Sensitivity analysis: slack (allowed range) for the objective function coefficients?

[FlorinAndrei]

With Pyomo, it's pretty easy to generate a sensitivity report that includes:

1. slack for constraints
2. shadow prices for constraints
3. range (slack) for the objective function coefficients (the intervals where they can change without changing the solution)

With OR-Tools I know how to do 1 and 2, but I'm not sure about 3.

Pyomo example:

### PROBLEM DATA ###

# load data
products = ['Doors', 'Windows']
plants = ['Plant1', 'Plant2', 'Plant3']
profit_rate_list = [3, 5]
hours_avail_list = [4, 12, 18]
hours_per_batch_list = [[1, 0], [0, 2], [3, 2]]

# parse lists into dictionaries
profit_rate = dict(zip(products, profit_rate_list))
hours_available = dict(zip(plants, hours_avail_list))
hours_per_batch = {
    pl: {pr: hours_per_batch_list[i][j] for j, pr in enumerate(products)}
    for i, pl in enumerate(plants)
}


### MODEL CONSTRUCTION ###
# Declaration
model = ConcreteModel()

# Decision Variables
model.weekly_prod = Var(products, domain=NonNegativeReals)

# Objective
model.profit = Objective(
    expr=sum(profit_rate[pr] * model.weekly_prod[pr] for pr in products),
    sense=maximize,
)

# Constraints
model.capacity = ConstraintList()
for pl in plants:
    model.capacity.add(
        sum(hours_per_batch[pl][pr] * model.weekly_prod[pr] for pr in products)
        <= hours_available[pl]
    )

### SOLUTION ###
solver = SolverFactory('glpk')
solver.solve(model)

### OUTPUT ###

# note that we're using f-strings for output here which is a little different and cleaner than in the video
print(f"Maximum Profit = ${1000*model.profit():,.2f}")
for j in products:
    print(f"Batches of {j} = {model.weekly_prod[j]():.1f}")

model.write('model.lp', io_options={'symbolic_solver_labels': True})
!glpsol -m model.lp --lp --ranges sensit.sen

# print sensitivity report
np.set_printoptions(linewidth=110)
f = open('sensit.sen', 'r')
file_contents = f.read()
print(file_contents)
f.close()

Output:

GLPK 5.0  - SENSITIVITY ANALYSIS REPORT                                                                         Page   1

Problem:    
Objective:  profit = 36 (MAXimum)

   No. Row name     St      Activity         Slack   Lower bound       Activity      Obj coef  Obj value at Limiting
                                          Marginal   Upper bound          range         range   break point variable
------ ------------ -- ------------- ------------- -------------  ------------- ------------- ------------- ------------
     1 c_u_capacity(1)_
                    BS       2.00000       2.00000          -Inf         .           -3.00000      30.00000 c_u_capacity(3)_
                                            .            4.00000        6.00000       4.50000      45.00000 c_u_capacity(2)_

     2 c_u_capacity(2)_
                    NU      12.00000        .               -Inf        6.00000      -1.50000      27.00000 c_u_capacity(1)_
                                           1.50000      12.00000       18.00000          +Inf      45.00000 weekly_prod(Doors)

     3 c_u_capacity(3)_
                    NU      18.00000        .               -Inf       12.00000      -1.00000      30.00000 weekly_prod(Doors)
                                           1.00000      18.00000       24.00000          +Inf      42.00000 c_u_capacity(1)_

GLPK 5.0  - SENSITIVITY ANALYSIS REPORT                                                                         Page   2

Problem:    
Objective:  profit = 36 (MAXimum)

   No. Column name  St      Activity      Obj coef   Lower bound       Activity      Obj coef  Obj value at Limiting
                                          Marginal   Upper bound          range         range   break point variable
------ ------------ -- ------------- ------------- -------------  ------------- ------------- ------------- ------------
     1 weekly_prod(Doors)
                    BS       2.00000       3.00000        .                -Inf        .           30.00000 c_u_capacity(3)_
                                            .               +Inf        4.00000       7.50000      45.00000 c_u_capacity(2)_

     2 weekly_prod(Windows)
                    BS       6.00000       5.00000        .             3.00000       2.00000      18.00000 c_u_capacity(2)_
                                            .               +Inf        6.00000          +Inf          +Inf

End of report

**Constraints**

Slack = slack

Marginal = shadow price

Slack: 2, 0, 0

Shadow price: 0, 1.5, 1

---

**Objective function**

Obj coef range = slack

Slack: 0 - 7.5, 2 - inf

---

OR-Tools example:

try:
    del solver
except Exception as e:
    pass

solver = pywraplp.Solver.CreateSolver('GLOP')
# solver = pywraplp.Solver.CreateSolver('PDLP')

weekly_prod = [solver.NumVar(0.0, solver.infinity(), p) for p in products]

objective = solver.Objective()
for i, prod in enumerate(products):
    objective.SetCoefficient(weekly_prod[i], profit_rate[prod])
objective.SetMaximization()

constraints = []
for i, plant in enumerate(plants):
    constraints.append(solver.Constraint(-solver.infinity(), hours_available[plant], plant))
    for j, prod in enumerate(products):
        constraints[i].SetCoefficient(weekly_prod[j], hours_per_batch[plant][prod])

wyn_status = solver.Solve()

print(f'profit: ${1000 * solver.Objective().Value():,.2f}')
for x in weekly_prod:
    print(f'{x.name()} {x.solution_value()}')

activities = solver.ComputeConstraintActivities()
o = [
    {
        'Name': c.name(),
        'shadow price': c.dual_value(),
        'slack': c.ub() - activities[i],
    }
    for i, c in enumerate(solver.constraints())
]
print(pd.DataFrame(o))

profit: $36,000.00
Doors 2.0
Windows 6.0

     Name  shadow price  slack
0  Plant1           0.0    2.0
1  Plant2           1.5    0.0
2  Plant3           1.0    0.0

How to obtain the slack for the objective function coefficients in OR-Tools?

[lperron]

From what I understand, this is only supported with glpk. It this true ?