1. Supplementary Chapter A Nonlinear and Non-Smooth Optimization

2. 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.

3. A Nonlinear Pricing Decision Model
Sales = × price
Total revenue = (price)(sales)
= price × ( × price )
= × price × price
Identify the price that maximizes total revenue, subject to any constraints that might exist.

4. Example A.1: Solving the Pricing Decision Model
Spreadsheet and Solver model
Premium Solver solution algorithm

5. 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.

6. 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 – (S)
where S = new Standard room price

7. Example 16.2 Continued
Model

8. Example A.2 Continued
Spreadsheet model

9. Example A.2 Continued
Solver model

10. 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.

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

12. 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).

13. 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.

14. 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.

15. 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

16. 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.

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

18. 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.

19. 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.

20. 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

21. 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%.

22. Example A.5 Continued
Spreadsheet and Solver model

23. Example A.5 Continued
Spreadsheet model formulas

24. 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.

25. Reorder Point Example
In the previous example, suppose lead time is t = 1 week (1/52 = years).
D =15,000/year
Reorder point = Demand during lead time
= Dt
= (15,000 units/year)( 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.

26. 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.

27. 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?

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

29. Example A.6 Continued
Optimization model

30. Example A.6 Continued
Spreadsheet and Solver model

31. 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.

32. 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.

33. 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.

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

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

36. Spreadsheet Model

37. Solver Model

38. 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.

39. 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.

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

41. 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%:

42. 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.

43. 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

44. 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.

45. 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).

46. Example A.10 Continued Optimization model:
For Evolutionary Solver, set bounds to restrict the search space.
X ≥ 0
Y ≥ 0
X ≤ 5
Y ≤ 5

47. Example A.10 Continued
Spreadsheet and Solver model

48. 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)

49. 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.

50. 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.

51. Example A.11 Continued
Spreadsheet model

52. Example A.11 Continued
Solver model
Minimize total tardiness

53. 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.

54. Example A.12: Touring American Baseball League Cities
A baseball fan living in Detroit wants to visit 14 ballparks of American League teams.

55. 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.

56. Example A.12 Continued
Spreadsheet formulas

57. Example A.12 Continued
Solver model and solution