identify redundant transformed (integer) variables in MILP problems from either a mathematical or computational point of view.

As F-ME, DF-ME could be a tool for reformulating and comparing different forms of the same problem, [18]. As suggested in [8, 16, 25] and shown in Example 3.2 of this paper, DF-ME could be used to eliminate only a subset of the constraints, or examine some particular models (e.g. Matching problems, Set Covering and Set Partitioning problems), in order to reformulate the models prior to applying a solution method. Like [18] for LP models, it might be of value to study DF-ME for MILP problems from a point of view of economics.

Acknowledgem ent. The authors would sincerely thank the referee for the helpful comments.

References

1. E.M.L. Beale, Cycling in the dual simplex algorithm, Naval Res. Logist. Quart.2 (1955), 269-276.

2. E.M.L. Beale, Mathematical Programming in Practice, Pitman, London, Reprinted, 1971.

3. G.H. Bradley and P.N. Wahi, An algorithm for integer linear programming: a combined algebraic and enumeration approach, Oper. Res. 21 (1973), 45-60.

4. A.V. Cabot, An enumeration alqorithm for knapsack problems, Oper. Res. 18 (1970), 306-311.


5. V. Chandru, Variable elimination in linear constraints, Comput. J. 36 (1993), 463-472.

6. R.A. Cook and L. Cooper, An algorithm for integer linear programming, Report No. AM65-2, Washington University, November 1965.

7. G.B. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, Third Printing, 1966.

8. G.B. Dantzig and B.C. Eaves, Fourier-Motzkin elimination and its dual, J. Combin. Theory (A), 14 (1973), 288-297.

9. L.E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea Publishing Company, New York, 1952.

10. R.J. Duffin, On Fourier’s analysis of linear inequality systems, Math. Programming Study, 1 (1974), 71-95.

11. B.C. Eaves and U. Rothblum, Dines-Fourier-Motzkin quantifier elimination and an application of corresponding transfer principles over ordered fields, Math. Programming, 53 (1992), 307-321.

12. R.S. Garfinkel and G.L. Nemhauser, Integer Programming, Wiley, New York, 1972.

13. S.I. Gass, Linear Programming: Methods and Applications, McGraw-Hill, New York, Third Edition, 1969.

14. F. Glover and D.A. Babayev, New results for aggregating integer-valued equations, Ann. Oper. Res. 58 (1995) 227-242.

15. M.G. Guan and H.D. Zheng, Linear Programming, Shandong Scientific Technology Press, Shandong, China, 1983, (in Chinese).

16. G.F. Knolmayer, Balance equations and modelling principles, 6th Annual International Mathematical Programming Symposium, Brussels, November 1976.

17. G.F. Knolmayer, Programmierungsmodelle fur die Produktionsprogrammpla- nung, Birkhaiiser, 1980.

18. G.F. Knolmayer, Computational experiments in the formulation of linear product-mix and non-convex production-investment models, Comput. and Ops. Res. 3 (1982), 207-219.

19. D.A. Kohler, Projections of convex polyhedral sets, Operations Research Centre Report, ORC 67-29, University of California, Berkeley 1967.

20. H.W. Kuhn, Solvability consistency for linear equations and inequalities, Amer. Math. Monthly, 63 (1956), 217-232.

21. R.D. Lee, An application of mathematical logic to the integer linear programming problem, Notre Dame J. Formal Logic, 23 (1972), 279-282.

22. D.M. Ryan, The solution of massive generalized set partitioning problems in aircrew rostering, J. Oper. Res. Soc. 43 (1992) 459-467.

23. H.P. Williams, Fourier-Motzkin elimination extension to integer programming problems, J. Combin. Theory (A), 21 (1976), 118-123.

24. H.P. Williams, Model Building in Mathematical Programming, Wiley, Chichester, 1978.

25. H.P. Williams, A characterisation of all feasible solutions to an integer program, Discrete Appl. Math. 5 (1983), 147-155.

26. H.P. Williams, Restricted vertex generation applied as a crashing procedure for linear programming, Comput. and Ops. Res. 4 (1984), 401-407.


27. H.P. Williams, Fourier’s method of linear programming and its dual, Amer. Math. Monthly, 93 (1986), 681-694.

28. L.A. Wolsey, Group-theoretic results in mixed integer programming, Oper. Res. 19 (1971), 1691-1697.

29. N. Zhu, On the Structure of Feasible Solutions to Mixed Integer Linear Programming Problems, Tech. Rep. The 2nd Scientific Discussion Conference, Sichuan Institute of Finance and Economics, Chengdu, China, September 1984, (in Chinese).

30. N. Zhu, A new algorithm for linear programming - the constraint reduction method, Caijing Kexue, no. 6 (1984) 52-56, (in Chinese); (Current Mathematical Publications 17 no. 14(1985), page 1996).

31. N. Zhu and S.L. Hu, A New Algorithm for a Parametric Linear Program, Tech. Rep., The Operations Research Conference of Sichuan Province, Chengdu, China, December 1984, (in Chinese).

32. N. Zhu and K. Broughan, On aggregating two linear Diophantine equations, Discrete Appl. Math. 82 (1998), 231-246.

Nan Zhu and Kevin Broughan Department of Mathematics University of Waikato Private Bag 3105 Hamilton N E W ZEALAN D [email protected] [email protected]

mailto:[email protected]

mailto:[email protected]

THE EQUIVALENT FORMS OF MIXED INTEGER LINEAR …€¦ · The Fourier-Motzkin elimination method...

Documents

Transcript of THE EQUIVALENT FORMS OF MIXED INTEGER LINEAR …€¦ · The Fourier-Motzkin elimination method...