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Production planning problem

This example is a simple production planning problem outlined in detail in our paper on (warped)-Gaussian process constrained optimization. The main idea is to determine production targets as to maximize profit over a number of time periods given a price at which the product can be sold and a cost of production. We assume that he sales price depends on the amount produced through an uncertain black-box function. Our paper outlines two approaches for modeling this black-box function based on data and incorporating it into the optimization problem:

  1. Use a [standard Gaussian Process] as a stochastic model for the uncertain black-box function.

  2. Use a [warped Gaussian Process] instead

We have implemented both approaches in ROmodel to make them more accessible. The (warped) Gaussian Process can be trained using the GPy framework. ROmodel contains two dedicated uncertainty sets, GPSet and WarpedGPSet, which take a GPY object as an input and construct corresponding uncertainty sets using the approaches outlined in our paper.

Formulation

The only decision variables of the problem are the production amounts in each time period. Parameters are the cost of production and the price at which the product can be sold:

# Number of planning periods
T = 6
# Construct model
m = pe.ConcreteModel()
# Decision variables
m.x = pe.Var(range(T), within=pe.NonNegativeReals, bounds=(xmin, xmax))
# Uncertainty set based on warped Gaussian process
m.uncset = ro.uncset.WarpedGPSet(gp, m.x, alpha)
# Uncertainty set based on standard Gaussian process
m.uncset = ro.uncset.GPSet(gp, m.x, alpha)
# Uncertain parameter
m.demand = ro.UncParam(range(T), uncset=m.uncset, bounds=(0, 1))
# Uncertain objective
profit = sum(m.x[t]*m.demand[t] for t in range(T))
profit -= sum(cost[t]*m.x[t] for t in range(T))
m.u = pe.Var()
m.profit = pe.Constraint(expr=profit >= m.u)
m.Obj = pe.Objective(expr=m.u, sense=pe.maximize)

Running the example

The production planning example is available as part of ROmodel and can be used as follows:

import romodel.examples as ex
import pyomo.environ as pe

m = ex.ProductionPlanning(alpha=0.9, warped=True)

solver = pe.SolverFactory('romodel.reformulation')
solver.options['solver'] = 'ipopt'
solver.solve(m)

Parameter alpha determines the confidence with which the solution will be feasible, while the warped parameter determines whether the standard or warped Gaussian Process is used. Solving the problem requires a non-linear, non-convex solver like ipopt.