Linear Programming and Optimization using Python

decision variables

That is, raw materials from the warehouse 3 and warehouse 1 for 2 and 6 tons, respectively, are brought to the first plant. And raw materials from a warehouse of 2 and warehouse 1 of 4 tons are brought to the second plant. That is, each plant will receive 8 tons of raw materials, as was necessary.

We also are touching upon how to formulate a LP using mathematical notations. 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. You can get the optimization results as the attributes of the model.

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It is basically like a text file containing the exact details of the optimization model as printed above. The first argument in the function represents the name we want to give to our model. The second argument tells our model whether we want to minimize or maximize our objective function. A step by step introduction to formulating and solving a linear optimization problem using PuLP library in Python. The coefficients of the linear objective function to be minimized. Line 5 defines the binary decision variables y and y held in the dictionary y.

Mathematical Optimization¶

That satisfies all of the constraints is called a feasible solution. The output of the above code is Optimal which tells us that our model has been able to find an optimal solution to the problem. We will define our decision variable as Xij which basically tells that X products should be delivered from Warehouse i to Customer j. Let’s say the company is Crocs which supplies only footwear, and the customers here are its distributors who need these crocs in bulk.


PuLP has a more convenient linear programming API than SciPy. You don’t have to mathematically modify your problem or use vectors and matrices. The third slack is 0, which means that the factory consumes all 90 units of the raw material B. That’s why the factory can’t produce the second or fourth product at all and can’t produce more than 45 units of the third product.

What Is Linear Programming?

Due to the transportation and storage constraints, the factory can consume up to one hundred units of the raw material A and ninety units of B per day. In the previous sections, you looked at an abstract linear programming problem that wasn’t tied to any real-world application. In this subsection, you’ll find a more concrete and practical optimization problem related to resource allocation in manufacturing. You no longer have the green line, only the points along the line where the value of x is an integer. The feasible solutions are the green points on the gray background, and the optimal one in this case is nearest to the red line.

  • Another example, As a marketing manager wants to allocate the optimum budget among alternative advertising media channels such as radio, television, newspaper, and magazine.
  • PuLP allows you to choose solvers and formulate problems more naturally.
  • Graph representation of a transportation problem and its optimal transport volume.
  • If the default algorithm does not break the degenerate cycle, it’s a good idea to try some other algorithms, for example the Dual-simplex Optimizer.

We return the newly linear optimization pythond problem and the variables for the problem. I am using miniconda to create an isolated environment – you can use any environment manager available for python like venv or full anaconda. In the code directory on github you will find the environment.yml file, after installing anaconda simply run the following commands on your terminal. While meeting our objective we have to make sure that we don’t voilate a couple of constraints. When calling a function or method, keyword arguments without a default value cannot be omitted. The objective is to maximize the value of y, subject to satisfying the inequalities , , and by finding the values of x1 and x2 that satisfy these conditions.

Step 1: Define the decision variables

A firm’s problem is to construct a plan that uses its 30 units of materials and 20 unites of labor to maximize its revenue. These constraints say that the allocation done for each customer or the j-th customer should be such that the demand of that customer is met. The first statement imports all the required functions that we will be using from the PuLP library. Pandas is a data manipulation library and Numpy is a library used majorly for working with multi-dimensional arrays in Python. I usually just import these libraries since they are mostly used in almost all data analysis projects. This section describes the available solvers that can be selected by the ‘method’ parameter.

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Have the specific name of sign restrictions or non-negativity constraints. As these variables can take any non-negative real number, they are called real variables, or continuous variables. The Interior Point is described as the method where the approximation of the constraints is considered as a set of boundaries covering a defined region. The approximations are then utilized for problems consisting of constraints, which are discontinuous or troublesome but have the possibility of modifying them so that they can be handled. From the data, I extract the costs of the two foods and the minimum and maximum nutrient values.

Section The Modern Diet Problem considers how to cope with nutritional problems, showing an example of an optimization problem with no solution. Linear programming is a mathematical optimization technique used to optimize a linear objective function subject to a set of linear constraints. It involves finding the values of decision variables that minimize or maximize the objective function while satisfying the given constraints. This linear-optimization relaxation can be solved easily by the simplex method.

Define the Constraints

Infeasible LP – A linear programming problem is infeasible if it doesn’t have a solution. This usually happens when no solution can satisfy all constraints at once. 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. With this, let us take a step back and understand what Discrete Optimization is! Discrete optimization is a part of optimization methodology that deals with discrete quantities, i.e., non-continuous functions.


We can now solve the problem at different price points as shown below. The above goal formulation might look a bit terse on mathematical notation `but we will break it down below. When the above program is executed, the following result is obtained. The results are shown in Table Optimal solution for the transportation problem and Figure Transportation problem.

Linear Programming Python Implementation

The following graph illustrates the firm’s constraints and iso-revenue lines. The cost of shipping matrix for Warehouse i to Customer j is as follows. Each value can also be represented as Cij suggesting Cost C to ship from Warehouse i to Customer j. It turns out that the optimal approach is to exclude the first product and to produce only the third one. The value 100 is large enough in this case because you can’t have more than 100 units per day. You use the sense parameter to choose whether to perform minimization or maximization (LpMaximize or -1).

In the following, to simplify the description, names of constraints are often omitted; but in fact it is safer to give an appropriate name. Graph representation of a transportation problem and its optimal transport volume. The first lines, not shown, report progress of the SCIP solver while lines 2 to 6 correspond to the output instructions of lines 14 to 16 of the previous program. This notebook serves as an introduction to Linear Programming with Python, covering both the concepts and practical applications through various popular optimization problems. I have no solution at the moment but will try to solve it in the next days. Not all items are desired, that is, they are worth less and / or have a high weighting which dilutes the pool I can draw from.


A variable is unbounded when one or both of its bounds is infinite. $$ overtime Finally, add an additional cost to the objective to penalize use of overtime. Converting hard constraints to soft is one way to resolve infeasibilities. Draw each of the constraints as you would draw any line in 2 dimensions. The objective is to maximize profit, and the company has to produce at least 100 of each type of phone.

The function value() and the corresponding method .value() return the actual values of the attributes. Prob.objective holds the value of the objective function, prob.constraints contains the values of the slack variables (we don’t require them but just a good to know fact). Mixed-integer linear programming allows for overcoming many demerits of linear programming. One can approximate non-linear functions withpiecewise linear functions, model-logical constraints (ayes/no problem like whether a customer will churn or not), and more. Capacity constraints are created in a similar way, in lines 13 to 15. For an example, consider now the same three production plants and five customers as before.

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The parameter c refers to the coefficients from the objective function. A_ub and b_ub are related to the coefficients from the left and right sides of the inequality constraints, respectively. You can use bounds to provide the lower and upper bounds on the decision variables.

The Cost function is the total cost of the food items, which we are trying to minimize. Because the cost should be minimal, at the same time, deriving the nutritional value from the combination of different food items should be maximum, considering the maximum and minimum constraints given in the data. A nutritional component’s minimum and maximum bounds define the inequality constraints. Scipy.optimize.linprog The SciPy linear programming library minimizes a linear objective function subject to linear equality and inequality constraints. For large-scale problems, a matrix forms is best because it simplifies the problem description and improves the speed of solution. Scipy.optimize.linprog is one of the available packages to solve Linear programming problems.

You can revise the basics of mathematical concepts in this article. That minimizes the cost function is called an optimal solution. In my previous article, I have demonstrated how to solve linear programming problems using the graphical method. This article will show you how to solve linear programming problems in Python using four different open-source libraries — Scipy, PuLP, Pyomo, and Google OR-Tools.

  • Minimization concerning linear programming in Python means to minimize the total cost of production whereas Maximization on the other hand means to maximize the company or organization’s profit.
  • In any solution to the dual, the values of the dual variables are known as the dual prices, also called shadow prices.
  • The optional parameter cat defines the category of a decision variable.
  • Because $x_$ has a negative coefficient in the objective, the optimization will minimize $x_$.
  • I now define the objective function and constraints using the same method as above.

As the simplex algorithm attempts to move in the direction of an improved objective value, it might happen that the algorithm starts cycling between non-optimal solutions with equivalent objective values. It is good practice to start with a descriptive model before attempting to write a mathematical model. In order to come up with a descriptive model, you should consider what the decision variables, objectives, and constraints for the business problem are, and write these down in words. We build the linear programming problem by adding the main objective function.

Hence, the efficiency of the convex optimization problem in time is much better. Memory and solution time may rise exponentially as you add more integer variables. Now that we are familiar with the terminologies let us throw some more light on Mixed-Linear integer programming. This type of problem is important, especially when we express quantities in integers like the number of products produced by the machine or the number of customers served.

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