A Comparative Study of Redundant Constraints Identification Methods in Linear Programming Problems

Abstract

The objective function and the constraints can be formulated as linear functions of independent variables in most of the real-world optimization problems. Linear Programming (LP) is the process of optimizing a linear function subject to a finite number of linear equality and inequality constraints. Solving linear programming problems efficiently has always been a fascinating pursuit for computer scientists and mathematicians. The computational complexity of any linear programming problem depends on the number of constraints and variables of the LP problem. Quite often large-scale LP problems may contain many constraints which are redundant or cause infeasibility on account of inefficient formulation or some errors in data input. The presence of redundant constraints does not alter the optimal solutions(s). Nevertheless, they may consume extra computational effort. Many researchers have proposed different approaches for identifying the redundant constraints in linear programming problems. This paper compares five of such methods and discusses the efficiency of each method by solving various size LP problems and netlib problems. The algorithms of each method are coded by using a computer programming language C. The computational results are presented and analyzed in this paper.

1. Introduction

Many researchers [1–17] have proposed different algorithms to identify the redundancies and removed them to get a reduced model for linear programming. In 1965, Zionts [17] suggested some improvements upon the implementation of Boot method, but not to the point where it achieved practical value. In addition, a number of other methods were developed that deal with redundancy, among which the geometric vertex enumeration method is the most well known. In geometric vertex enumeration method, the essential characteristic is the establishment of a number of situations in which redundancy can be recognized immediately without further computations.

In 1971, Lisy [12] used the rules given by Ziont to identify all redundant constraints in systems of linear constraints. Gal [7] enlarged this approach by adding rules for situations in which constraints can be identified immediately as being nonredundant. Gal proposed another method to classify constraints as redundant or necessary. They produce results that are unconditionally correct; they perform iterations of an active set linear programming algorithm. Later Caron et al. [6] appended the above methods by adding rules to deal with degeneracy.

Brearly et al. [4] proposed a simple method to identify redundant constraints from a system of linear constraints. This method involves the lower and upper bounds of the variables. Telgan [15] proposed a deterministic method to identify redundant constraints by using minimum ratio criteria as in simplex method. Stojković and Stanimirović [13] proposed a method to identify redundant constraints by applying the maximum and minimum principle. Paulraj et al. [14] proposed a heuristic method to identify redundant constraints by using the intercept matrix of constraints of a linear programming problem. Gutman and Ioslovich [8] described a new approach to preprocess nonnegative large-scale problems so as to reduce the dimensions considerably by defining and removing redundant constraints and variables. This test is applicable to all nonnegative large-scale linear programming problem with group constraints. Group constraints only contain zeros and ones coefficients. Constraints and variables are removed by primal and dual tests. This method is applicable to constraints of knapsack problems.

A brief introduction to the redundant constraints of linear programming problems is presented in Section 2. Section 3 discusses the methods for identifying redundant constraints in linear problems. Section 4 deals with the computational results of the methods, and Section 5 concludes the paper.

2. Redundant Constraints

A redundant constraint is a constraint that can be removed from a system of linear constraints without changing the feasible region.

Consider the following system of 𝑚 nonnegative linear inequality constraints and 𝑛 variables (𝑚≥𝑛): 𝐴𝑋≤𝑏,𝑋≥0, (2.1) where 𝐴∈𝑅^𝑚𝑥𝑛,𝑏∈𝑅^𝑚,𝑋∈𝑅^𝑛, and 0∈𝑅^𝑛.

Let 𝐴_𝑖𝑋≤𝑏_𝑖 be the 𝑖th constraint of the system (2.1) and let 𝑆={𝑋∈𝑅^𝑛/𝐴_𝑖𝑋≤𝑏_𝑖,𝑋≥0} be the feasible region associated with system (2.1).

Let 𝑆_𝑘={𝑋∈𝑅^𝑛/𝐴_𝑖𝑋≤𝑏_𝑖,𝑋≥0,𝑖≠𝑘} be the feasible region associated with the system of equations 𝐴_𝑖𝑋≤𝑏_𝑖,𝑖=1,2,…,𝑚,𝑖≠𝑘. The 𝑘th constraint 𝐴_𝑘𝑋≤𝑏_𝑘(1≤𝑘≤𝑚) is redundant for the system (2.1) if and only if 𝑆=𝑆_𝑘.

Definition 2.1. Redundant constraints can be classified as weakly and strongly redundant constraints.

Weakly Redundant Constraints
The constraint 𝐴_𝑖𝑋≤𝑏_𝑖 is weakly redundant if it is redundant and 𝐴_𝑖𝑋=𝑏_𝑖 for some 𝑋∈𝑆.

Strongly Redundant Constraints
The constraint 𝐴_𝑖𝑋≤𝑏_𝑖 is strongly redundant if it is redundant and 𝐴_𝑖𝑋<𝑏_𝑖 for all 𝑋∈𝑆.

Binding Constraint
Binding constraint is the one which passes through the optimal solution point. It is also called a relevant constraint.

Nonbinding Constraint
Nonbinding constraint is the one which does not pass through the optimal solution point. But it can determine the boundary of the feasible region.

Example 2.2. Consider the following linear inequality constraints:
(1) 2𝑥₁+1𝑥₂≤8,
(2) 4𝑥₁+0𝑥₂≤15,
(3) 1𝑥₁+3𝑥₂≤9,
(4) 1𝑥₁+2𝑥₂≤14,
(5) 1𝑥₂≤4,
(6) 1𝑥₁+1𝑥₂≤5,
where 𝑥₁+𝑥₂≤0.
In Figure 1, the region OABCD is the feasible region and the vertex 𝐶 is the optimal point. The constraints (1), (3), and (6) are binding, (4) and (5) are strictly redundant. The 2nd constraint is non-binding. Among the binding constraints, (6) is weakly redundant.

Figure 1 

3. Methods for Identification of Redundant Constraints

Many methods are available in the literature to identify the redundant constraints in linear programming problems. In this paper, the following five methods are discussed and compared (1)bounds method [4](2)linear programming method [6](3)deterministic method [15](4)stojković and Stanimirović method [13](5)heuristic method [14].

3.1. Bounds Method

Brearly et al. [4] proposed a simple method for identifying redundant constraints for a Linear Programming Problem (LPP) with bounded variables. This method involves the lower and upper bounds of the variables. The upper and lower bounds of each constraint are computed and compared with the right-hand side of that constraint to decide if it is a redundant constraint or not.

Procedure of the Method.
The general form of an LPP with bounded variables is max𝑧=^𝑛_𝑗=1𝑐_𝑗𝑥_𝑗,Subjectto^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗≤𝑏_𝑖(),𝑙𝑖=1,2,…,𝑚_𝑗≤𝑥_𝑗≤𝑢_𝑗𝑗=1,2,…,𝑛. (3.1)

Step 1. Compute upper and lower bounds for each constraint by 𝑈_𝑖=_{𝑗∈𝑃𝑖}𝑎_𝑖𝑗𝑢_𝑗+_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑙_𝑗,𝐿_𝑖=_{𝑗∈𝑃𝑖}𝑎_𝑖𝑗𝑙_𝑗+_{𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑢_𝑗, (3.2) where 𝑃_𝑖={𝑗;𝑎_𝑖𝑗>0}, and 𝑁_𝑖={𝑗;𝑎_𝑖𝑗<0}, 𝐿_𝑖 may be 0, and 𝑈_𝑖 may be +∞.

Step 2. Test whether U_𝑖≤𝑏_𝑖,𝑖=1,2,…,𝑚. The 𝑖th constraint ∑^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗≤𝑏_𝑖 is redundant if U_𝑖≤𝑏_𝑖.

Example 3.1. Consider the following LPP: max𝑧=4𝑥₁+2𝑥₂+𝑥₃,subjectto2𝑥₁+𝑥₂+𝑥₃≤30,3𝑥₁+𝑥₂+𝑥₃𝑥≤26,₂+𝑥₃𝑥≤13,₁+2𝑥₂+𝑥₃≤45,0≤𝑥₁≤8.67,0≤𝑥₂≤13,0≤𝑥₃≤13. (3.3)

Solution 1. Step 1. Define 𝑃₁={1,2,3},𝑁₁𝑃=𝜑,₂={1,2,3},𝑁₂𝑃=𝜑,₃={2,3},𝑁₃𝑃=𝜑,₄={1,2,3},𝑁₄=𝜑, (3.4) where 𝜑 is the empty set.Step 2. Compute 𝐿₁=0,U₁𝐿=17.34+13+13=43.34,₂=0,U₂𝐿=26.01+13+13=52.01,₃=0,U₃𝐿=13+13=26,₄=0,U₄=8.67+26+13=47.67. (3.5) Since all U_𝑖≥𝑏_𝑖, 𝑖=1,2,3,4, there is no redundant constraint found by this method.

3.2. Linear Programming Method

Caron et al. [6] developed an algorithm for identifying redundant constraints. This method will take more computational effort to identify the redundant constraints. To identify the redundant constraints, the left-hand side of each constraint is optimized subject to the remaining constraints. The optimal objective functional value is compared with the right-hand side value of corresponding constraints to decide if it is redundant or not. In this method, the objective function of the original LPP is not considered.

Let 𝐏 denote the given linear programming problem.

Let 𝑃_𝑖 denote the LP Problem without the 𝑖th constraint ∑^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗≤𝑏_𝑖 of 𝑃 and let the objective function of LP problem 𝑃_𝑖 be max𝑧_𝑖=∑^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗.

Step 1. Find the optimal objective function value to the problem 𝑃_𝑖, 𝑖=1,2,…,𝑚, by using the simplex method. Let 𝑧_𝑖 be the optimal objective function value of problem 𝑃_𝑖.

Step 2. Check whether 𝑧_𝑖≤𝑏_𝑖. The 𝑖th constraint ∑^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗≤𝑏_𝑖 is redundant if 𝑧_𝑖≤𝑏_𝑖.
Otherwise, it is not redundant.

Example 3.2. Consider the Example 3.1 presented in Section 3.1.

Solution 2. By solving the above Example 3.2, we get 𝑧=43.33,  𝑥₁=4.33,  𝑥₂=13.00, and 𝑥₃=0.Step 1. For 𝑖=1,2,3,4, consider the problem 𝑃_𝑖 as follows: max𝑧_𝑖=left-handsideofconstraint𝑖,subjectto3𝑥₁+𝑥₂+𝑥₃𝑥≤26,₂+𝑥₃𝑥≤13,₁+2𝑥₂+𝑥₃𝑥≤45,₁,𝑥₂,𝑥₃≥0. (3.6)
Using the simplex method,the solution of problem 𝑃₁ is 𝑧₁=21.67, 𝑥₁=4.33, 𝑥₂=13.00, and 𝑥₃=0.00.the solution of problem 𝑃₂ is 𝑧₂=45, 𝑥₁=15, 𝑥₂=0.00, and 𝑥₃=0.00.the solution of problem 𝑃₃ is 𝑧₃=26, 𝑥₁=0.0, 𝑥₂=19.00, and 𝑥₃=7.00.the solution of problem 𝑃₄ is 𝑧₄=30.33, 𝑥₁=4.33, 𝑥₂=13.0, and 𝑥₃=0.00.Step 2. Here 𝑧₁<𝑏₁,𝑧₂>𝑏₂,𝑧₃>𝑏₃, and 𝑧₄<𝑏₄. Therefore, the constraints (1) and (4) are redundant.

3.3. Deterministic Method

Telgan [15] developed an algorithm for identifying redundant constraints and implicit equalities in system of linear constraints using minimum ratio criteria as in the simplex method.

Procedure of the Method
Assume that a basic feasible solution is given, and the corresponding contracted simplex tableau is set up. Let 𝑦_𝑖𝑗 be the entries of this tableau with 𝑦_𝑖0 the right-hand side coefficients, and let 𝑥^𝐵_𝑗 and 𝑥^𝑁_𝑗 represent the basic and nonbasic variables, respectively. Let 𝐻 be the set of all indices of constraints to be identified as either redundant or nonredundant.
Let 𝑢_𝑘 be the slack variable corresponding to the 𝑘th constraint.
Step 1. If the solution is nondegenerate, all 𝑢_𝑘=𝑥^𝑁_𝑗 correspond to nonredundant inequalities, remove 𝑘 from 𝐻 and continue with Step 3.Step 2. In a degenerate solution check all nonbasic variables 𝑢_𝑘=𝑥^𝑁_𝑝 with 𝑘∈𝐻. Check the property 𝑦_𝑖𝑝≥0 for all𝑖 with 𝑦_𝑖0=0. If this holds, then the constraint 𝐴_𝑘𝑥≤𝑏_𝑘 is not redundant, and remove 𝑘 from 𝐻.Step 3. Check all basic variables 𝑢_𝑘=𝑥^𝐵_𝑟 with 𝑘∈𝐻 for the property 𝑦_𝑖𝑗≤ 0 for all 𝑗. If this holds, then the constraint 𝐴_𝑘𝑥≤𝑏_𝑘 is redundant, and remove 𝑘 from 𝐻.Step 4. Check all basic variables 𝑢_𝑘=𝑥^𝐵_𝑖 with 𝑘∈𝐻 for the property 𝑦_𝑟0/𝑦_𝑟𝑠=min{𝑦_𝑖𝑜/𝑦_𝑖𝑠,𝑦_𝑖𝑠>0} is attained at a unique 𝑟 for some 𝑠. If this holds, the constraint 𝐴_𝑘𝑥≤𝑏_𝑘 is not redundant, and remove 𝑘 from 𝐻.Step 5. If 𝐻=𝜑, then stop. Else, go to Step 6.Step 6. If there is no basic variable 𝑢_𝑘=𝑥^𝐵_i with 𝑘∈𝐻, then introduce a nonbasic variable 𝑢_𝑘=𝑥^𝑁_𝑗 with 𝑘∈𝐻 (e.g., the one with the smallest index 𝑘) into the basis and continue with Step 1.Step 7. Select a basic variable 𝑢_𝑘=𝑥^𝐵_𝑗 with 𝑘∈𝐻 (e.g., the one with the smallest index 𝑘) and perform a feasible pivot step in column 𝑝 with 𝑦_𝑖𝑝=max𝑦_𝑖𝑗. Continue with Step 1.

Example 3.3. Consider Example 3.1 presented in Section 3.1.Solution 3. Iteration. Contracted simplex table, see Table 1.Step 1. 𝑥₁≥0,𝑥₂≥0, and 𝑥₃≥0 are not redundant.Step 4. Now, divide the RHS values by the first column and take the minimum of it min{15,8.66,−,45}=8.66, which corresponds to the 2nd row. Therefore, constraint (2) is not redundant. Now, 𝐻={1,3,4}.Step 7. Select 𝑢₁ and find the maximum {2,1,1}=2, which corresponds to the 1st column. Therefore, pivoting on 𝑦₂₁, we obtain, see what is Table 2.Iteration. Divide the RHS values by the 2nd column and take minimum of it. min{38,26,13,21.8}=13, which corresponds to the 3rd row. Therefore, constraint (3) is not redundant. Now, 𝐻={1,4}.Step 4. Selecting 𝑢₁,𝑢₃ we cannot pivot. So select 𝑢₄ and the maximum {−1/3,5/3,2/3}=5/3 corresponds to the 2nd column. Therefore, pivoting on 𝑦₃₂, we obtain, see what is Table 3. Step 3. In the first and last row, all the coefficients are ≤0.Step 5. 𝐻=𝜑. Therefore, Constraints (1) and (4) are redundant.

Table 1 

Table 2 

Table 3 

3.4. Stojković and Stanimirović Method

This method is proposed by Stojković and Stanimirović [13]. It is a simple method. It verifies the existence of the saddle point of payoff matrix for the game problem by applying the maxmin and minmax principles.

Procedure of the Method
Step 1. Compute 𝑑_𝑖𝑗=𝑎_𝑖𝑗/𝑏_𝑖𝑐_𝑗,𝑖=1,2,…,𝑚, and 𝑗=1,2,…,𝑛. Step 2. If max_{1≤𝑖≤𝑚}min_{1≤𝑗≤𝑛}𝑑_𝑖𝑗=min_{1≤𝑖≤𝑛}max_{1≤𝑖≤𝑚}𝑑_𝑖𝑗, then there are no redundant constraints. Stop.
Else, if there exist 𝑘 and 𝑙 such that 𝑑_𝑘𝑗≤𝑑_𝑙𝑗, for all 𝑗=1,2,…, then the 𝑘th constraint is redundant.

Example 3.4. Consider the Example 3.1 presented in Section 3.1.Solution 4. Step 1. 𝑑_𝑖𝑗=2112016033011041520261521261132180190=45min0.016660.016660.03330.028850.019230.03850.00.03850.07690.00550.02220.02220.016660.019230.00.0055.max0.028850.03850.0769 (3.7)Step 2. max_1≤𝑖≤4min_1≤𝑗≤3𝑑_𝑖𝑗=0.01923,min_1≤𝑗≤3max_1≤𝑖≤4𝑑_𝑖𝑗=0.02855,max_1≤𝑖≤3min_1≤𝑗≤4𝑑_𝑖𝑗≠min_1≤𝑗≤4max_1≤𝑖≤4𝑑_𝑖𝑗. (3.8) There must be at least one redundant constraint in the above problem.
Here 𝑑_1𝑗≤𝑑_2𝑗, 𝑗=1,2,3, hence the constraint (1) is redundant.

3.5. Heuristic Method

Paulraj et al. [14] proposed a heuristic method to identify redundant constraint by using the intercept matrix of constraints of a linear programming problem.

Procedure of the Method
Step 1. Let 𝐼 be the set of subscripts associated with the initial basic variables (slack variables). Initially let that set be 𝐼={1,2,…,𝑚}.
Let 𝐽 be the set of subscripts associated with the initial decision variables. Initially let that set be 𝐽={1,2,…,𝑛}.Step 2. Construct an intercept matrix “𝜃” using the following relationship 𝜃_𝑗𝑖=𝑏_𝑖𝑎_𝑖𝑗,𝑎_𝑖𝑗>0,for𝑗∈𝐽,𝑖∈𝐼. (3.9)Step 3. Determine the entering variables making use of the following steps.(i)Calculate 𝑧_𝑗−𝑐_𝑗=𝐶_𝐵𝐵⁻¹𝑎_𝑗−𝑐_𝑗 for all nonbasic variables 𝑎_𝑗=(𝑎_1𝑗,𝑎_2𝑗,…,𝑎_𝑚𝑗)^𝑇.(ii)Let 𝛽_𝑗=min{𝜃_𝑗𝑖}, for𝑗∈𝐽,𝑖∈𝐼.(iii)Compute 𝑧^′_𝑗−𝑐^′_𝑗=𝛽_𝑗(𝑧_𝑗−𝑐_𝑗) for𝑗∈𝐽.Step 4. (i) Let 𝑧^′_𝑘−𝑐^′_𝑘=min_𝑗∈𝐽{𝑧^′_𝑗−𝑐^′_𝑗}.
(ii) If 𝑧^′_𝑘−𝑐^′_𝑘≥0, then the problem has no redundant constraints and stop.
(iii) Otherwise, take away the element 𝑘 from the set 𝐽, that is, 𝐽=𝐽−{𝑘}.
(iv) Let 𝜃_𝑘𝑙=min_𝑖∈𝐼{𝜃_𝑘𝑖}=𝛽_𝑘.
(v) Take away the element l from the set 𝐼, that is, 𝐼=𝐼−{𝑙}.
(vi) Find 𝑝 such that min {𝜃_𝑝𝑙}=𝛽_𝑝 for 𝑝∈𝐽. If so, take away such 𝑝 elements from the set 𝐽, that is, 𝐽=𝐽−{𝑝}.Step 5. If 𝐽=𝜑, then go to Step 6. Otherwise, go to Step 4.Step 6. If 𝐼=𝜑, then the problem has no redundant constraints and stop. Otherwise, all the constraints 𝑖∈𝐼 whose intercepts on coordinate axis 𝑗 satisfy the condition 𝜃_𝑗𝑖≥max{𝛽𝑗} where 𝑖∈𝐼 and 𝑗=1,2,…,𝑛 are redundant. Stop.

Example 3.5. Consider the Example 3.1 presented in Section 3.1Solution 5. Step 1. (i) 𝐼={1,2,3,4}, (ii) 𝐽={1,2,3}.Step 2. [see Table 4].Step 3. [see Table 4]. Step 4. [see Table 5]. Step 5. 𝐽=𝜑. Constraints (1) and (4) are redundant constraints.

Table 4 

Table 5 

4. Numerical Results

The comparative results of the five methods are presented in the following tables. Table 6 shows the comparison results of small-scale problems, Table 7 shows the comparison results of medium-scale problems, and Table 8 shows the comparison results of netlib problems [18]. Comparison results of large size problems from OR library [19] are presented in Table 9.

Table 6 

Comparison of five methods: small-scale problems.

Table 7 

Comparison of five methods: medium-scale problems.

Table 8 

Comparison of five methods: netlib problems.

Table 9 

Comparison of five methods: Large Size Problems.

In Figures 2, 3, 4 and 5, the comparative results of small scale problems, medium scale problems, netlib

Figure 2 

Figure 3 

Figure 4 

Figure 5 

problems and large size problems are shown graphically. Figure 2 shows that the heuristic and linear programming methods identify more redundant constraints than the other three methods. Figure 3 indicates that the Stojkovic and stanimirovic method identifies no redundant constraints where as deterministic method identifies very low redundant constraints compared with other methods. Figure 4 indicates that the deterministic method identifies nothing; brearly’s, Stojkovic and stanimirovic methods identify very low redundant constraints and the remaining methods more or less coincide. Figure 5 shows that the Stojkovic and stanimirovic method identifies very less redundant constraints where as heuristic method identify more redundant constraints than the others.

The tables deal with the identification of the number of redundant constraints in linear programming problems by using the five methods. It is very easy to identify quickly the best method in finding redundant constraints of LP problems. Heuristic method seems to be less time consuming, and it requires less computational effort. It also finds more redundant constraints when compared with the other four methods. So this method would be easy and reliable method for identifying redundant constraints. Even though the LP method identifies more redundant constraints, it needs more computational work and takes more time. Brearly’s method identifies less redundant constraints with less computational effort than heuristic and LP methods. Deterministic methods identify more redundant constraints with more computational effort. So time consumption is bigger when compared with heuristic method. Stojković and Stanimirović identified a smaller number of redundant constraints than the others.

The efficiency of the algorithms was also tested by solving the first set of Linear Programming Problems mentioned before and after removing the redundant constraints, identified by each method. Table 6 gives the computational results.

5. Conclusions

In this paper, the heuristic approach [14] for identifying redundant constraints has been compared with other four methods. Each method has its own role in viewing computational effort and time factor. Linear programming method [6] and deterministic method [15] more or less coincide with heuristic method in most of the problems. Brearly et al. [4] method depends on the upper and lower bounds of the decision variables for identifying the redundant constraints. Hence, the heuristic method is more useful than the other methods.