With the globalization of markets and the focus on customer satisfaction, supply chains are becoming more complex. The response of many companies to these challenges is to focus on their core business and outsource other components of their supply chain, such as the supply of raw material, transportation and warehousing.

Recognizing the importance of the topic, a large volume of research has been devoted to the optimal design of supply chain networks under different conditions. This paper considers two important elements that have usually been neglected in the literature, but are often present when the transportation and warehousing service is outsourced: discrete transportation costs and warehouse term-contracts. Efficient mixed-integer linear programming (MILP) formulations for handling these aspects are proposed for a 2-echelon, multi-commodity, multi-period problem with multiple modes of transportation in each link.

In most of the research reported on the facility location problem, transportation costs are considered to be a linear function of the transported amounts, with optional fixed costs. However, in several real-life applications the cost function is piece-wise constant, because there is a fixed cost per each transportation unit used (e.g.: trucks), and if the unit is full or half-full the transportation cost is the same. When the quantity exceeds the unit capacity, another unit is used and the cost rises to another step in the function. Five different approaches to model this situation are analyzed and compared. The first approach considers integer variables to compute the number of units of each mode of transportation used in each link at each time period. The second approach treats the cargo loading as an assignment problem in which the maximum number of units per each mode of transportation is known beforehand. The other three approaches are based on representing the stepwise nature of the cost function using SOS2 variables and General Disjunctive Programming, with its respective Big-M and Convex Hull reformulations. For a midsize problem with 3 plants, 6 potential warehouses, 8 customers, 5 products and 36 periods only the formulation with integer variables was able to solve the problem to a 0.5% gap in less than 2 minutes using the default options of the solver Gurobi 6.0.2. The other four approaches were not able to find a feasible solution after 15 minutes. The partial relaxation of the formulation with integer variables, in which the transportation units are allowed to take continuous values presents a 4% relaxation gap and solves the problem in less than 10 seconds, making it useful to obtain good lower bounds.

The second element included is warehouse contracting policies. When the service is provided by an external company there are minimum length clauses in the contracts and also if a contract is not immediately renewed, there is a minimum wait period to start a new contract. These clauses prevent the warehouse to be empty for short periods of time, making it easier for the provider to find other customers to fill the vacant periods. In order to address these requirements two extra discrete variables were added to the formulation, one to indicate when a new contract is started and another to indicate when a contract is finished. Using these variables and the original discrete variables to indicate monthly warehouse use, minimum contract length and minimum constraints were formulated. The introduction of these new variables and constraint increased the total number of discrete variables by 5% and the solution time by 20%. Finally, we report results for a large-scale industrial application.

The paper shows that important aspects present when the transportation and the warehousing is outsourced can be efficiently modeled maintaining the solution times suitable for large scale applications and to deal with uncertain parameters.

**References**

[1] Cordeau, Jean-François, Federico Pasin, and Marius M. Solomon. "An integrated model for logistics network design." *Annals of Operations Research *144.1 (2006): 59-82.

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