AUGMECON-Py: A Python framework for multi-objective linear optimisation under uncertainty

maximize or minimize

Mixed-integer linear programming problems are solved with more complex and computationally intensive methods like the branch-and-bound method, which uses linear programming under the hood. Some variants of this method are the branch-and-cut method, which involves the use of cutting planes, and the branch-and-price method. Often, when people try to formulate and solve an optimization problem, the first question is whether they can apply linear programming or mixed-integer linear programming.

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We also are touching upon how to formulate a LP using mathematical notations. The values of the decision variables that minimizes the objective function while satisfying the constraints. In a profit maximizing problem such as this one, these parallel lines are often called isoprofit lines, because all the points along such a line represent the same profit.

Unbounded Linear Programming Problem

The function is the minimization of the sum of transportation costs. The first constraint requires that the demand is satisfied, and the second constraint ensures that factory capacities are not exceeded. One of the important features of linear optimization problems is that they are easy to solve. Common texts on mathematical optimization describe in lengthy detail how a linear optimization problem can be solved.

The department has 60 hours of labor available each day, 16 hours of machine time, and 400 board feet of wood. A consultant has developed a linear programming model for the department. OR-Tools comes with its own linear programming solver, called GLOP . It is an open-source project created by Google’s Operations Research Team and written in C++. You can define variable names in your model to make your model look more intuitive to the person who will be reading it later. Hence, we create indices for our decision variables which will be defined later.

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For the sake of simplicity, and to maintain brevity, I am planning to define only five constraints. Now the lpSum method helps in calculating the sum of the linear expressions, so we will use it to define the constraint of Calories in the data. The motive here is to find out the optimal values of x & y such that the inequalities are given in red, blue & yellow, and inequalities x ≥ 0 and y ≥ 0 are satisfied. The optimal values are those that maximize the objective function. In this article we covered how you can solve a linear programming problem using Gurobi Python interface with gurobipy library. What we need is some way of generating integers for the \(x_1\) and \(x_2\) decision variables.

minimum and maximum

The second slack is 40 because the factory consumes 60 units of raw material A out of a potential 100 units. The first slack is 0, which means that the values of the left and right sides of the manpower constraint are the same. The factory produces 50 units per day, and that’s its full capacity. Rhs_ineq holds the right-side coefficients from the inequality constraints. Lhs_ineq holds the left-side coefficients from the inequality constraints. For each unit of the first product, three units of the raw material A are consumed.

Gurobi: Always Free for Academics

Based on weights, the knapsack can then be appropriately filled by a collection that is optimal in the context of weight as the special attribute. Let us assume that an entrepreneur is interested in the wine making company and would like to buy its resources. The entrepreneur then needs to find out how much to pay for each unit of each of the resources, the pure-grape wines of 2010 A, B and C. This can be done by solving the dual version of the model that we will discuss next. Let us formalize this as an optimization problem with mathematical formulas. This process of describing a situation algebraically is called the formulation of a problem in mathematical optimization.

Let’s say the linear optimization python is Crocs which supplies only footwear, and the customers here are its distributors who need these crocs in bulk. In the above code, you define tuples that hold the constraints and their names. LpProblem allows you to add constraints to a model by specifying them as tuples.


It starts with an initial feasible solution, and then tests to see if it can improve the result of the objective function. It continues until the objective function cannot be further improved. This graphic shows an example of an LP with multiple optimal solutions. This can happen when the slope of the objective function is the same as the slope of one of the constraints, in this case line AB. All the points on line AB are optimal solutions, with the same objective value, because they are all extreme points within the feasible region.

7. Duality¶

A graphical representation of the constraints and objective is shown in the figure below. The contour lines are the objective and the vertices are labeled. The duality theorems provide the foundations of enlightening economic interpretations of linear programming problems.

You can determine which constraints are binding in a solution by examining the slack values with DOcplex. The dual price of a binding constraint can help you make decisions regarding the constraint. In LP, the optimal objective values of D and P are equivalent, and occurs where these bounds meet. While infeasibility implies a model where constraints are too limiting, unboundedness implies a model where an important constraint is either missing or not restrictive enough.


In addition, you’ll have to consider some restrictions like budget and the variety of food. Suppose you are a dietician and is responsible to advise one of your customer on the best possible food plan he should follow to attain “optimum nutrition”. But we might come across problems where the feasible regions are not bounded, or the problem is infeasible. Due to the limited number of workers, a company has maximum of 100 labour hours.

TORA. But, the knowledge of performing optimization programmatically is second to none! This means that at least one of your variables isn’t constrained and can reach up to positive or negative infinity, making the objective infinite as well. For a convex function, it is just opposite – line segment lies on or below the graph of the function f. A function is called convex- if a line segment is drawn from any point (x, f) to another point (y, f), from x to y — lies on or above the graph of f, as shown in the figure below.

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Linear functions are convex, so linear programming problems are convex problems. Constraints – The restrictions on the decision variables in an optimization problem are known as constraints. In the above example, the inequalities in red, blue & yellow are the constraints. Note that these could very well be equations or equality constraints. In any linear programming problem we need to correctly identify the decision variables. Decision variables are variables that represent a decision made in the problem.

  • For each unit of the first product, three units of the raw material A are consumed.
  • This notebook contains 8 mini problems from various optimization topics.
  • Scipy.optimize.linprog The SciPy linear programming library minimizes a linear objective function subject to linear equality and inequality constraints.
  • Finally, you’ll look at resources and libraries to help further your linear programming journey.

This graphic shows an example of an infeasible constraint set for the telephone production problem. Assume in this case that the person entering data had accidentally entered lower bounds on the production of 1100 instead of 100. The arrows show the direction of the feasible region with respect to each constraint. This data entry error moves the lower bounds on production higher than the upper bounds from the assembly and painting constraints, meaning that the feasible region is empty and there are no possible solutions. It is good practice to start with a descriptive model before attempting to write a mathematical model.

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