Business Analytics with Nonlinear Programming Chapter 4 Business Analytics with Nonlinear Programming Business Analytics with Management Science Models and Methods Beni Asllani University of Tennessee at Chattanooga

Chapter Outline Chapter Objectives Prescriptive Analytics in Action Introduction: Challenges to NLP Models Local optimum versus global optimum The solution is not always found at an extreme point Multiple feasible areas Example1: World Class furniture Example2: Optimizing an Investment Portfolio Exploring Big Data with Nonlinear Programming Wrap up

Chapter Objectives Explain what are nonlinear programming models and why they are more difficult to solve than LP models Demonstrate how to formulate nonlinear programming models Show how to use Solver to reach solution for nonlinear programming models Explain how to read the answer report and how to perform sensitivity analysis for nonlinear programming models Offer practical recommendations when using nonlinear models in the era of big data

Prescriptive Analytics in Action Flood risk of Netherlands Recommendation to increase protection standards tenfold Expensive cost To determine economically efficient flood protection standards for all dike ring areas Minimize the overall investment cost Ensure protection Maintain fresh water supplies Use of nonlinear programming model Able to find optimal standard levels for each of 53 disk ring area Only three of them need to be changed Allow the government to effectively identify strategies and establish standards with a lower cost

Introduction Linear Programming Nonlinear Programming (NLP) The objective function and constraints are linear equations Both proportional and additive Nonlinear Programming (NLP) To deal with not proportional or additive business relationships Same structure: objective function and a set of constraints More challenging to solve Necessity of using NLP Difficult to use But more accurate than linear programming

Challenge to NLP Models Represented with curved lines or curved surface Complicated to represent relationship with a large number of decision variables

Local Optimum versus Global Optimum Area of Feasible Solution with Local and Global Maximum

The Solution Is Not Always Found at an Extreme Point Possible Optimal Solution For NLP Model

Multiple Feasible Areas Multiple Areas of Feasible Solutions

Challenge to NLP Models Three Challenges NLP is Facing: Local Optimum versus Global Optimum The Solution Is Not Always Found at and Extreme Point Multiple Feasible Areas Solutions developed to deal with the challenges Advanced heuristics such as genetic algorithms Simulated annealing Generalized reduced gradient (GRG) method Quadratic programming Barrier methods However, these algorithms often are not successful

Example1: World Class Furniture Stores five different furniture categories Economic Order Quantity (EOQ) model Allow to optimally calculate the amount of inventory with the goal of minimizing the total inventory cost Does not consider: Storage capacity (200,000 cubic feet) purchasing budget ($1.5 million)

Formulation of NLP Models Define decision variables Formulate the objective function Holding cost: Ordering cost: Total cost:

Formulation of NLP Models  

Solving NLP Models with Solver Step 1: Create an Excel Template Figure 4-4 shows the what-if template for the WCF inventory problem. The economic order quantity (Raw 15) calculates the optimal order quantity, the amount in each order that minimizes the overall inventory cost. The following classic EOQ formula [41] can be used in cells B15, C15, D15, E15, and F15: 𝐸𝑂𝑄=√(2𝑜𝐷/ℎ) If there are no storage or budget constraints, the amounts shown in cells B15, C15, D15, E15, and F15, respectively 335, 642, 526, 526, and 194, would be the optimal order amounts. However, due to operational constraints, these values may not be feasible. The inventory manager must seek the best possible feasible solutions and store these values in cells B16:F16. The initial inventory amount shown in Figure 4-4 correspond to a trial solution of ordering 10 units per order for each product as indicated in cells B16 through F16. The template also calculates the Average Inventory, Average Number of Orders per Week, Total Supply Available, Maximum Cubic Foot Storage Required, Ordering Cost per Week, Holding Cost per Week, Inventory Operating Cost per Week, and Total Inventory Value for each product. The specific formulas used to calculate each of these values for the last product (Bookcases in column F) are shown in the respective adjacent cells (G17:G24). Finally, the following measures are calculated: actual capacity usage (H20), ordering cost per week (H21), holding cost per week (H22), value of the objective function (H23), and total inventory value (H24) which includes purchasing, ordering, and holding costs. The formulas for these cells are shown in the adjacent cells (I20:I24).

Solving NLP Models with Solver Step2: Apply Solver

Solving NLP Models with Solver Step3: Interpret Solver Solution Objective Cell Variable Cells Constraints

Sensitivity Analysis for NLP Models Reduced Cost Reduced gradient Shadow Price Lagrange multiplier Valid only at the point of the optimal solution

Example2: Optimizing an Investment Portfolio Trade-off between return on investment and risk is an important aspect in financial planning Smart Investment Services (SIS) designs annuities, IRAs, 401(k) plans and other products of investment Prepare a portfolio involving a mix of eight mutual funds

Investment Portfolio Problem Formulation Define decision variables 2. Formulate objective function

Investment Portfolio Problem Formulation  

Solving the Portfolio Problem The what-if template for this investment problem

Solving the Portfolio Problem

Exploring Big data with NLP Volume The availability of more data allows organization to explore, formulate and solve previously unsolvable problem Variety and Velocity Offer significant challenges for optimization models Advanced software programs Used to navigate trillions of permutations, variables and constraints Such as Solver

Wrap up The NLP formulation shares the same with LP model GRG algorithm is best suited for NLP models A risk that the algorithm will result in a local optimum Provide a good starting point in the trial template Add a non-negativity constraint for decision variables Pay close attention when selecting Solver parameters

Wrap up