Styjun

Given transport costs, per single unit of delivery, for a supermarket from three distribution centers to ten separate stores.

Note: Please look in the #data section of my code to see the data that I'm not allowed to post in photo form. ALSO note while my costs are a vector with 30 entries. Each distribution centre can only access 10 costs each. So DC1 costs = entries 1-10, DC2 costs = entries 11-20 etc..

I want to minimize the transport cost subject to each of the ten stores demand (in units of delivery).

This can be done by inspection. The the minimum cost being $150313. The problem being implementing the solution with Python and Gurobi and producing the same result.

What I've tried is a somewhat sloppy model of the problem in Gurobi so far. I'm not sure how to correctly index and iterate through my sets that are required to produce a result.

This is my main problem: The objective function I define to minimize transport costs is not correct as I produce a non-answer.

The code "runs" though. If I change to maximization I just get an unbounded problem. So I feel like I am definitely not calling the correct data/iterations through sets into play.

My solution so far is quite small, so I feel like I can format it into the question and comment along the way.

from gurobipy import *

#Sets
Distro = ["DC0","DC1","DC2"]
Stores = ["S0", "S1", "S2", "S3", "S4", "S5", "S6", "S7", "S8", "S9"]
D = range(len(Distro))
S = range(len(Stores))

Here I define my sets of distribution centres and set of stores. I am not sure where or how to exactly define the D and S iteration variables to get a correct answer.

#Data
Demand = [10,16,11,8,8,18,11,20,13,12]
Costs = [1992,2666,977,1761,2933,1387,2307,1814,706,1162,
 2471,2023,3096,2103,712,2304,1440,2180,2925,2432,
 1642,2058,1533,1102,1970,908,1372,1317,1341,776]

Just a block of my relevant data. I am not sure if my cost data should be 3 separate sets considering each distribution centre only has access to 10 costs and not 30. Or if there is a way to keep my costs as one set but make sure each centre can only access the costs relevant to itself I would not know.

m = Model("WonderMarket")
#Variables
X = 
for d in D:
 for s in S:
 X[d,s] = m.addVar()

Declaring my objective variable. Again, I'm blindly iterating at this point to produce something that works. I've never programmed before. But I'm learning and putting as much thought into this question as possible.

#set objective
m.setObjective(quicksum(Costs[s] * X[d, s] * Demand[s] for d in D for s in S), GRB.MINIMIZE)

My objective function is attempting to multiply the cost of each delivery from a centre to a store, subject to a stores demand, then make that the smallest value possible. I do not have a non zero constraint yet. I will need one eventually?! But right now I have bigger fish to fry.

m.optimize()

I produce a 0 row, 30 column with 0 nonzero entries model that gives me a solution of 0. I need to set up my program so that I get the value that can be calculated easily by hand. I believe the issue is my general declaring of variables and low knowledge of iteration and general "what goes where" issues. A lot of thinking for just a study exercise!

Appreciate anyone who has read all the way through. Thank you for any tips or help in advance.

Your objective is 0 because you do not have defined any constraints. By default all variables have a lower bound of 0 and hence minizing an unconstrained problem puts all variables to this lower bound.

A few comments:

Unless you need the names for the distribution centers and stores, you could define them as follows:

D = 3
S = 10
Distro = range(D)
Stores = range(S)

You could define the costs as a 2-dimensional array, e.g.

Costs = [[1992,2666,977,1761,2933,1387,2307,1814,706,1162],
 [2471,2023,3096,2103,712,2304,1440,2180,2925,2432],
 [1642,2058,1533,1102,1970,908,1372,1317,1341,776]]

Then the cost of transportation from distribution center d to store s are stored in Costs[d][s].

You can add all variables at once and I assume you want them to be binary:

X = m.addVars(D, S, vtype=GRB.BINARY)

(or use Distro and Stores instead of D and S if you need to use the names).

Your definition of the objective function then becomes:

m.setObjective(quicksum(Costs[d][s] * X[d, s] * Demand[s] for d in Distro for s in Stores), GRB.MINIMIZE)

(This is all assuming that each store can only be delivered from one distribution center, but since your distribution centers do not have a maximal capacity this seems to be a fair assumption.)

You need constraints ensuring that the stores' demands are actually satisfied. For this it suffices to ensure that each store is being delivered from one distribution center, i.e., that for each s one X[d, s] is 1.

m.addConstrs(quicksum(X[d, s] for d in Distro) == 1 for s in Stores)

When I optimize this, I indeed get an optimal solution with value 150313.