1. 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

2. 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

3. 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

4. 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

5. 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

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

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

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

9. Multiple Feasible Areas
Multiple Areas of Feasible Solutions

10. 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

11. 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)

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

13. Formulation of NLP Models

14. 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).

15. Solving NLP Models with Solver
Step2: Apply Solver

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

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

18. 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

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

20. Investment Portfolio Problem Formulation

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

22. Solving the Portfolio Problem

23. 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

24. 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

25. Wrap up