Integral Logistics Management — Operations Management and Supply Chain Management Within and Across Companies

3.2.2 Location Selection and Location Configuration with Linear Programming

Intended learning outcomes: Produce an overview on linear programming (LP). For the design option of “in part decentralized production for the global market”, explain the use of mixed-integer linear programming (MILP).

Probably the most challenging configuration of a production network as shown in Figure is in part decentralized production for the global market. In this case especially the task of location configuration is difficult to solve, but it is also not insignificant in the other cases. It is the task of determining a global production plan: What products and — in the face of limits of capacity — how much of what product will be manu­factured for what markets at what level at what locations? A similar question can also arise for decentralized distribution or decentralized service: What customers will be served by what distribution and service locations?

Very many influencing variables soon lead to a complex problem. Decision making can be supported — often with simplified model assumptions — by linear programming (LP).

In linear programming (LP) the task is to solve a problem that can be expressed as in Figure

Fig.        Problem formulation in linear programming: Maximize the objective function OF and solve for x, subject to the constraints.

If the number of decision variables is two, the problem can be solved using a simple graphical method. With a greater number of variables, the use of an algorithm is recommended, such as, for example, the simplex algorithm. The complexity of the problem increases with increasing values of the number of variables (n) and the number of constraints (m). Computation time does not increase polynomially with n and m: the simplex algorithm is what is called an “NP-hard” algorithm. With high values of n and m, the procurement of data is also a problem.

It is mostly larger companies that use this quantitative method in practice.

  • The automobile manufacturer Daimler uses software called “network analyzer” to determine, among other things, what products should be manufactured at what locations. The software uses mixed-integer linear programming (MILP). The constraints are, for example, market guidelines that must be met and the limits of capacity that must be considered. Entering into the objective function are, among other things, the production costs at the locations and the transport costs from the locations to the markets.
  • In a similar way, the cement company Holcim uses MILP for location selection of their works. A special application was, for example, the case where a cement works had to be closed because of exhausted raw materials. The issue was whether the other works could deliver the required quantities. In the optimization, the increased transport costs stood vis-à-vis the fixed costs to set up a new works and the different production costs in the other works.

For some years now, MS Excel has offered a Solver tool that can be used to solve an LP problem with (in the current release) 200 variables.

A word of caution: A basic problem is that new roads and concentration centers are always being built. Moreover, im­portant customers can move away, or the political and economic business environment can change. Then, any one selected location can prove to be suboptimal. If the high building and equipment costs have not yet been written off, the facility cannot simply be changed to fit the new data, or in other words, it cannot simply be moved to a new location or re-equipped. In the long-term view, therefore, simple, robust methods do not always have to be a priori at a disadvantage compared to complicated optimization algorithms (for example, nonlinear programming or heuristics as well).

Course section 3.2: Subsections and their intended learning outcomes