Finding multiple solutions to general integer linear programs

Abstract

Integer linear programming (ILP) problems occur frequently in many applications. In practice, alternative optima are useful since they allow the decision maker to choose from multiple solutions without experiencing any deterioration in the objective function. This study proposes a general integer cut to exclude the previous solution and presents an algorithm to identify all alternative optimal solutions of an ILP problem. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method.

Introduction

Integer Programming (IP) is an important technique for dealing with problems that arise frequently in diverse fields such as capital budgeting, production planning, capacity planning, scheduling and chemical engineering process. These applications are extensively surveyed in Djerdjour, 1997, Salkin and Mathur, 1989, Simons, 1996, Taha, 2003. Biegler and Grossmann (2004) provided a retrospective article on mathematical programming models and optimization techniques that have been applied in process systems engineering. They indicated that operations problems give rise to mixed-integer linear programming (MILP) and mixed-integer non-linear programming (MINLP) models for scheduling and supply chain problems. Besides, the design and synthesis problems and the hybrid systems in the control problems have also been formulated as MILP and MINLP problems. Since the enterprise can achieve a cost advantage by reducing costs from the current solution, optimization techniques have received extensive attention from the practitioners and the researchers in the last few decades.

A number of applications require to model discrete variables, such as number of units or batches, “yes or no” decision, sequences in scheduling, etc. Generally, a problem with integer variables, zero-one variables and mixed variables are called IP problems. Because of their importance in formulating many practical problems, there has been a pronounced increase in the development of optimization approaches on IP models. These approaches can mainly be classified into stochastic and deterministic as follows:

• Stochastic methods: The stochastic methods involve random elements in their search and converge to the global optimum depending on a statistical argument. For instance, Salcedo et al. (1990) proposed an improved random search algorithm for solving non-linear optimization problems. Cardoso et al. (1996) solved non-convex non-linear integer programming problems with simulated annealing. Rosen and Harmonosky (2005) improved the performance of simulated annealing for discrete variable simulation optimization by basing portion of the search procedure on inferred statistical knowledge of the system instead of using a strict random search. Wang and Liao (1998) developed methods for solving polynomial integer programs by the genetic algorithm. Xiong and Rao (2004) proposed a mixed-discrete fuzzy non-linear programming approach to solve fuzzy optimization problems with mixed-discrete design variables through hybrid genetic algorithm. Although these stochastic methods have the advantage of easy implementation and little prior knowledge of the optimization problem, they are suitable for problems where the function evaluations are cheap, and can not guarantee rigorous global optimality (Biegler and Grossmann, 2004). Moreover, the probability of finding the global solution decreases when the problem size increases.

• Deterministic methods: In a general survey of optimization techniques (Biegler and Grossmann, 2004, Grossmann and Biegler, 2004), many deterministic methods for IP problems have been reviewed. The most common algorithm for solving MILP problems is the LP-based branch and bound method (Dakin, 1965). Johnson et al. (2000) presented the progress in MILP methods through 1998. They discussed about modeling, preprocessing and the methodologies of branch-and-cut (Caprara and Fischetti, 1997) and branch-and-price (Barnhart et al., 1998). Grossmann (2002) provided a unified overview and derivation of MINLP techniques. Major methods for MINLP problems include Branch and Bound (BB) (Borchers and Mitchell, 1994, Leyffer, 2001, Stubbs and Mehrotra, 1999), Generalized Benders Decomposition (GBD) (Geoffrion, 1972), Outer-Approximation (OA) (Duran and Grossmann, 1986, Fletcher and Leyffer, 1994, Quesada and Grossmann, 1992), and Extended Cutting Plane Method (ECP) (Westerlund and Pettersson, 1995, Westerlund and Pörn, 2002). Biegler and Grossmann (2004) investigated the important extensions of MINLP methods for dealing with quadratic master problems, equalities and non-convexities.

IP is a very powerful technique for treating problems involving continuous and discrete variables. In this study, we consider a pure Integer Linear Programming (ILP) problem where all the variables are integer. The mathematical formulation of an ILP problem can be expressed as follows:
ILP:

$$\begin{array}{cc}\hfill \text{Minimize}& f(X)\hfill \\ \hfill \text{Subject to}& g_k(X)⩽0\text{,}k=1\text{,}2\text{,}\dots \text{,}K\text{,}\hfill \end{array}$$

$$X=(x_1\text{,}\dots \text{,}{x}_n)\text{,}{x}_i\in \{0\text{,}1\}\text{for}1⩽i⩽m\text{,}{x}_i\in Z\text{for}m+1⩽i⩽n\text{,}$$

$$0⩽{\underline{x}}_i⩽x_i⩽{\overline{x}}_i\text{,}i=m+1\text{,}m+2\text{,}\dots \text{,}n\text{,}$$

where $X=(x_1\text{,}\dots \text{,}{x}_n)$ is a vector of variables, f and g_k are linear functions, and ${\underline{x}}_i$ and ${\overline{x}}_i$ are lower and upper bounds of the integer variable x_i, respectively.

The alternative optima of an ILP problem can be found if more than one solution can satisfy the same optimal value of the objective function. In practice, alternative optima are useful because they allow the decision maker to choose from many solutions without experiencing any deterioration in the objective function. For instance, if multiple product-mix situations are available to reach the maximum revenue, it may be advantageous from the standpoint of sales competition to produce two products rather than one (Taha, 2003). Therefore, finding all alternative solutions with the same optimal objective value of an ILP problem is prominent.

For deriving all optimal solutions of an ILP problem, an integer cut is added to the original model to make the previous solution infeasible, and the model is solved again to find another optimum. For special case of 0–1 variables, Balas and Jeroslow (1972) introduced the well-known binary cut involving no additional variables and only one constraint. Duran and Grossmann (1986) used this binary cut in their OA algorithm to exclude binary combinations. Tawarmalani and Sahinidis (2002) mentioned that BARON can identify the K best solutions for a mixed-integer non-linear program, where K is an option specified by the user. This paper proposes a global optimization approach on finding all solutions of a general ILP program. First, a general integer cut is presented to derive an alternative solution under the same optimal objective value. The original ILP problem with a general integer cut is reformulated into another MILP problem solvable by the conventional IP techniques to obtain a globally optimal solution. Then, we develop an algorithm to locate all alternative optimal solutions of the ILP problem.

The rest of this paper is organized as follows. In Section 2, we discuss some theoretical propositions about employing a general integer cut to derive an alternative optimum of an ILP problem. Section 3 describes the proposed algorithm for finding all solutions of an ILP problem. Section 4 demonstrates some numerical examples. Finally, concluding remarks are made in Section 5.