Business analytics methods, models and decisions evans analytics2e ppt 14

Chapter 14
Applications of Linear Optimization


Applications of Linear Optimization


Building Linear Optimization Models
 Building optimization models is more of an art than a science.

◦ Learning how to build models requires logical thought facilitated by studying examples and
observing their characteristics.

 Key issues:

◦ Formulation
◦ Spreadsheet implementation
◦ Interpreting results
◦ Scenario and sensitivity analysis
◦ Gaining insight for making good decisions


Types of Constraints in Optimization Models
 Simple Bounds

◦ Constraints on values of a single variable

 Limitations

◦ Allocation of scarce resources

 Requirements

◦ Minimum levels of performance

 Proportional Relationships

◦ Requirements for mixtures or blends of materials or strategies

 Balance Constraints

◦ Ensure the flow of material or money is accounted for at locations or between time periods: input =
output


Process Selection Models
 Process selection models generally involve choosing among different types of
processes to produce a good.

◦ Example: make or buy decisions


Example 14.1: Camm Textiles
 A mill that operates on a 24/7 basis produces three types of fabric on a make-to-order basis.
 The key decision is which type of loom to use for each fabric type over the next 13 weeks.
 The mill has 3 dobbie looms and 15 regular looms.
 If demand cannot be met, the fabric is outsourced.


Example 14.1 Continued
 Model Formulation

 Di = yards fabric i to produce on dobbie loom
 Ri = yards fabric i to produce on regular loom
 Pi = yards fabric i to outsource
 Objective

 Minimize total cost of milling and outsourcing
 Constraints

 Requirements: Total production and outsourcing of each fabric = demand
 Limitations: Production on each type of loom cannot exceed the available production time
 Nonnegativity


Example 14.1 Continued
 Demand constraints
 Production + outsourcing = demand

◦ Fabric 1: D1 + P1 = 45,000
◦ Fabric 2: D2 + R2 + P2 = 76,500
◦ Fabric 3: D3 + R3 + P3 = 10,000


Example 14.1: Continued
 Loom capacity limitation constraints
 First convert yards/hour into hours/yard.
E.g., for fabric 1 on the dobbie loom:
hours/yard = 1/(4.7 yards/hour) = 0.213 hours/yard

 Capacity of the three dobbie looms:

(24 hours/day)(7 days/week)(13 weeks)(3 looms) = 6,552 hours

 Constraint on available production time on dobbie looms:
0.213D1 + 0.192D2 0.227D3 ≤ 6,552

 Constraint for regular looms:
0.192R2 + 0.227R3 ≤ 32,760


Example 14.1 Continued
 Full Model


Spreadsheet Design
 Camm Textiles model

 Decision variables
 Objective


Solver Model

Set cell C14 to zero as a constraint because fabric 1 cannot be produced on a
regular loom. Whenever you restrict a single decision variable to equal a value or set
it as a ≤ or ≥ type of constraint, Solver considers it as a simple bound.


Example 14.2: Interpreting Solver Reports for the Camm Textiles
Problem
 Answer Report

Example 14.2: Interpreting Solver Reports for the Camm Textiles
Problem
 Sensitivity Report


Solver Output and Data Visualization
 Solver requires some technical knowledge of linear optimization concepts and
terminology, such as reduced costs and shadow prices.

 Data visualization can help analysts present optimization results in forms that are more
understandable and can be easily explained to managers and clients in a report or
presentation.


Answer Report Visualization
 Camm Textiles


Sensitivity Report Visualization
 Reduced costs: how much the unit production or purchasing cost must be changed to force the
value of a variable to become positive in the solution.


Visualizing Allowable Ranges
 Unit cost coefficients: use an Excel Stock Chart (see text for details).

◦

A stock chart typically shows the “high-low-close” values of daily stock prices; here we can compute the
maximum-minimum-current values of the unit cost coefficients.

For those lines that have no maximum limit (the
blue dash) such as with Fabric 1 Purchased,
the unit costs can increase to infinity; for those
that have no lower limit (the red triangle) such
as Fabric 1 on Dobbie, the unit costs can
decrease indefinitely.


Visualizing Shadow Prices
 Shadow prices show the impact of changing the right-hand side of a binding constraint. Because the plant operates
on a 24/7 schedule, changes in loom capacity would require in “chunks” (i.e., purchasing an additional loom) rather
than incrementally.

 However, changes in the demand can easily be assessed using the shadow price information.


Visualizing Allowable Ranges for Shadow Prices
 Stock Chart


Blending Models
 Blending problems involve mixing several raw materials that have different
characteristics to make a product that meets certain specifications.

◦ Dietary planning, gasoline and oil refining, coal and fertilizer production, and the production of
many other types of bulk commodities involve blending.

 We typically see proportional constraints in blending models.


Example 14.3: BG Seed Company
 BG Seed Company is developing a new birdseed mix.

◦ Nutritional requirements specify that the mixture contain at least 13% protein, at least 15% fat, and
no more than 14% fiber.

◦ BG’s objective is to determine the minimum cost mixture that meets nutritional requirements.


Example 14.3 Continued
 Formulating the model

 Define Xi = pounds of ingredient i in 1 pound of mix
 Objective function

 minimize 0.22X1 + 0.19X2 + 0.10X3 + 0.10X4 + 0.07X5 + 0.05X6 + 0.26X7 + 0.05X6 + 0.26X7 +
0.11X8


Example 14.3 Continued
 Protein constraint
 Total pounds of protein provided/total pounds of mix ≥ 0.13

◦

(0.169X1 + 0.12X2 + 0.085X3 + 0.154X4 + 0.085X5 + 0.12X6 + 0.18X7 + 0.119X8)/(X1 + X2 + X3 + X4 + X5 +
X6 + X7 + X8) ≥ 0.13

 Add constraint X + X + X + X + X + X + X + X = 1
1
2
3
4
5
6
7
8

◦
◦

Protein constraint simplifies to
0.169X1 + 0.12X2 + 0.085X3 + 0.154X4 + 0.085X5 + 0.12X6 + 0.18X7 + 0.119X8 ≥ 0.13

Formulate other nutritional constraints in a similar
way.


Example 14.3 Continued
 Complete model