Business analytics methods, models and decisions evans analytics2e ppt SC a

Example A.1: Solving the Pricing Decision Model  Spreadsheet and Solver model Premium Solver solution algorithm...  Solver provides Answer, Sensitivity, and Limits reports for nonlinea

Supplementary Chapter A Nonlinear and Non-

Smooth Optimization

 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.

Modeling and Solving

Nonlinear Optimization Problems

Example A.1: Solving the Pricing

Decision Model

 Spreadsheet and Solver model

Premium Solver solution algorithm

 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: A Hotel Pricing Model

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

 Model

Example 16.2 Continued

 Spreadsheet model

Example A.2 Continued

 Solver model

Example A.2 Continued

 Solver provides Answer, Sensitivity, and Limits

reports for nonlinear optimization models

 However, the Sensitivity report is quite different from that for linear models

Interpreting Solver Reports for

Nonlinear Optimization Models

 Answer Report

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

 Sensitivity Report

Example A.3 Continued

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).

 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:

Locating Central Facilities

Straight line (Euclidean) distance is the hypotenuse of the triangle.

Rectilinear distance is the sum of the left and bottom sides of the triangle.

 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: Finding the Best Location for a Medical Laboratory

 Distance between each hospital and the new

facility is assumed to be straight line

 Define (X i , Y i ) as the coordinates of hospital i

(X c ,Y c ) as the coordinates of the laboratory

Example A.4 Continued

 Spreadsheet model

Example A.4 Continued

The Solver model minimizes the total distance in cell C20 by

changing the decision variables in cells B23 and C23.

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

Example A.4 Continued

 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.

The Economic Order-Quantity (EOQ)

Model

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 Assumptions

 Q = order quantity

 D = annual demand

 C = unit cost of the item

 C 0 = cost per order placed

 i = inventory carrying charge per unit

EOQ Model Development

 Annual demand = 15,000 units.

 Ordering costs = $200 per order.

 Purchase cost = $22 per item.

 Carrying charge rate = 20%.

Example A.5: Solving the EOQ Model

 Spreadsheet and Solver model

Example A.5 Continued

 Spreadsheet model formulas

Example A.5 Continued

 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

The When-to-Order Decision

 In the previous example, suppose lead time is

 If we place an order when the inventory level

reaches 288, then the order will arrive when the

stock level falls to zero

Reorder Point Example

 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

Using Empirical Data for Nonlinear

Optimization Modeling

 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: A Model for Advertising

Strategy

Example A.6 Continued

Using Trendlines, logarithmic functions fit the data:

 Optimization model

Example A.6 Continued

 Spreadsheet and Solver model

Example A.6 Continued

 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.

Practical Issues Using Solver for

Nonlinear 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)

= ax 2 + 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.

Quadratic Optimization

 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

The Markowitz Portfolio Model

 x i = fraction of the portfolio invested in stock i

 The risk of a portfolio is weighted sum of the variances and covariances

Model Development

 An investor is considering 3 stock investments.

Example A.7: An Example of the Markowitz Model

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.

 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

Parameter Analysis

 Solver Parameter Analysis Results

Example A.8: Analyzing Risk versus Reward

 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%:

Example A.8 Continued

 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

Evolutionary Solver for

Non-Smooth Optimization

 Alternate spreadsheet model for K&L Designs

example with fixed costs (Chapter 15)

Spreadsheet Models with Non-Smooth Excel Functions

P i ≥ 0

I j ≥ 0

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.

 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.9: Using Evolutionary Solver for the K&L Design Fixed-Cost Problem

 Edwards Manufacturing is studying where to locate a tool bin on the factory floor.

for tools are shown below.

 Distances to the tool bin are rectilinear (parallel to the coordinate system).

Example A.10: A Rectilinear Location

Model

 Spreadsheet and Solver model

Example A.10 Continued

 Job-sequencing problems involve finding an

optimal sequence by which to process jobs

 Lateness (L i) is the difference between completion

time (C i ) and due date (D i):

L i = C i – D i (A.10)

 Tardiness (T i) is the amount of time by which

completion time exceeds due date:

T i = max {0, L i} (A.11)

Optimization Models for Sequencing and Scheduling

 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

Sequencing Rules

 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: Finding Optimal Job

Sequences

 Spreadsheet model

Example A.11 Continued

 Solver model

Example A.11 Continued

Minimize total tardiness

 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

The Traveling Salesperson Problem

(TSP)

 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

Example A.12 Continued