Business analytics methods, models and decisions evans analytics2e ppt 15

Chapter 15 Integer Optimization Integer Optimization    An integer linear optimization model (integer program) has some or all variables restricted to being whole numbers A mixed-integer linear optimization model has only a subset of variables restricted to being integer while others are continuous A special type of integer problem is one in which variables can be only or 1; these are used to model logical yes-or-no decisions Solving Models with General Integer Variables    Decision variables that we force to be integers are called general integer variables Algorithms for integer optimization models first solve the LP relaxation (no integer restrictions imposed) and gradually enforce integer restrictions using systematic searches Solver has a default integer tolerance of 0.05 so it will stop if it finds an integer solution within 5% of the optimal solution ◦ To find the guaranteed optimal integer solution, Integer Tolerance must be set to Click the Options button in the Solver Parameters dialog and ensure that the value of Integer Optimality (%) is Example 15.1: Sklenka Skis Revisited   The optimal solution was to produce 5.25 pairs of Jordanelle skis and pairs of 10.5 Deercrest skis Feasible integer solutions are shown on the accompanying graph Example 15.1 Continued  To enforce integer restrictions on variables using Solver, click on Integers under the Constraints list and then click the Add button In the Add Constraint dialog, enter the variable range in the Cell Reference field and choose int from the drop-down box The maximum value of the objective function for the model with integer restrictions is smaller than the linear optimization solution Whenever you add a constraint to a model, the value of the objective function can never improve and usually worsens Example 15.1 Continueed   Graphical illustration of optimal integer solution Note that rounding the original solution is not optimal Sensitivity Analysis for Integer Optimization   Because integer models are discontinuous by their very nature, sensitivity information cannot be generated in the same manner as for linear models; therefore, no Sensitivity report is provided by Solver— only the Answer report is available To investigate changes in model parameters, it is necessary to re-solve the model Example 15.2: A Cutting-Stock Problem     A company makes 110-inch wide rolls of thin sheet metal and slices them in smaller rolls of 12, 15, and 30 inches A cutting pattern is a configuration of the number of smaller rolls of each type that are cut from the raw stock Six different cutting patterns are used Demands for the coming week are 500 12-inch rolls, 715 15-inch rolls, and 630 30-inch rolls The problem is to develop a model that will determine how many 110-inch rolls to cut into each of the six patterns to meet demand and minimize scrap Example 15.2 Continued  Model development ◦ Xi = number of 110-inch rolls to cut using pattern i  Xi needs to be a whole number (general integer variable) because each roll that is cut generates a different number of end items ◦ The only constraints are end-item demand, nonnegativity, and integer restrictions Spreadsheet Implementation and Solver Model for the Cutting-Stock Model Example 15.9 Continued  Spreadsheet and Solver model Mixed-Integer Optimization models   Many practical applications of optimization involve a combination of continuous variables and binary variables A common example is a plant location and distribution model in which a company must decide which plant to build (binary variables) and then how to best ship the product from the plant to the distribution centers (continuous variables) ◦ E.g.: With increased demand that exceeds capacity at Marietta and Minneapolis, GAC is considering adding a new plant in either Fayetteville or Chico Example 15.10: A Mixed-Integer Plant Location Model    Define a binary variable for the decision of which plant to build: Y1 = if the Fayetteville plant is built and Y2 = if the Chico plant is built Define normal variables Xij, representing the amount shipped from plant i to distribution center j Objective function: Example 15.10 Continued  Capacity constraints ◦ Capacity constraints for the Marietta and Minneapolis plants remain as before However, for Fayetteville and Chico, we can allow shipping from those locations only if a plant is built there ◦ Note that if the binary variable is zero, then the right-hand side of the constraint is zero, forcing all shipment variables to be zero also If, however, a particular Y-variable is 1, then shipping up to the plant capacity is allowed Example 15.10 Continued  Demand constraints  Build exactly one plant  Nonnegativity for Xij Spreadsheet Model Solver Model Binary Variables, IF Functions, and Nonlinearities in Model Formulation  In the plant location model, why not use IF functions instead of these constraints?  From a spreadsheet perspective, there is nothing wrong with this ◦ However, from a linear optimization perspective, the use of an IF function no longer preserves the linearity of the model (technically, the model would be called nonsmooth) and we would get an error message in trying to solve the model ◦ Similarly, using the constraint X31Y1 + X32Y1 + X33Y1 + X34Y1 ≤ 1500 is logically correct, but multiplying the two variables together results in a nonlinear function Fixed-Cost Models   Many business problems involve fixed costs; they are either incurred in full or not at all Binary variables can be used to model such problems in a similar fashion as we did for the plant location model Example 15.11: Incorporating Fixed Costs into the K&L Designs Model  K&L Designs model was developed in Chapter 14:  Suppose that the company must rent some equipment, which costs $65 for months The equipment can be rented or returned each quarter, so if nothing is produced in a quarter, it makes no sense to incur the rental cost Example 15.11 Continued  Model development ◦ Define binary variables for each season: Yi = if production occurs during season i Yi = if not  Objective function Example 15.11 Continued  Material balance constraints  We must ensure that whenever a production variable, P, is positive, the corresponding Y variable is equal to 1; conversely, if the Y variable is (you don’t rent the equipment), then the corresponding production variable must also be  Production variables are nonnegative, and Y variables are binary Spreadsheet Implementation Solver Model Modeling Issue  This model does not preclude feasible solutions in which a production variable is while its corresponding Y-variable is E.g., PA could be zero even if YA = 1, ◦ This implies that we incur the fixed cost even though no production is incurred during that time period   Although such a solution is feasible, it can never be optimal, because a lower cost could be obtained by setting the Y-variable to without affecting the value of the production variable, and the solution algorithm will always ensure this Therefore, it is not necessary to explicitly try to incorporate this in the model ... complex, problem in many businesses such as food service, hospitals, and airlines Typically a huge number of possible schedules exist and customer demand varies by day of week and time of day, further... from the raw stock Six different cutting patterns are used Demands for the coming week are 500 12-inch rolls, 715 15-inch rolls, and 630 30-inch rolls The problem is to develop a model that will... constraints are end-item demand, nonnegativity, and integer restrictions Spreadsheet Implementation and Solver Model for the Cutting-Stock Model Workforce-Scheduling Models   Workforce scheduling