Mathematical Techniques

Mathematical techniques are based on the representation of the essential aspects of an actual system using mathematical languages. Basically, a mathematical model needs to contain enough details to answer the questions for a certain problem [14]. Mathematical techniques may include linear programming, nonlinear programming, MIP, and Lagrangian Relaxation [15-17). Different mathematical techniques have been adopted to solve logistics problems, including linear programming models [18-22], MIP models [23-39], and Lagrangian Relaxation models [40-43].

Mathematical programming models have been demonstrated to be useful analyt­ical tools in optimizing decision-making problems such as those encountered in LP |44,45]. Linear programming was first proposed in 1947 and has been widely used in solving constrained optimization problems. "Programming" in this case is appli­cable when all of the underlying models of the real-world processes are linear [17,46]. MIP is used when some of the variables in the model are real values and others are integer values (0, 1). Mixed-integer linear programming (MILP) occurs when objective function and all the constraints are linear in form; otherwise, it is mixed-integer nonlinear programming (MINLP), which is harder to solve [16]. The idea behind the Lagrangian Relaxation methodology is to relax the problem by removing the constraints that make the problem difficult to solve, putting them into the objective function, and assigning a weight to each constraint [47]. Each weight represents a penalty that is added to a solution that does not satisfy the particular constraint.

All of the mathematical techniques are fully matured and are thus guaranteed to produce the optimal solution (or near-optimal solutions) for a certain type of prob­lem [12]. However, for two reasons this technique has limited application in solv­ing complex logistics problems. First, mathematical equations are not always easy to formulate, and the associated complexities in the development of mathematical algorithms increase as the number of variables and constraints increase [12,48]. Because the majority of logistics networks are complex with the presence of large numbers of variables and constraints, mathematical methods may not be very effec­tive in solving real-world LP problems [12,15]. Second, even if it is possible to translate a difficult LP problem into mathematical equations, the problem would become intractable or NP-hard because of the exponential growth of the model size and complexity [12,49]. The drawbacks of mathematical techniques may make it almost impossible to employ them for solving real-life, large-scale LP problems unless the problems are oversimplified.