Integer programming formulation

Many combinatorial optimisation problems can be formulated as the problems in integer programming in which all decision variables are required to take integral values. We first model the demand-responsive container rail scheduling problem as an integer programme.

We consider the day divided into hourly slots for weekly booking and scheduling. The following notation will be used:

Sets:

T : set of schedulable timeslots t M : set of customers j

Sj : set of potential booking timeslots for customer j Ct : set of potential customers for departure timeslot t R : set of service restrictions for departure timeslots

Decision variables:

xt : 1, if a train departs in timeslot t, 0 otherwise

ytj : 1, if customer j is served by the train departing in timeslot t , 0 otherwise

Parameters:

FC is a fixed cost of running a train

wtj : customer jsatisfaction in departure timeslot t Nj: demand of customer j (number of containers)

rt : train congestion cost in departure timeslot t gt : staff cost in departure timeslot t

P1 : minimum train loading (number of containers) P2 : capacity of a train (number of containers)

The IP formulation of the container rail scheduling problem is:

Minimise

∑ ∑ ∑ ∑ ( )

The objective is to minimise the generalised cost representing the operating costs and the virtual loss of future revenue. The first term in the objective function aims to minimise the number of trains on a weekly basis. The fewer trains, the greater reduction on operating costs a rail carrier can expect. The second term is to maximise the total customer satisfaction using values from a customer satisfaction functions (Figure 3.3 - 3.6). Each customer holds the

highest satisfaction at a preferred booking time range, the satisfaction then decreases probabilistically to the lowest satisfaction at the last alternative booking time range, i.e.

departure later than the preferred booking time range would cause a decrease in the future demand, and the rail carrier is expected to take a loss in future revenue. For the evaluation of a schedule, the probability of customer satisfaction is then multiplied by demand N_j. The last term in the objective function aims to minimise the timeslot-operating cost. A rail carrier is likely to incur additional costs in operating a demand responsive schedule, in which departure times may vary from week to week. This may include train congestion cost and staff cost. The train congestion cost reflects an incremental delay resulting from interference between trains in a traffic stream. The rail carrier calculates the marginal delay caused by an additional train entering a particular set of departure timeslots, taking into account the speed-flow relationship of each track segment. The over-time cost for crew and ground staff would also be paid when evening and night trains are requested.

Constraints (3.4) ensure that no customer will be left uncovered and each customer can only be served by one train. These hard constraints are crucial for a rail carrier. This is because it does not make good business sense in a competitive market to offer a demand responsive service, requiring customers to state both their preferred, and a set of alternative, time ranges regarding their practical container operations (either at the terminal of departure or destination terminal, or at both ends) and business strategies and then to decline their business. In addition, customers’ shipment cannot be split in multiple trains. This is because the destination terminal (inland container depot: ICD) located in the heart of the capital city is relatively small and all arrival containers must be stacked in the designated area.

Expensive equipment is required to move the containers from the train to the ICD and later onto transport to the final destination. Multiple shipments are likely to result in container

stacks having to be “shuffled” to assemble a particular customer’s load; thereby imposing substantial container handling cost.

Constraints (3.5) ensure that the demand assigned to a departure time slot must not be less than the minimum train loading P₁. Setting a minimum train loading ensures satisfactory revenue for a rail carrier and spreads out the capacity utilisation on train services. The carrier may want to set the minimum train loading as high as possible, ideally equal to the capacity of a train. Section 4.4 explains how sensible values for P₁ can be determined.

Constraints (3.6) ensure that the demand assigned to a departure time slot must not exceed the capacity of a train. These are hard operational constraints; if load exceeds the capacity, running the train can damage the locomotive engine and railway infrastructure, e.g. tracks.

Note that in case of a single customer’s demand being more than the capacity of a train, we allow splitting this demand over multiple trains and treat the demand as different sub-customers; however, this particular case rarely occurs in practice.

Constraints (3.7) ensure that if timeslot t is selected for customer j, a train does depart at that timeslot. On the other hand, if departure timeslot t is not selected for customer j, a train may or may not run at that time.

Constraints (3.8) are a set of banned departure times. The restrictions may be pre-specified so that a railway planner schedules trains to achieve a desirable headway or to avoid congestion at the container terminal.

Constraints (3.9) require that all decision variables in the model are binary.

The container rail scheduling problem was initially solved by an integer programming branch and bound method (CPLEX Solver, ILOG 2002). We failed to find an optimal solution within the time limit 12 hours, particularly when the minimum train loading P₁ is close to the capacity of a train. Since all decision variables in our container rail scheduling model are binary (constraints 3.9) and fractional solutions to linear relaxations are meaningless, tight bounds on the objective function value cannot be obtained and used to reduce the size of the search space.

Note that the matrix of coefficients in (3.4) consists of only zeroes and ones and that the constraints are equations. Thus there is a relaxation of the container rail scheduling problem to a set partitioning problem (SPP). Formally, the SPP is the problem of partitioning the rows i (i=1,...,M) of a zero-one matrix (a_ij) by distinct subsets of the columns j ( j =1 K, ,N)

Constraints (3.11) ensure that each row is covered (served) by exactly one column and (3.12) are the integrality constraints. The SPP is known to be NP-hard (Balas and Padberg, 1976);

hence the container rail scheduling problem is NP-hard. In addition, the container rail scheduling problem incorporates other constraints that require the consistency between

different sets of decision variables which will make it more complex than SPP. Many current approaches for NP-hard problems focus on finding good solutions within a reasonable time using various local search heuristic methods.