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Supplementary Chapter A Nonlinear and NonSmooth Optimization Modeling and Solving Nonlinear Optimization Problems The objective and constraint functions for nonlinear optimization models not have a linear structure like linear and integer optimization models ◦ Building models relies on fundamental modeling principles, business logic, functional relationships, and data-fitting techniques Nonlinear models are generally more difficult to solve ◦ Solver provides efficient solution procedures A Nonlinear Pricing Decision Model Sales = -2.9485 × price + 3240.9 Total revenue = (price)(sales) = price × (-2.9485 × price + 3240.9) = 22.9485 × price2 + 3240.9 × price Identify the price that maximizes total revenue, subject to any constraints that might exist Example A.1: Solving the Pricing Decision Model Spreadsheet and Solver model Premium Solver solution algorithm Example A.2: A Hotel Pricing Model The Marquis Hotel is considering a major remodeling effort and needs to determine the best combination of rates and room sizes to maximize revenues while keeping the number of rooms at or below the current capacity of 450 Example A.2 Continued Projected number of rooms sold for a given room type = For example, using a standard room: [250 – 1.5(S – 85)(250)]/85 = 625 – 4.41176(S) where S = new Standard room price Example 16.2 Continued Model Example A.2 Continued Spreadsheet model Example A.2 Continued Solver model Interpreting Solver Reports for Nonlinear Optimization Models Solver provides Answer, Sensitivity, and Limits reports for nonlinear optimization models However, the Sensitivity report is quite different from that for linear models Spreadsheet Models with Non-Smooth Excel Functions Alternate spreadsheet model for K&L Designs example with fixed costs (Chapter 15) Pi ≥ Ij ≥ In this way, there is no need for the binary variables and the additional constraints that involve them, which are more difficult to logically understand and model Example A.9: Using Evolutionary Solver for the K&L Design Fixed-Cost Problem The Evolutionary Solver algorithm requires that all variables have simple upper and lower bounds to restrict the search space to a manageable region Thus, we set upper bounds of 600 (the total demand) and lower bounds of for each of them Example A.10: A Rectilinear Location Model Edwards Manufacturing is studying where to locate a tool bin on the factory floor X,Y coordinates of the production areas and demand for tools are shown below Distances to the tool bin are rectilinear (parallel to the coordinate system) Example A.10 Continued Optimization model: For Evolutionary Solver, set bounds to restrict the search space X≥0 Y≥0 X≤5 Y≤5 Example A.10 Continued Spreadsheet and Solver model Optimization Models for Sequencing and Scheduling Job-sequencing problems involve finding an optimal sequence by which to process jobs Lateness (Li) is the difference between completion time (Ci) and due date (Di): Li = Ci – Di (A.10) Tardiness (Ti) is the amount of time by which completion time exceeds due date: Ti = max {0, Li} (A.11) Sequencing Rules Shortest processing time (SPT) sequencing of jobs minimizes the average completion time for all jobs Earliest due date (EDD) sequencing of jobs minimizes the maximum number of tardy jobs Other criteria such as average tardiness, total tardiness, or total lateness are also of interest Evolutionary Solver can be used for such problems Example A.11: Finding Optimal Job Sequences A custom manufacturing company has 10 jobs waiting to be processed Processing times and due dates are shown below A sequence of integers for the job ordering is called a permutation The objective is to find the permutation that optimizes the chosen criteria Example A.11 Continued Spreadsheet model Example A.11 Continued Solver model Minimize total tardiness The Traveling Salesperson Problem (TSP) A salesperson needs to visit n different cities and return home in the minimum total distance A tour is a route that visits each city once and returns to the start Applications include FedEx, UPS, and soft drink vendors that deliver goods to customers With n customers or cities, there are (n−1)! tours If n = 14, more than billion tours are possible Example A.12: Touring American Baseball League Cities A baseball fan living in Detroit wants to visit 14 ballparks of American League teams Example A.12 Continued Number the cities from to 13 City will be the starting/ending point and any city can be assigned this position The 13 decision variables are the city to visit next (from cities to 12) City 13 is assigned to return to city Use the alldifferent constraint for 13 decision variables so that each city is visited only once Example A.12 Continued Spreadsheet formulas Example A.12 Continued Solver model and solution ... that the minimum variance increases to 0.020, a 66.67% increase Parameter Analysis Using spreadsheet models and Solver, it is easy to systematically vary a parameter of a model and investigate... ◦ Recall from algebra that a quadratic function is f(x) = ax2 + bx + c ◦ In other words, a quadratic function has only constant, linear, and squared terms Quadratic optimization models can be... Solver again from the current solution to try to find a better solution Quadratic Optimization A quadratic optimization model is one that has a quadratic objective and all linear constraints
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