Business analytics methods, models and decisions evans analytics2e ppt 15

 An integer linear optimization model integer program has some or all variables restricted to being whole numbers..  A mixed-integer linear optimization model has only a subset of va

Chapter 15

Integer Optimization

 An integer linear optimization model (integer

program) has some or all variables restricted to being

whole numbers.

 A mixed-integer linear optimization model has only a

subset of variables restricted to being integer while

others are continuous.

 A special type of integer problem is one in which

variables can be only 0 or 1; these are used to model logical yes-or-no decisions.

Integer Optimization

 Decision variables that we force to be integers are called

general integer variables.

 Algorithms for integer optimization models first solve the

LP relaxation (no integer restrictions imposed) and

gradually enforce integer restrictions using systematic searches.

 Solver has a default integer tolerance of 0.05 so it will

stop if it finds an integer solution within 5% of the optimal solution.

◦ To find the guaranteed optimal integer solution, Integer Tolerance must

be set to 0 Click the Options button in the Solver Parameters dialog and ensure that the value of Integer Optimality (%) is 0.

Solving Models with General Integer

Variables

 The optimal solution

 To enforce integer restrictions on variables using Solver, click on

Integers under the Constraints list and then click the Add button In the Add Constraint dialog, enter the variable range in the Cell

Reference field and choose int from the drop-down box.

Example 15.1 Continued

The maximum value of the objective function for the model with integer restrictions is smaller than the linear optimization solution Whenever you add

a constraint to a model, the value of the objective function can never improve and usually worsens.

 Because integer models are discontinuous by their very nature, sensitivity information cannot be generated in the same manner as for linear models; therefore, no

Sensitivity report is provided by Solver— only the

Answer report is available

 To investigate changes in model parameters, it is

necessary to re-solve the model.

Sensitivity Analysis for Integer

Optimization

 A company makes 110-inch wide rolls of thin sheet metal and slices them in smaller rolls of 12, 15, and 30 inches

 A cutting pattern is a configuration of the number of smaller rolls of each type that are cut from the raw stock Six different cutting

patterns are used

 Demands for the coming week are 500 12-inch rolls, 715 15-inch rolls, and 630 30-inch rolls

 The problem is to develop a model that will determine how many 110-inch rolls to cut into each of the six patterns to meet demand and minimize scrap

Example 15.2: A Cutting-Stock Problem

 Model development

◦ X i = number of 110-inch

rolls to cut using pattern i

 X i needs to be a whole

number (general integer

variable) because each

roll that is cut generates a

different number of end

Spreadsheet Implementation and Solver

Model for the Cutting-Stock Model

 Workforce scheduling is a practical, yet highly complex, problem in many businesses such as food service,

hospitals, and airlines.

 Typically a huge number of possible schedules exist and customer demand varies by day of week and time of day, further complicating the problem of assigning workers to time slots.

Workforce-Scheduling Models

 Brewer Services schedules customer

service workers on Monday to Friday, 8

a.m to 5 p.m

 Staffing requirements are shown in the

table

 5 permanent employees work all day

 Part-time employees work 4-hour shifts

 What is the minimum number of part-time

employees needed to meet staffing

requirements?

Example 15.3: Brewer Services

 Model development

◦ X i = integer number of part-time workers that start on the i th 4-hour shift

(i = 1 for 8 a.m., …, i = 6 for 1 p.m.)

◦ Constraints: For each hour, we need to ensure that the total number of

part-time employees who work that hour is at least as large as the

minimum requirements.

Example 15.3 Continued

Net staff requirements for each hour

Spreadsheet Implementation for Brewer Services

Solver Model for Brewer Services

 The solution has 9 excess employees at

8 a.m., which may not be desirable.

 Most scheduling problems have multiple

optimal solutions.

 An easy way to find an alternative

optimal solution is to add an additional

constraint with the objective function

equal to the optimal solution value:

and then set the new objective function

to minimize the number of excess

employees at 8 a.m

Example 15.4: Finding Alternative Optimal

Solutions for Brewer Services Model

 Modified Solver model and Alternative Optimal Solution

Example 15.4 Continued

However, now we have 9 excess employees at noon.

 A better approach would be to

define additional constraints to

restrict the excess number of

employees in the range E21:E29

to be less than or equal to some

maximum number k and then

attempt to minimize the original

objective function

◦ The smallest value of k that results in

a feasible solution is k = 3.

Example 15.4 Continued

 A binary variable x is a general integer variable that is

restricted to being between 0 and 1:

◦ We usually just write this as x = 0 or 1.

 Binary variables are used to model logical decisions.

Integer Optimization Models with

Binary Variables

 Project selection model.

 Five potential projects are being considered Each

project is expected to generate a return (given by the net present value) but requires a fixed amount of cash and personnel.

 The available budget and human resources do not allow selection of all 5 projects.

 The objective is to maximize expected return.

Example 15.5: Hahn Engineering

 Personnel limitation: 5X 1 + 3X 2 + 2X 3 + 5X 4 + 3X 5 ≤ 12

Example 15.5 Continued

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 Spreadsheet implementation and Solver model

Example 15.5 Continued

 Sensitivity analysis for integer optimization requires solving the model for changes in the data.

re- The heat map below shows the impact of various levels

of available resources (cash and personnel) on the

expected return

◦ The upper-left-hand corner of each colored region (shown boxed

in the figure)represents the lowest amount of resources required

to achieve that return

Sensitivity Analysis for Hahn

Engineering

 Examples

◦ If project 1 is selected, then project 4 must also be selected

◦ Excel IF functions, while logical, cannot be used in linear

optimization

◦ Model this as X 4 ≥ X 1

◦ If X 1 = 1, then X 4 is forced to be 1

 Other types of logical constraints:

Using Binary Variables to Model Logical Constraints

Suppose project managers want to ensure that:

 If project 1 is selected, then project 4 is also selected: X 4 − X 1 ≥ 0

 At most one of projects 1 and 3 can be selected: X 1 + X 3 ≤ 1

 Define a cell for each constraint function (cells B17 and B18)

Example 15.6: Adding Logical Constraints into the Project-Selection Model

 Location model: find a subset of locations that cover a service area.

 Anderson Fire Department serves 7 districts

 The village wants to determine where to locate fire stations so that response times to residents in all 7 districts is no more than 8

minutes

 Response times:

Example 15.7: Anderson Village Fire

Department

 Model development

 Define X j as binary

X j = 1 if a fire station is located in district j

X j = 0 if a fire station is not located in district j

 Minimize the number of fire stations:

 Each district must be reachable within 8 minutes by some fire station:

Example 15.7 Continued

 Spreadsheet implementation and Solver model

◦ Construct a matrix by converting all response times that are within 8 minutes to 1s and those that exceed 8 minutes to 0s Then the constraint functions for each district are simply the SUMPRODUCT of the decision variables and the rows of this matrix

Example 15.7 Continued

 The village’s board of trustees wants to better

understand the trade-offs between response time and required number of fire stations.

◦ Vary response times from 5 to 10 minutes to see the effect on the number of fire stations required

◦ Analytic Solver Parametric Analysis

Example 15.8: Parameter Analysis for

Response Times

A response time

of 5 minutes requires 3 fire stations.

A response time

of 9-10 minutes requires 1 fire station.

 A company has numerous potential locations for

distribution centers to ship products to customers.

 A single-sourcing policy specifies that each customer can only be supplied from one distribution center.

 The problem is to determine which customers to assign

to each distribution center

A Customer-Assignment Model for

Supply Chain Optimization

 Cij = total cost of satisfying

the demand of customer j

from distribution center i

 k = number of distribution

centers to be selected

Customer-Assignment Optimization Model

 Distributes restaurant supplies in 5 major cities.

have been identified.

configuration that minimizes sourcing costs.

Example 15.9: Paul & Giovanni Foods

 Spreadsheet and Solver model

Example 15.9 Continued

 Many practical applications of optimization involve a combination of continuous variables and binary variables

 A common example is a plant location and distribution model in

which a company must decide which plant to build (binary variables) and then how to best ship the product from the plant to the

distribution centers (continuous variables)

◦ E.g.: With increased demand that exceeds capacity at Marietta and

Minneapolis, GAC is considering adding a new plant in either Fayetteville

or Chico

Mixed-Integer Optimization models

 Define a binary variable for the decision of which plant to

build: Y1 = 1 if the Fayetteville plant is built and Y2 = 1 if the Chico plant is built.

 Define normal variables Xij, representing the amount

shipped from plant i to distribution center j.

 Objective function:

Example 15.10: A Mixed-Integer Plant

Location Model

 Capacity constraints

◦ Capacity constraints for the Marietta and Minneapolis plants

remain as before However, for Fayetteville and Chico, we can allow shipping from those locations only if a plant is built there

◦ Note that if the binary variable is zero, then the right-hand side of the constraint is zero, forcing all shipment variables to be zero also If, however, a particular Y-variable is 1, then shipping up to the plant capacity is allowed

Example 15.10 Continued

 Demand constraints

 Build exactly one plant

 Nonnegativity for Xij

Example 15.10 Continued

Spreadsheet Model

Solver Model

 In the plant location model, why not use IF functions

instead of these constraints?

 From a spreadsheet perspective, there is nothing wrong with this

◦ However, from a linear optimization perspective, the use of an IF function no longer preserves the linearity of the model

(technically, the model would be called nonsmooth) and we would

get an error message in trying to solve the model

◦ Similarly, using the constraint X 31 Y 1 + X 32 Y 1 +X 33 Y 1 + X 34 Y 1 ≤ 1500

is logically correct, but multiplying the two variables together

results in a nonlinear function

Binary Variables, IF Functions, and

Nonlinearities in Model Formulation

 Many business problems involve fixed costs; they are either incurred in full or not at all.

 Binary variables can be used to model such problems in

a similar fashion as we did for the plant location model.

Fixed-Cost Models

 K&L Designs model was developed in Chapter 14:

 Suppose that the company must rent some equipment, which costs

$65 for 3 months The equipment can be rented or returned each quarter, so if nothing is produced in a quarter, it makes no sense to incur the rental cost

Example 15.11: Incorporating Fixed

Costs into the K&L Designs Model

 Model development

Yi = 1 if production occurs during season i

Yi = 0 if not

 Objective function

Example 15.11 Continued

 Material balance constraints

 We must ensure that whenever a production variable, P, is positive, the corresponding Y variable is equal to 1; conversely, if the Y

variable is 0 (you don’t rent the equipment), then the corresponding production variable must also be 0

 Production variables are nonnegative, and Y variables are binary

Example 15.11 Continued

Spreadsheet Implementation

Solver Model

 This model does not preclude feasible solutions in which a production variable is 0 while its

corresponding Y-variable is 1 E.g., P A could be

zero even if Y A = 1,

◦ This implies that we incur the fixed cost even though

no production is incurred during that time period

 Although such a solution is feasible, it can never be optimal, because a lower cost could be obtained by

setting the Y-variable to 0 without affecting the

value of the production variable, and the solution algorithm will always ensure this

 Therefore, it is not necessary to explicitly try to

incorporate this in the model

Modeling Issue