Business analytics methods, models and decisions evans analytics2e ppt 14

 Solver requires some technical knowledge of linear optimization concepts and terminology, such as reduced costs and shadow prices..  Reduced costs: how much the unit production or pur

Chapter 14

Applications of Linear Optimization

Applications of Linear Optimization

 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

Building Linear Optimization Models

 Process selection models generally involve choosing among different types of

processes to produce a good.

◦ Example: make or buy decisions

Process Selection Models

 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: Camm Textiles

 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

 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

Example 14.1 Continued

 Camm Textiles model

 Decision variables

 Objective

Spreadsheet Design

 Answer Report

Example 14.2: Interpreting Solver Reports for the Camm Textiles Problem

 Sensitivity Report

Example 14.2: Interpreting Solver Reports for the Camm Textiles Problem

 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

Solver Output and Data Visualization

 Camm Textiles

Answer 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

Sensitivity Report Visualization

 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

Visualizing Allowable Ranges

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.

 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 Shadow Prices

 Stock Chart

Visualizing Allowable Ranges for Shadow Prices

 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.

Blending Models

 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: BG Seed Company

 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

 Complete model

Example 14.3 Continued

Spreadsheet Implementation of BG Seed Company

Solver Model for BG Seed Company

 Solver solution shows the model is infeasible!

 Solver Feasibility Report

Dealing with Infeasibility

A conflict exists in trying to meet both fat and fiber constraints.

Only sunflower seeds and safflower contain enough fat but they also have a lot of fiber.

 Lower the fat requirement or raise the fiber limitation

What-If Scenarios

1st Scenario:

Fat requirement is lowered from 15% to 14.5% 2nd Scenario:

Fiber limitation is raised from 14% to 14.5%.

Optimal Cost per pound:

$0.148 if fat requirement lowered

$0.152 if fiber limitation raised

 Many types of financial investment problems are modeled and solved using linear

 Innis Investments manages 6 mutual funds A client wants to invest a $500,000 inheritance The objective is to minimize risk.

 Constraints:

 Invest no more than $200,000 in any one fund

 Invest at least $50,000 each in the multinational and balanced funds

 Invest at least 40% combined in the income equity and balanced funds

 Achieve an average return of at least 5%

Example 14.4: Innis Investments

 Model Formulation

 Define Xi = dollar amount invested in fund i

◦ The total risk would be measured by the weighted risk of the portfolio, where the weights are the proportion of the total investment in any fund (Xj/500,000)

Example 14.4 Continued

 Constraints

◦ Invest all money:

◦ Achieve required return:

◦ Have at least 40% in income equity and balanced funds:

◦ At least $50,000 in each of multinational and balanced funds:

◦ Restrict each investment to $200,000, and include nonnegativity:

Example 14.4 Continued

Spreadsheet Implementation for Innis Investments

Solver Model for Innis Investments

 A poorly scaled model is one that computes values of the objective, constraints, or intermediate

results that differ by several orders of magnitude.

 Poor scaling can cause Solver engines to return messages such as “Solver could not find a

feasible solution,” “Solver could not improve the current solution,” or even “The linearity

conditions required by this Solver engine are not satisfied,” or it may return results that are

suboptimal

◦ In the Solver options, you can check the box Use Automatic Scaling

◦ The best way to avoid scaling problems is to carefully choose the “units” implicitly used in your model so that all computed results are within a few orders of magnitude of each other

Scaling Issues in Using Solver

 Little Investment Advisors is working with a client on determining an optimal portfolio of bond

funds The client has $350,000 to invest and wants to achieve the largest weighted percentage return and keep the weighted risk measure no greater than 5.00.

Example 14.6: Little Investment Advisors

 Premium Solver solution without scaling, resulting in an incorrect solution!

Example 14.6 Continued

 Solver solution after scaling the variables

Example 14.6 Continued

 The transportation problem involves determining how much to ship from a set of

sources of supply (factories, warehouses, etc.) to a set of demand locations

(warehouses, customers, etc.) at minimum cost.

Transportation Models

 GAC produces refrigerants at 2 plants and ships to 5 distribution centers.

 Define the decision variables as: Xij = amount shipped from plant i to distribution center j

 The objective is to minimize the total cost of shipping between plants and distribution centers.

◦ minimize 12.60X11 + 14.35X12 + 11.52X13 + 17.58X14 + 9.75X21 + 16.26X22 + 8.11X23 + 17.92X24

Example 14.7: General Appliance Corporation

 Constraints

◦ The amount shipped from each plant cannot exceed its capacity.

◦ Demand at each distribution center is met.

◦ Nonnegativity

Example 14.7 Continued

GAC Spreadsheet Implementation and Solver Model

 Depending on how cells in your spreadsheet model are formatted, the Sensitivity report produced

by Solver may not reflect the accurate values of reduced costs or shadow prices because an

insufficient number of decimal places may be displayed.

 We highly recommend that after you save the Sensitivity report to your workbook, you select the

reduced cost and shadow price ranges and format them to have at least two or three decimal places.

Formatting the Sensitivity Report

Example: GAC Sensitivity Report

Original Sensitivity Report Reformatted Sensitivity Report

 Reduced costs tell how much the unit

shipping cost would have to be reduced to

make it attractive to ship along a route.

 We cannot increase the demand at any

distribution center without creating an

infeasible problem The shadow prices

reflect the cost savings that would occur

for a unit decrease in demand at one of the

distribution centers.

Example 14.8: Interpreting Sensitivity Information for the GAC Model

 The GAC solution exhibits a phenomenon called degeneracy A solution is degenerate if the

right-hand-side value of any constraint has a zero Allowable Increase or Allowable Decrease

◦ Degeneracy can impact the interpretation of sensitivity analysis information For example, reduced costs and shadow prices may not be unique, and you may have to change objective function coefficients beyond their allowable increases or decreases before the optimal solution will change

Degeneracy

 The basic decisions are how much to produce in each time period to meet anticipated demand

over each period

 Although it might seem obvious to simply produce to the anticipated level of sales, it may be

advantageous to produce more than needed in earlier time periods when production costs may

be lower and store the excess production as inventory for use in later time periods, thereby letting lower production costs offset the costs of holding the inventory.

Multiperiod Production Planning Models

 K&L Designs makes hand-painted jewelry boxes.

 Forecasted sales are 150 in autumn, 400 in winter, and 50 in spring

 Unpainted boxes cost $20 and each box takes 2 hours to complete

 The cost of capital is 6% per quarter

 Holding cost per item = 0.06(20) = $1.20/quarter

 Labor rates are $5.50, $7.00, and $6.25 per hour during autumn, winter, and spring, respectively

 Minimize the combined cost of production and inventory holding costs.

Example 14.9: K&L Designs

 Decision variables

◦ Pi = amount to produce in quarter i (1 = autumn; 2 = winter; 3 = spring)

◦ Ii = inventory at the end of quarter i

Example 14.9 Continued

 Constraints

◦ Satisfy demand using production in a quarter and the inventory held from the previous time quarter Any amount

in excess of the demand is held to the next quarter

◦ Therefore, the constraints take the form of inventory balance equations:

Example 14.9 Continued

 Complete model

Example 14.9 Continued

Spreadsheet Implementation for K&L Designs

Solver Model for K&L Designs

 To ensure that demand is satisfied, we can set the cumulative production in each quarter to be at

least as great as the cumulative demand.

◦ This eliminates inventory variables

Example 14.10: An Alternative Optimization Model for K&L Designs

Alternative Spreadsheet Model

Alternative Solver Model

Comparison of Sensitivity Reports

 The company’s financial manager needs to ensure that funds are available to pay expenses yet needs to maximize investment income.

 Three short-term investments are being considered:

 1-month CD paying 0.25%

 3-month CD paying 1.00% at maturity

 6-month CD paying 2.30% at maturity

 The net expenditures for the next 6 months are forecast as $50,000, ($12,000), $23,000,

($20,000), $41,000, and ($13,000)

 A cash balance of $10,000 must be maintained Currently the cash balance is $200,000.

Example 14.11: D.A Branch & Sons

 Model development

◦ Ai = amount ($) to invest in a 1-month CD at the start of month i

◦ Bi = amount ($) to invest in a 3-month CD at the start of month i

◦ Ci = amount ($) to invest in a 6-month CD at the start of month i

Example 14.11 Continued

 Optimization model

Example 14.11 Continued

Spreadsheet Model for D.A Branch & Sons

Spreadsheet Model Formulas for D.A Branch & Sons

Solver Model for D.A Branch & Sons

 Solver handles simple lower bounds (e.g., C ≥ 500) and upper bounds (e.g., D ≤ 1,000) quite

differently from ordinary constraints in the Sensitivity report.

 Lower and upper bounds are treated in a manner similar to nonnegativity constraints, which also

do not appear explicitly as constraints in the model.

 This makes it more difficult to interpret the sensitivity information, because we no longer have the

shadow prices and allowable increases and decreases associated with these constraints.

Models with Bounded Variables

 J&M Manufacturing makes 4 models of gas grills

 Determine how many grills to produce in order to maximize profit.

Example 14.12: J&M Manufacturing

 Model development

◦ Define A, B, C, and D number of units of models A, B, C, and D to produce, respectively

◦ The objective function is to maximize the total net profit:

maximize (250 - 210)A + (300 - 240)B + (400 - 300)C + (650 - 520)D = 40A + 60B + 100C + 130D

 Constraints include limitations on the amount of production hours available in each department,

the minimum sales requirements, and maximum sales potential limits.

◦ Watch the dimensions carefully!

Example 14.12 Continued

 Constraints:

◦ Department capacity

◦ Minimum sales requirements and maximum sales potential

Example 14.12 Continued

Spreadsheet Implementation for J&M Manufacturing

Spreadsheet Model Formulas for J&M Manufacturing

Solver Model for J&M Manufacturing

J&M Manufacturing Answer Report

J&M Manufacturing Sensitivity Report

Note that none of the bound constraints appear in the Constraints section.

 For product B, the lower bound constraint is B ≥ 0 How much more would the profit on B have to be in order for it to

be economical to produce anything other than the minimum amount required?

◦ The answer is given by the reduced cost The unit profit on B would have to be reduced by at least - $1.905 (that is, increased by at least +

$1.905).

 Product D is at its upper bound

◦ The reduced costs of $19.29 tells how much the unit profit have to be lowered before it is no longer economical to produce the maximum amount

Interpreting Reduced Costs

 Increasing the right-hand-side value of the bound constraint, B ≥ 0, by 1 unit will result in a profit reduction of $1.905.

 Increasing the right-hand side of the constraint D ≤ 1,000 by 1 will increase the profit by $19.29

◦ The reduced cost associated with a bounded variable is the same as the shadow price of the bound constraint However, we no longer have the allowable range over which we can change the constraint values.

Interpreting Reduced Costs as Shadow Prices