Supplementary Chapter A Nonlinear and Non-Smooth Optimization

Modeling and Solving Nonlinear Optimization Problems The objective and constraint functions for nonlinear optimization models do 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 For example, using a standard room: 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.

Example A.3: Interpreting Solver Reports for the Hotel Pricing Model Answer Report

Example A.3 Continued Sensitivity Report Lagrange multipliers are similar to shadow prices but are only approximations. If the number of rooms increased by 1, the approximate revenue increase would be $12.08 (the actual increase found by re-solving is $11.87).

Locating Central Facilities A common problem in designing service systems is to locate a facility in a “central” location with respect to other facilities to minimize some measure of distance from the central location to each of the other facilities. Distance measures: Straight line (Euclidean) distance is the hypotenuse of the triangle. Rectilinear distance is the sum of the left and bottom sides of the triangle.

Example A.4: Finding the Best Location for a Medical Laboratory A medical testing laboratory collects blood samples from 5 local hospitals. Managers want to determine the best location for a new testing facility, taking into consideration both distance and number of trips per month.

Example A.4 Continued Distance between each hospital and the new facility is assumed to be straight line Define (Xi , Yi) as the coordinates of hospital i (Xc ,Yc) as the coordinates of the laboratory

Example A.4 Continued Spreadsheet model The Solver model minimizes the total distance in cell C20 by changing the decision variables in cells B23 and C23.

Example A.4 Continued Bubble chart of hospital and optimal laboratory (star) locations. Size of bubbles correspond to the number of trips/month.

The Economic Order-Quantity (EOQ) Model The EOQ model is used to optimize inventories of retail goods such as groceries and commodities that have stable demand over time. Inventory costs: Purchase costs—unit costs per item to purchase from suppliers Order preparation costs—costs involve the time spent preparing and placing orders, such as clerical, telephone, receiving, and inspection time Inventory-holding cost—all expenses associated with carrying inventory, such as rent on storage space, utilities, insurance, taxes, and the cost of capital Shortage costs—additional costs for shipping, invoicing, and labor for back orders or lost profit opportunities and possible future loss of revenues because of lost sales The economic order quantity is the amount to order that minimizes the total cost of ordering and holding.

EOQ Model Assumptions 1. Only a single inventory item is considered. 2. The entire quantity arrives at one time. 3. The demand for the item is constant over time. 4. No shortages are allowed.

EOQ Model Development Q = order quantity D = annual demand C = unit cost of the item C0 = cost per order placed i = inventory carrying charge per unit

Example A.5: Solving the EOQ Model Annual demand = 15,000 units. Ordering costs = $200 per order. Purchase cost = $22 per item. Carrying charge rate = 20%.

Example A.5 Continued Spreadsheet and Solver model

Example A.5 Continued Spreadsheet model formulas

The When-to-Order Decision The EOQ model does not specify when to order. The reorder point is the inventory level when a new order is placed. Lead time is how long it takes to receive an item once the order is placed. Set the reorder point to be the inventory level that provides enough stock to satisfy all demand during the lead time.

Reorder Point Example In the previous example, suppose lead time is t = 1 week (1/52 = 0.0192 years). D =15,000/year Reorder point = Demand during lead time = Dt = (15,000 units/year)(0.0192 years) = 288 units If we place an order when the inventory level reaches 288, then the order will arrive when the stock level falls to zero.

Using Empirical Data for Nonlinear Optimization Modeling For many applications of nonlinear optimization, the form of the objective or constraint functions are derived from empirical data. We can use line-fitting techniques to establish the functions.

Example A.6: A Model for Advertising Strategy DTP Corporation produces two major products. The total budget for advertising is $500,000. Experimental data has been collected on profits resulting from various advertising expenditures. How should DTP allocate the $500,000 between the two products, assuming that at least $50,000 must be spent on each product?

Example A.6 Continued Using Trendlines, logarithmic functions fit the data:

Example A.6 Continued Optimization model

Example A.6 Continued Spreadsheet and Solver model

Practical Issues Using Solver for Nonlinear Optimization Solver cannot guarantee finding the absolute best solution (global optimal solution) to nonlinear problems. A local optimal solution is one for which all points close by are no better than the solution The message, “Solver has found a solution” indicates at least a local optimum. If you get the message “Solver has converged to the current solution. All constraints are satisfied.” then you should run 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. 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 solved using the Standard LP/Quadratic solving method within Solver.

The Markowitz Portfolio Model The Markowitz portfolio model is a classic quadratic optimization model in finance that seeks to minimize risk of an investment portfolio subject to a constraint on the portfolio’s expected return. Risk is measured using variances and covariances of the individual investments.

Model Development xi = fraction of the portfolio invested in stock i The risk of a portfolio is weighted sum of the variances and covariances

Example A.7: An Example of the Markowitz Model An investor is considering 3 stock investments. Variance-covariance matrix

Spreadsheet Model

Solver Model

Sensitivity Report The Lagrange multiplier predicts that the minimum variance will increase by 63.2% if the target return is increased from 10% to 11%. If you re-solve the model, you will find 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 its impact on the solution. For example, in the Markowitz model, we might be interested in understanding the relationship between the minimum risk and the target return.

Example A.8: Analyzing Risk versus Reward Solver Parameter Analysis Results

Example A.8 Continued Alternative modeling approach – maximize the return subject to a constraint on risk. For example, suppose the investor wants to maximize expected return subject to a risk (variance) no greater than 1%:

Evolutionary Solver for Non-Smooth Optimization Excel functions IF, ABS, MAX, MIN lead to non- smooth models (that violate linearity conditions). Solver’s Standard Evolutionary algorithm can be used to solve non-smooth models using a heuristic solution approach. Heuristics are intelligent rules to search among solutions and find better solutions.

Spreadsheet Models with Non-Smooth Excel Functions Alternate spreadsheet model for K&L Designs example with fixed costs (Chapter 15) 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. Pi ≥ 0 Ij ≥ 0

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 0 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 5 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 6 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 0 to 13. City 0 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 0 to 12). City 13 is assigned to return to city 0. 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