Common use of Instances Clause in Contracts

Instances. Data about the location of facilities and of customers, costs, demand and contract terms were randomly generated based on the following assumptions and parameter values. • The shipper operates two warehouses. • The warehouses and customer areas are randomly uniformly located in a 2-dimensional (1000×1000) grid. • The carriers’ facilities are randomly located in a (700 × 700) centered area of the grid. • For each carrier e, the size of Ie is randomly chosen from the uniform discrete distribution on an interval [i−, i+]: U [i−, i+] (see Table 2). • Customers can only be served from carriers’ facilities that are within an Euclidean distance of 500 units. No carrier can supply all customers. • The long-haul cost and cross-docking services Fi,l for operating facility i at capacity level l is of the form Fi,l = Ci + Ti,l, where Ci is the fixed operational cost of facility i, and Ti,l is the cost charged by the carrier for (full truckload) shipment requiring l trucks from the shipper’s warehouses to facility i and for operating it at level l. • The fixed operational cost Ci is set randomly from a discrete uniform distribution U [500, 1000]. • For a given operating level l, the variable cost Ti,l for long-haul transportation is proportional to the Euclidean distance from facility i to the closest warehouse. As a function of the level l, it is modelled by a staircase function with decreasing marginal costs. • The cost charged by the parcel delivery company for transporting one unit of good from facility i to customer area k, that is, Ui,k, is taken equal to the Euclidean distance from i to k. • The planning horizon is subdivided into three demand seasons, starting with low, then high, and finally mid season. For each customer region k ∈ K and each period t ∈ T , the demand quantity t (in weight units) is issued from a uniform distribution which depends on the season: U [0.1, 0.4] in low season, U [0.35, 0.65] in mid season, and U [0.6, 0.9] in high season. • In a given instance, all contracts have the same fixed duration He ∈ {2, 3, 4}, for all e ∈ E. • Each carrier has enough available capacity to meet the demand of all customer areas that can be served from its facilities. Σ

Instances. Data about the location of facilities and of customers, costs, demand and contract terms were randomly generated based on the following assumptions and parameter values. • The shipper operates two warehouses. • The warehouses and customer areas are randomly uniformly located in a 2-dimensional (1000×1000) grid. • The carriers’ facilities are randomly located in a (700 × 700) centered area of the grid. • For each carrier e, the size of Ie is randomly chosen from the uniform discrete distribution on an interval [i−, i+]: U [i−, i+] (see Table 2). • Customers can only be served from carriers’ facilities that are within an Euclidean distance of 500 units. No carrier can supply all customers. • The long-haul cost and cross-docking services Fi,l for operating facility i at capacity level l is of the form Fi,l = Ci + Ti,l, where Ci is the fixed operational cost of facility i, and Ti,l is the cost charged by the carrier for (full truckload) shipment requiring l trucks from the shipper’s warehouses to facility i and for operating it at level l. • The fixed operational cost Ci is set randomly from a discrete uniform distribution U [500, 1000]. • For a given operating level l, the variable cost Ti,l for long-haul transportation is proportional to the Euclidean distance from facility i to the closest warehouse. As a function of the level l, it is modelled by a staircase function with decreasing marginal costs. • The cost charged by the parcel delivery company for transporting one unit of good from facility i to customer area k, that is, Ui,k, is taken equal to the Euclidean distance from i to k. • The planning horizon is subdivided into three demand seasons, starting with low, then high, and finally mid season. For each customer region k ∈ K and each period t ∈ T , the demand quantity t (in weight units) is issued from a uniform distribution which depends on the season: U [0.1, 0.4] in low season, U [0.35, 0.65] in mid season, and U [0.6, 0.9] in high season. • In a given instance, all contracts have the same fixed duration He ∈ {2, 3, 4}, for all e ∈ E. • Each carrier has enough available capacity to meet the demand of all customer areas that can be served from its facilities. ΣΣ The MPC Mt is equal to 10% of the total capacity reservation fee, that is, the minimum fee that would be charged by carrier e at period t if it were assigned all the demand ( k:e∈Ek can possibly handle through its network of facilities Dt ) that it The main parameters that determine the size of the instances are the number of carriers |E|, the number of periods N = |T|, the number of customer areas |K|, and the range [i−, i+] of the number of facilities per carrier. The instances are partitioned into three classes according to the value of these parameters, namely small, medium and large instances. In each class, 6 combinations of parameter values are considered as displayed in Table 2. Moreover, we are also interested in analyzing the effect of the contract duration on the difficulty to solve the MDPC model. Hence, three different values of He, namely, He = 2, 3, 4, are considered for each instance class. For each combination of parameters (|E|, |T|, |K|, [i−, i+], He), a set of five instances was randomly generated, for a grand total of 3 × 6 × 3 × 5 = 270 instances.