Introduction Introduction to Linear Programming and Integer Linear Programming ► Outline - brief introduction to… § Modeling § Linear Programming RMIT, Melbourne, Australia § Integer Linear Programming § Solution Methods for LP and ILP Helena Ramalhinho Lourenço § AMPL LP & ILP introduction 1 LP & ILP introduction 2 Real Problem OR Methodology Linear Programming Data Analysis ► Linear programming (LP) is a widely used mathematical Set of Models modeling technique designed to help managers in Optimization Model planning and decision making relative to resource maker - Solution Methods Results & Interpretation allocation. ► Integer Linear Programming (ILP) Simulation Decision § An integer programming model is one where one or more of the decision variables has to take on an integer or binary values in Solution & Analysis OR provides many methods the final solution. and techniques to analyze Scenarios ► Combinatorial Optimization (next) the consequences of decisions Make a decision before their introduction to reality.LP & ILP introduction 3 LP & ILP introduction 4 OR Methodology OR Methodology ► Identify a problem ► Identify the problem ► Get to know the problem context (get Data) § Describe in detail the problem, and identify all the components. § Identify the qualitative and quantitative aspects of the problem. ► Build a Model (Mathematical Model) ► Get to know the problem context ► Obtain a solution to the model (Algorithm) § Identify the relationships of the problem with the context of the ► Understand the solution the real context organization. ► Take the decision § Obtain the relevant data. § Verify the data. LP & ILP introduction 5 LP & ILP introduction 6 © Helena Ramalhinho-Lourenço, (2019) 1 OR Methodology OR Methodology ► Build a model ► Obtain a solution to the model § A Model is an abstract representation of a real world system § Which are the adequate solution methods to be applied to the § Simplification is the very essence of Modeling model? § Which components should be included in the model? * Exact Methods § Types of models: * Heuristics and Metaheuristics • Linear programming § Which is the best software to be applied? • Integer Linear Programming * Commercial software • Combinatorial Optimization * Algorithm design and code • Networks • Non-linear • Simulation • Etc. LP & ILP introduction 7 LP & ILP introduction 8 OR Methodology Linear Programming ► Understand the solution the real context ► Linear programming (LP) is a widely used mathematical § Does the solution obtained makes sense in the real life? modeling technique designed to help managers in § Did we have considered all relevant components? planning and decision making relative to resource ► Take the decision allocation. § Evaluate the impact of the solution § Evaluate the decision process (model + method) § Review frequently the impact of the decision. LP & ILP introduction 9 LP & ILP introduction 10 Linear Programming Linear Programming ► The Linear Programming Models have 4 properties in ► Formulating a linear program involves developing a common: mathematical model to represent the managerial problem. § All problems seek to maximize or minimize some quantity (the ► The steps in formulating a linear program are: objective function). § Completely understand the managerial problem being faced. § Restrictions or constraints that limit the degree to which we can § Identify the objective and the constraints. pursue our objective are present. § Define the decision variables. § There must be alternative courses of action from which to choose. § Use the decision variables to write mathematical expressions for § The objective and constraints in problems must be expressed in the objective function and the constraints. terms of linear equations or inequalities. LP & ILP introduction 11 LP & ILP introduction 12 © Helena Ramalhinho-Lourenço, (2019) 2 Linear Programming Methodology Example 2: Call Center ► Understand well the problem (by words) ► A call center is hiring personnel since it is § What is the problem expanding to a 24h working period. § What is the decision we have to take? Can we make quantitative? § The call center works 24h a day, and needs personnel to § What are the constraints on the decisions? attend the customers every hour. § What is the objective or how we are going to evaluate the solution? § The human recourses and the operations directors have ► Define the decision variables (math) estimate the number of persons in each interval of time. ► Define the constraints on the variables (math) § There are 6 intervals of time (4 hours each). ► Define the objective function (math) § The contract of the new employees is for 8h in a row. ► Solve it!!! LP & ILP introduction 13 LP & ILP introduction 14 Example 2: Call Center Example 2: Call Center ► Need of additional employeesInterval for ofinterval times of time: ► Describe the problem by words. § What is the problem? Period 1 2 3 4 5 6 § What is the decision we have to take? Can we make quantitative? 0:00 – 4:00 – 8:00 – 12:00 – 16:00 – 20:00 – 4:00 h 8:00 h 12:00 h 16:00 h 20:00 h 24:00 h § What are the constraints on the decisions? Employees § What is the objective or how we are going to evaluate the needed 9 8 3 7 5 4 solution? ► Which is the minimal number of employees that should be hired? LP & ILP introduction 15 LP & ILP introduction 16 Example 2: Call Center Ejemplo 1: Personal laboratorio ► Define the decision variables: ► Define the constraints § Number of employees to be hired to start at the beginning of § In period 1, there is the need of 9 employees… period 1 (x1) § In period 2, there is the need of 8 employees… § Number of employees to be hired to start at the beginning of § In period 3, there is the need of 3 employees… period 2 (x2) Periods § … 1 2 3 4 5 6 § Number of employees to be hired to start at the beginning of 0:00-4:00 4:00-8:00 8:00-12:00 12:00-16:00 16:00-20:00 20:00-24:00 period 3 (x3) 0:00 X1 X1 4:00 X2 X2 § Number of employees to be hired to start at the beginning of 8:00 X3 X3 period 4 (x4) 12:00 X4 X4 16:00 X5 X5 § … 20:00 X6 X6 Personnel ≥ ≥ ≥ ≥ ≥ ≥ 9 8 3 7 5 4 LP & ILP introduction 17 LP & ILP introduction 18 © Helena Ramalhinho-Lourenço, (2019) 3 Example 2: Call Center Example 2: Call Center ► The constraints… ► Define the constraints: § The number of persons that start at beginning of period 1 + the § X6 + X1 ≥ 9 ones that continue from period 6 must be greater or equal to 9: § X1 + X2 ≥ 8 * X6 + X1 ≥ 9 § X2 + X3 ≥ 3 § The number of persons that start at beginning of period 2 + the § X3 + X4 ≥ 7 ones that continue from period 1 must be greater or equal to 8. § X4 + X5 ≥ 5 * X1 + X2 ≥ 8 § X5 + X6 ≥ 4 § … § Xj ≥ 0, j= 1,...,6. LP & ILP introduction 19 LP & ILP introduction 20 Example 2: Call Center Example 2: Call Center ► Define the objective function ► The complete model ► Min Z= X1 + X2 + X3 + X4 + X5 + X6 ► Min Z= X1 + X2 + X3 + X4 + X5 + X6 § Subject to: * X6 + X1 ≥ 9 * X1 + X2 ≥ 8 * X2 + X3 ≥ 3 * X3 + X4 ≥ 7 * X4 + X5 ≥ 5 * X5 + X6 ≥ 4 * Xj ≥ 0, j= 1,...,6. LP & ILP introduction 21 LP & ILP introduction 22 Solution Methods Solution Methods ► Obtain the optimal solution of the LP Model ► LINEAR PROGRAMMING SOFTWARE SURVEY ► Exact Method § https://www.informs.org/ORMS-Today/Public-Articles/June- Volume-44-Number-3/Linear-Programming-SoFtware-Survey § The PL models are “easy” to be solved § June 2017 § Simplex Method (Dantzig 1947) * https://www.inForms.org/ORMS-Today/OR-MS-Today-SoFtware- * Primal simplex Surveys/Linear-Programming-SoFtware-Survey * Dual simplex § Vendors § Interior Point Method (Karmarkar 1984) * GLPK (GNU Linear Programming Kit), Gurobi OptimiZer, IBM ILOG § ALGORITMS / SOFTWARE CPLEX OptimiZation Studio, LINDO & LINGO, Premium Solver Pro, * Excel Solver etc * Commercial Software § Prices * Open Software * Free to $4000/licence LP & ILP introduction 23 LP & ILP introduction 24 © Helena Ramalhinho-Lourenço, (2019) 4 Simplex Method Simplex Method ► Dantzig (1947) ► In Euclidean space, an object is convex if for every pair of § The set of solutions is a convex set. points within the object, every point on the straight § If there is a optimal solution, there exist an optimal solution in a line segment that joins them is also within the object. corner point (or extreme point) § An extreme point always have at least two adJacent extreme points. § If a extreme point has no adJacent extreme points with better value for the obJective function, then it is the optimal solution. LP & ILP introduction 25 LP & ILP introduction 26 Simplex Method Simplex Method ► Basic theorem ► Example of Linear Programming Maximizar Z =60x +40x ► If exist a optimal solution there exist one in a extreme 1 2 sa point. + x1 x2 £8 ► Simplex method 2x2 £12 § Look only in the extreme point (or corner points of the polytope) 4x1+2x2 £22 § There are a finite numBer of extreme points x1³0, x2 ³0 LP & ILP introduction 27 LP & ILP introduction 28 Only five extreme Infinite number of points feasible solutions LP & ILP introduction 29 LP & ILP introduction 30 © Helena Ramalhinho-Lourenço, (2019) 5 Simplex Method Initialization Find initial feasible extreme-point solution Look for the optimal ► An iterative procedure solution in the 5 § Move from extreme extreme/corner point to extreme point Is the current Yes points feasible extreme- Stop point solution § Optimality test in the the actual optimal? solution is Simplex Method: optimal * If a extreme-corner No solution has no adjacent solutions that Move to a better are better, then it must adjacent feasible extreme-point be an optimal solution solution LP & ILP introduction 31 LP & ILP introduction 32 step 1. Solution (0,0) step 2. Adjacent with z=0. solution(0,6) with z=240. LP & ILP introduction 33 LP & ILP introduction 34 Step 3. Adjacent solution(2,6) with Step 4.