gurobi objective function: cost coefficients dependent on sum of solution variables

[Monique Bakker]

Hi,

I am trying to solve a model (python) where the coefficients in the linear objective function depend on the sum of certain solution variables.

All variables are binary.

I get the error 'Divisor must be a constant'.

Is there a way around this? 

Thanks. Code:

from gurobipy import *

import math

import numpy

import openpyxl

# physicians

physicians = ['DrA', 'DrB']

# halfdays in planspan

planspan = ['MoAM', 'MoPM', 'TuAM', 'TuPM', 'WeAM', 'WePM', 'ThAM', 'ThPM', 'FrAM', 'FrPM']

# activities

activities = ['Admin1', 'Frac2', 'Spec3', 'Fleb4', 'Latr5', 'Endo6', 'Surg7', 'Surg8',

     'Noth9', 'Annl10']

# Model

m = Model("motest1")

# Assignment variables: x[j,a,t] ==1 if physician j is assigned to activity a

# on half day t.

# Just as in workforce1.py, since an assignment model always produces integer solutions, we use

# continuous variables and solve as an LP.

x = []

n = 1

for j in physicians:

    for t in planspan:

        for a in activities:

            x.append(m.addVar(vtype=GRB.BINARY, name="x%d%s_%s_%s" % (n,j,t,a)))

            n = n+1

m.update()

# Add constraints to ensure that EXACTLY one of the activities is assigned

# note that sum(x[0:9]) does not include the 9th item labeled x10 

m.addConstr(sum(x[0:10]) == 1)

m.addConstr(sum(x[10:20]) == 1)

m.addConstr(sum(x[20:30]) == 1)

m.addConstr(sum(x[30:40]) == 1)

m.addConstr(sum(x[40:50]) == 1)

m.addConstr(sum(x[50:60]) == 1)

m.addConstr(sum(x[60:70]) == 1)

m.addConstr(sum(x[70:80]) == 1)

m.addConstr(sum(x[80:90]) == 1)

m.addConstr(sum(x[90:100]) == 1)

# j=2

m.addConstr(sum(x[101:110]) == 1)

m.addConstr(sum(x[110:120]) == 1)

m.addConstr(sum(x[120:130]) == 1)

m.addConstr(sum(x[130:140]) == 1)

m.addConstr(sum(x[140:150]) == 1)

m.addConstr(sum(x[150:160]) == 1)

m.addConstr(sum(x[160:170]) == 1)

m.addConstr(sum(x[170:180]) == 1)

m.addConstr(sum(x[180:190]) == 1)

m.addConstr(sum(x[190:200]) == 1)

M = array([[ 2.        ,  2.        ],

       [ 0.83333333,  1.16666667],

       [ 2.16666667,  2.        ],

       [ 0.        ,  0.        ],

       [ 0.5       ,  0.16666667],

       [ 0.5       ,  0.83333333],

       [ 1.5       ,  1.16666667],

       [ 0.83333333,  1.33333333],

       [ 1.66666667,  1.33333333],

       [ 0.2       ,  0.2       ]])

C = array([[ 0.69230769,  0.69230769],

       [ 0.72137137,  0.63005549],

       [-0.67160603,  0.60671454],

       [ 0.        ,  0.        ],

       [ 0.06355356, -0.31430663],

       [ 0.76585138,  1.32377788],

       [-0.43737702, -0.64233165],

       [ 1.79534889,  0.61685841],

       [ 0.        ,  0.        ],

       [ 0.        ,  0.        ]])

obj = LinExpr()

for j in range(len(physicians)):

    for a in range(len(activities)-2): # activity 2-8 (index starts at 0)

        if j == 0:        

            rng = x[a:100:10]  # this sum is the number of instances of each activity 

        elif j == 1:

            rng = x[a+100:200:10]

        d = sum(rng)

        if d >= 0:

            for i in range(len(rng)):

                obj += 1/d * (d - M[a][j] + C[a][j])  * rng[i]  

        elif d == 0:

            for i in range(len(rng)):

                obj += (d - M[a][j] + C[a][j]) * rng[i] 

[Eric Berger]

Is there a typo? Should

m.addConstr(sum(x[101:110]) == 1)

be 

m.addConstr(sum(x[100:110]) == 1)

i.e. 100 instead of 101 ?

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[Sonja Mars]

Hi,

The problem is this line:

> obj += 1/d * (d - M[a][j] + C[a][j]) * rng[i]

And especially this term: 1/d. d is a linear expression, it is a sum of variables that is generated here:

> if j == 0:
> rng = x[a:100:10] # this sum is the number of instances of each activity
> elif j == 1:
> rng = x[a+100:200:10]
> d = sum(rng)

When I ran your code d looked like this:

<gurobi.LinExpr: x1DrA_MoAM_Admin1 + x11DrA_MoPM_Admin1 + x21DrA_TuAM_Admin1 + x31DrA_TuPM_Admin1 + x41DrA_WeAM_Admin1 + x51DrA_WePM_Admin1 + x61DrA_ThAM_Admin1 + x71DrA_ThPM_Admin1 + x81DrA_FrAM_Admin1 + x91DrA_FrPM_Admin1>

With "Divisor must be a constant" Gurobi is telling you that you cannot have something like 1/(c^T*x) where x is a vector of variable in your objective. This is nonlinear. You will have to reformulate your model to be linear or quadratic otherwise Gurobi cannot solve it.

Best regards,
Sonja

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Dr. Sonja Mars
Gurobi Optimization