在商品贸易问题中错误指定 puLP 中的某些内容（混合整数规划）

Question

I'm struggling to understand what I'm doing wrong in a toy optimization problem (mixed integer programming).

Let's say we have three people with different numbers of some good. They currently have 10, 0, and 50 units of it, respectively. They NEED 15, 0, and 46 units, respectively, and I'm trying to use puLP to come up with the optimal transfers between the people to minimize unmet need (NEED - SUPPLY AFTER REDISTRIBUTING) for all three people.

A trivial solution here could be that person 3 could give person 1 a total of 4 units, so that the supplies after redistributing become 14, 0, 46, and now person 1 only has one unit of unmet need (they want 15 units).

When I run the following puLP code I get the result that person 1 should give person 2 10 units, and that person 3 should give person 1 50 units.

Clearly I'm doing something wrong, could you help point out my mistake?

I didn't use loops or list comprehensions because I wanted to make everything super explicit and clear to find my mistake. The only constraints I have are that people can't give away more units than then currently have.

Thanks!

from pulp import *
import pandas as pd


# Creates a list of the unique people
people = ['1','2','3']

# Creates a dictionary for the number of units of units currently in supply
current = {
    '1': 10,
    '2': 0,
    '3': 50
}

# Creates a dictionary for the number of units needed
need = {
    '1': 15,
    '2': 0,
    '3': 46
}


# Creates the prob variable to contain the problem data
prob = LpProblem("Goods redistribution problem", LpMinimize)

# Creates a list of tuples containing all the possible trades
Routes = [(s,t) for s in people for t in people]

# Remove any tuples that are self trades
Routes = [element for element in Routes if (element[0] != element[1])]


# A dictionary called route_vars is created to contain the referenced variables (the routes)
# Make sure there isn't a route to itself
route_vars = {
'1': {'2': LpVariable("1_2",0,None,LpInteger), '3': LpVariable("1_3",0,None,LpInteger)},
'2': {'1': LpVariable("2_1",0,None,LpInteger), '3': LpVariable("2_3",0,None,LpInteger)},
'3': {'1': LpVariable("3_1",0,None,LpInteger), '2': LpVariable("3_2",0,None,LpInteger)}
}


# The objective function is added to prob first
#prob += lpSum([(need[t] - (current[s] + route_vars[t][s])) for (s,t) in Routes]), "Sum of unmet need after trading"
prob += (need['1'] - (current['1'] + route_vars['2']['1'] + route_vars['3']['1'])) \
    + (need['2'] - (current['2'] + route_vars['1']['2'] + route_vars['3']['2'])) \
    + (need['3'] - (current['3'] + route_vars['1']['3'] + route_vars['2']['3'])), "Sum of unmet need after trading"


# The amount each source trades cannot exceed the amount they currently have
prob += (route_vars['1']['2'] + route_vars['1']['3']) <= current['1'], f"Can't trade more than current inventory for person 1"
prob += (route_vars['2']['1'] + route_vars['2']['3']) <= current['2'], f"Can't trade more than current inventory for person 2"
prob += (route_vars['3']['1'] + route_vars['3']['2']) <= current['3'], f"Can't trade more than current inventory for person 3"

prob.solve()

pd.Series({v: int(v.varValue) for v in prob.variables()})

Answer 1

The problem is with the objective function which in this case always will evaluate to 1 which is the total unmet supply. The reason why the model behaves this way is because overstocked demand contributes negatively in the objective function

As an example we can take your optimal solution which gives us a total balance for each person:

total = {
'1': 50,
'2': 10,
'3': 0 }

The objective function is essentially giving us lpSum(need[p] - total[p] for p in people) which means

15 - 50 = -35
0 - 10 = -10
46 - 0 = 46

So we have

(-35) + (-10) + 46 = 1

What is missing is a term that cancels out the negative contribution of overstocking in the objective function, like so:

lpSum(need[p] - total[p] + overstock[p] for p in people )

Taking kabdulla's comment about multi-hope trading into account and adding decision variables to keep track of overstocking and totals (strictly for convenience and readability), a correct model can be formulated.

Check it out and let me know if you need clarification on anything

 from pulp import *
import pandas as pd

# Creates a list of the unique people
people = ['1','2','3']

# Creates a dictionary for the number of units of units currently in supply
current = {
    '1': 10,
    '2': 0,
    '3': 50
}

# Creates a dictionary for the number of units needed
need = {
    '1': 15,
    '2': 0,
    '3': 46
}

# Creates the prob variable to contain the problem data
prob = LpProblem("Goods redistribution problem", LpMinimize)


overstock = LpVariable.dicts("overstock",
                             [p for p in people],
                             lowBound = 0)

route_vars = LpVariable.dicts("Trades",
                     [(i,j) for i in people for j in people if i != j],
                     lowBound = 0, 
                     cat = "Integer")

total = LpVariable.dicts("Total", 
                   [p for p in people],
                   lowBound = 0)

#obj
prob += lpSum(need[p] - total[p] + overstock[p] for p in people )

#s,t
for p in people:
    prob += total[p] - need[p] <= overstock[p] , "overstock definition for person" + str(p)

for p in people:
    prob += lpSum(route_vars[(p,j)] for j in people if p != j) + current[p] - lpSum(route_vars[(i,p)] for i in people if i != p) == total[p], "definition of total stock for person" + str(p)

for i in people:
    prob += lpSum(route_vars[(i,j)] for j in people if i != j) <= current[i] , "trading constraint person" + str(i)

prob.solve()

#Print optimal values in console
for v in prob.variables():
    print(v.name, " = ",  v.varValue)

#Print optimised objective function in console
print("Unmet supply:", prob.objective.value())

#Print solution status in console
print("Status:", LpStatus[prob.status])