# Pulp problem gets the right objective value but fails to find the minimal solution

I converted an excel solver problem to Pulp. However, while the pulp program calculates the objective function value correctly, it doesn't arrive at the right answer. I can't figure out what is wrong in the code. Can anyone identify what I may be doing wrong here?

I know the objective functions in pulp and excel are both the same, since if I plug in the pulp solution in the excel model, excel gives me the same objective function value as pulp. But for some reason, pulp cannot find the low cost that excel does.

Thank you again for your help!

from pulp import *
import pandas as pd

init_inventory = 500
init_workers = 100

reg_work_hours = 160
max_ot_hours = 20

cost_hiring = 1600
cost_firing = 2000

monthly_wage = 1500
ot_wage_rate = 13

labor_hours_per_shoe = 4
mat_cost_per_shoe = 15
hold_cost_per_month = 3

Months = ["Month 1", "Month 2", "Month 3", "Month 4"]

ShoesNeeded = dict(zip(Months, [3000, 5000, 2000, 1000]))

prob = LpProblem("ProductionPlanning", LpMinimize)

hire = LpVariable.dicts("hire", Months, 0, None, LpInteger)
fire = LpVariable.dicts("fire", Months, 0, None, LpInteger)

avail_workers = {Months : init_workers + hire[Months] - fire[Months]}

for m in range(1,4):
avail_workers[Months[m]] = avail_workers[Months[m-1]] + hire[Months[m]] - fire[Months[m]]

reg_hours = {m: avail_workers[m] * reg_work_hours for m in Months}
max_ot_avail = {m: avail_workers[m] * max_ot_hours for m in Months}

ot_used = LpVariable.dicts("ot_used", Months, 0, None, LpInteger)

total_labor_hours = {m: reg_hours[m] + ot_used[m] for m in Months}

prod_capacity = {m: total_labor_hours[m] / labor_hours_per_shoe for m in Months}

shoes_produced = LpVariable.dicts("Shoes_produced", Months, 0, None, LpInteger)

invent_ending = {Months : init_inventory + shoes_produced[Months] - ShoesNeeded[Months]}

for m in range(1,4):
invent_ending[Months[m]] = invent_ending[Months[m-1]] + shoes_produced[Months[m]] - ShoesNeeded[Months[m]]

prob += lpSum(hire[m] * cost_hiring +
fire[m] * cost_firing +
avail_workers[m] * monthly_wage +
ot_used[m] * ot_wage_rate +
shoes_produced[m] * mat_cost_per_shoe +
invent_ending[m] * hold_cost_per_month for m in Months), "Total Cost"

for m in Months:
prob += ot_used[m] <= max_ot_avail[m], f"Max OT for {m}"
prob += shoes_produced[m] <= prod_capacity[m], f"Production capacity for {m}"
prob += invent_ending[m] >= ShoesNeeded[m], f"Minimum quantity needed for {m}"

prob.solve()

for v in prob.variables():
print(v.name, "=", v.varValue)

print("Total Cost = ", value(prob.objective))

• It would be great if, you can show what mathematical formulation you are trying to solve instead of writing only the python code. Feb 7 at 5:58
• The avail_workers, don't you think it should be a variable and should be formulated as a constraint? Feb 7 at 14:30
• @A.Omidi thank you for replying. I am not sure how to formulate the problem in mathematical notation, but I attached an excel file with the problem formulated for solver, and the solver solution has a much lower cost than the pulp solution. Feb 8 at 15:02

How do you define the "right" answer?

If the objective function value is computed correctly and reaches minimum value, then it is possible that the optimal solution is not unique, and that the solver you have used with pulp (CBC) simply found another optimal solution. There is no reason to say it is not "the right answer", as long as the minimal value is reached and that all constraints are satisfied.

If you wish to have a different optimal value, you can add no good cuts for integer variables to exclude the current solution.

You could also add some additional constraints to enforce part of the solution you found in Excel, and check if that guides the solver in pulp (CBC) to the same solution.

• Thank you for your answer. The link in my post is to the same problem setup in excel with solver, and it gets a much more optimal solution (i.e., the cost is much lower). I know the objective function in both cases is the same, since if I plug in the final decision variable values from pulp into the excel model, I get the same suboptimal value as I am getting in pulp. This is why I think I may be making a mistake in setting up the constraints in pulp. Feb 8 at 14:55

Avail_workers seems to be an array of variable, indexed by month. It's like a derived decision variable as it depends upon two other decision variables- hire and fire
Add a constraint with $$availworker_m$$ being a variable (no need to declare it integer)
$$availWorker_m = availWorker_{m-1}+hire_m-fire_m \ \ \forall m \in$$ Months[1:]
$$availworker_0 = initWorker+hire_0$$
$$shoesproduced_m\times labourHoursPerShoe \le availWorker_m\times regWorkHours +otUsed_m$$