#### SCHEDULE SETS AND PARAMETERS ### set cities; set links within {c1 in cities, c2 in cities: c1 <> c2}; # Set of cities, and set of intercity links param last > 0 integer; # Number of time intervals in a day set times := 1..last; # Set of time intervals in a day set schedule within {c1 in cities, t1 in times, c2 in cities, t2 in times: (c1,c2) in links}; # Member (c1,t1,c2,t2) of this set represents # a train that leaves city c1 at time t1 # and arrives in city c2 at time t2 ### DEMAND PARAMETERS ### param section > 0 integer; # Maximum number of cars in one section of a train param demand {schedule} > 0; # For each scheduled train: # the smallest number of cars that # can meet demand for the train param low {(c1,t1,c2,t2) in schedule} := ceil(demand[c1,t1,c2,t2]); # Minimum number of cars needed to meet demand param high {(c1,t1,c2,t2) in schedule} := max (2, min (ceil(2*demand[c1,t1,c2,t2]), section*ceil(demand[c1,t1,c2,t2]/section) )); # Maximum number of cars allowed on a train: # 2 if demand is for less than one car; # otherwise, lesser of # number of cars needed to hold twice the demand, and # number of cars in minimum number of sections needed ### DISTANCE PARAMETERS ### param dist_table {links} >= 0 default 0.0; param distance {(c1,c2) in links} > 0 := if dist_table[c1,c2] > 0 then dist_table[c1,c2] else dist_table[c2,c1]; # Inter-city distances: distance[c1,c2] is miles # between city c1 and city c2 ### VARIABLES ### var U 'cars stored' {cities,times} >= 0; # u[c,t] is the number of unused cars stored # at city c in the interval beginning at time t var X 'cars in train' {schedule} >= 0; # x[c1,t1,c2,t2] is the number of cars assigned to # the scheduled train that leaves c1 at t1 and # arrives in c2 at t2 ### OBJECTIVES ### minimize cars: sum {c in cities} U[c,last] + sum {(c1,t1,c2,t2) in schedule: t2 < t1} X[c1,t1,c2,t2]; # Number of cars in the system: # sum of unused cars and cars in trains during # the last time interval of the day minimize miles: sum {(c1,t1,c2,t2) in schedule} distance[c1,c2] * X[c1,t1,c2,t2]; # Total car-miles run by all scheduled trains in a day ### CONSTRAINTS ### account {c in cities, t in times}: U[c,t] = U[c, if t > 1 then t-1 else last] + sum {(c1,t1,c,t) in schedule} X[c1,t1,c,t] - sum {(c,t,c2,t2) in schedule} X[c,t,c2,t2]; # For every city and time: # unused cars in the present interval must equal # unused cars in the previous interval, # plus cars just arriving in trains, # minus cars just leaving in trains satisfy {(c1,t1,c2,t2) in schedule}: low[c1,t1,c2,t2] <= X[c1,t1,c2,t2] <= high[c1,t1,c2,t2]; # For each scheduled train: # number of cars must meet demand, # but must not be so great that unnecessary # sections are run