Formulating constraints of a LP in Pulp Python

Question

I have this LP problem, and I'm trying to solve it using PuLP in Python-3. One option that I can think of is to write all variable explicitly, but I want to avoid it. Is there a way that I can use lists/dicts in this problem? (I did refer to https://pythonhosted.org/PuLP/CaseStudies/a_sudoku_problem.html where dicts were being used, but didn't quite understand the entire solution)

Assume wt{i,j,type} denotes the number of traded goods between person[i] and person[j] of type.

LP Problem:

(Here, cost{i,j} is a known cost of pairing for all (i,j) pairs.

subject to:

I would be really grateful for any help, as I'm a beginner to both optimization and python/pulp.

Answer 1

The 'lists/dicts' is a way to define variables over domains (indexed variables). The indexs argument of LpVariable.dicts() defines the domain - cartesian product of the supplied sets. See also documentation of PuLP - LpVariable.

The code sample below does not contain all your constraints, but I believe you can easily fill-in the remaining ones. Constraint 1 (with const1 and const2) is treated via the lower and upper bound of the wt variable instead.

from pulp import LpProblem, LpVariable, LpMaximize, LpInteger, lpSum, value

prob = LpProblem("problem", LpMaximize)

# define 'index sets'
I = range(10) # [0, 1, ..., 9]
J = range(10)
T = range(3)

# define parameter cost[i,j]
cost = {}
for i in I:
    for j in J:
        cost[i,j] = i + j # whatever

# define wt[i,j,t]
const1 = 0 # lower bound for w[i,j,t]
const2 = 100 # upper bound for w[i,j,t]
wt = LpVariable.dicts(name="wt", indexs=(I, J, T), lowBound=const1, upBound=const2, cat=LpInteger)

# define assign[i,j]
assign = LpVariable.dicts(name="assign", indexs=(I, J))

# contraint 
for i in I:
    for j in J:
        prob += assign[i][j] == lpSum(wt[i][j][t] for t in T), ""

# objective
prob += lpSum(cost[i,j] * assign[i][j] for i in I for j in J)

prob.solve()

for i in I:
    for j in J:
        for t in T:
            print "wt(%s, %s, %s) = %s" % (i, j, t, value(wt[i][j][t]))

for i in I:
    for j in J:
        print "assign(%s, %s) = %s" % (i, j, value(assign[i][j]))