083: Transshipment problem

language Essence 1.3

given n_warehouses, n_transshipment, n_customer : int(1..)

letting
    Warehouse     be domain int(1..n_warehouses),
    Transshipment be domain int(1..n_transshipment),
    Customer      be domain int(1..n_customer)

given

    $ cost of transporting items from a Warehouse to a Transshipment point
    costWT : function (total) tuple (Warehouse, Transshipment) --> int(1..),

    $ cost of transporting items from a Transshipment point to a Customer
    costTC : function (total) tuple (Transshipment, Customer) --> int(1..),

    $ available stock at each Warehouse
    stock : function (total) Warehouse --> int(1..),

    $ demand for each customer
    demand : function (total) Customer --> int(1..)

letting maxStock be max(range(stock))
letting maxDemand be max(range(demand))

find amountWT : function (Warehouse, Transshipment) --> int(1..maxStock)
find amountTC : function (Transshipment, Customer)  --> int(1..maxDemand)

$ preservation of flow
such that
    forAll t : Transshipment .
        sum([ amount | ((w, t'), amount) <- amountWT, t = t' ]) =
        sum([ amount | ((t', c), amount) <- amountTC, t = t' ])

$ meet the demand
such that
    forAll c : Customer .
        sum([ amount | ((t, c'), amount) <- amountTC, c = c' ]) = demand(c)

$ stay within the stock
such that
    forAll w : Warehouse .
        sum([ amount | ((w', t), amount) <- amountWT, w = w' ]) <= stock(w)

given maxCost : int(1..)

$ calculate the total cost
find totalCost : int(1..maxCost)
such that
    totalCost = sum([ costWT(key) | (key, amount) <- amountWT ])
              + sum([ costTC(key) | (key, amount) <- amountTC ])