040: A Distribution Problem with Wagner-Whitin Costs

language Essence 1.3

$ Problem Wagner-Whitin Distribution
$
$ Problem details available at http://www.csplib.org/Problems/prob040/
$
$ Essence model by Andrew Martin
$
$ Licenced under CC-BY-4.0 : http://creativecommons.org/licenses/by/4.0/

$ in this model, rather than define multiple levels, children of each node are defined explicitly

given numNodes : int(1..)

$ leaves are always first numLeaves nodes
$ leaves cannot have any children
given numLeaves : int(1..)

$ period 0 is empty, all stock is 0
given numPeriods : int(1..)

$ used to provide bound to output
given maxStock : int(1..)

letting dNodes be domain int(1..numNodes)
letting dLeaves be domain int(1..numLeaves)
letting dAllPeriods be domain int(0..numPeriods)
letting dPeriods be domain int(1..numPeriods)

given holdingCost : matrix indexed by [dNodes] of int(0..)

given procCost : matrix indexed by [dNodes] of int(0..)

given demand : matrix indexed by [dLeaves, dPeriods] of int(0..)

$ used to determine where supply comes goes to
given children : matrix indexed by [dNodes] of set of dNodes

find orders : matrix indexed by [dNodes, dPeriods] of int(0..maxStock)

$ AUX find statement
find stock : matrix indexed by [dNodes, dAllPeriods] of int(0..maxStock)

$ minimising the cost
$ holding*stock per period per node plus proc if any orders were placed per period per node
minimising (sum t : dPeriods . (sum i : dNodes . (holdingCost[i] * stock[i][t] + procCost[i] * (orders[i][t] > 0))))


such that

$ stock starts at 0
    forAll i : dNodes .
        stock[i][0] = 0
,
$ non-leaf constraints - orders coming from children must be fulfilled
    forAll t : dPeriods .
        forAll i : int(numLeaves+1..numNodes) .
            stock[i][t] = stock[i][t-1] + orders[i][t] - (sum m in children[i] . orders[m][t])
,
$ leaf constraints - demands must be fullfilled
    forAll t : dPeriods .
        forAll i : dLeaves .
            stock[i][t] = stock[i][t-1] + orders[i][t] - demand[i][t]