Chapter 14 applications of linear optimization

? Solver requires some technical knowledge of linear optimization concepts and terminology, such as reduced costs and shadow prices giá mờ-see note.. ? Reduced costs: how much the unit

Chapter 14 Applications of Linear Optimization

Lecturer: Nguyen Van Dung Ph.D.

Slides are based on slides accompanied the book “Business Analytics: Methods, Models, and Decisions”, with improvement from the lecturer

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

🞂Simple Bounds (giới hạn)

◦ Constraints on values of a single variable

◦ Ensure the flow of material or money is accounted for at

locations or between time periods: input = output

Types of Constraints in 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 (vải) on a make-to-order basis.

🞂 The key decision is which type of loom (khung cửi dệt

vải) to use for each fabric (vải) 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

🞂 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

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

🞂 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 (giá mờ-see note)

🞂 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 (gia tăng, tăng thêm)

🞂 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

(hạt dùng cho chim ăn) 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

🞂 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 X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 = 1

◦ Protein constraint simplifies to

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

Example 14.3 Continued

Formulate other nutritional

constraints in a similar way.

🞂 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

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 optimization

🞂 Such portfolio investment models problems have

the basic characteristics of blending models.

Portfolio Investment Models

🞂 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 X i = 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

🞂 Innis Investments

◦ Allowable Increase and Allowable Decrease values for the

weighted return are very small, 0.00906 and 0.00111,

respectively; so any changes in the target return will require solving the model

re-Example 14.5: Risk versus Reward

🞂 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

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

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 (sự thoái hoá) 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

🞂 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

🞂 Người quản lý tài chính của công ty cần đảm bảo rằng có đủ tiền để thanh toán chi phí nhưng vẫn cần tối đa hóa thu nhập

từ đầu tư.

🞂 Ba khoản đầu tư ngắn hạn đang được xem xét như sau:

🞂 Chứng chỉ tiền gửi 1 tháng trả 0,25%

🞂 Chứng chỉ tiền gửi 3 tháng thanh toán 1,00% khi đáo hạn

🞂 Chứng chỉ tiền gửi 6 tháng thanh toán 2,30% khi đáo hạn

🞂 Chi phí ròng trong 6 tháng tới được dự đoán là $50.000,

🞂 Phát triển mô hình

◦ Ai = số tiền ($) để đầu tư vào CC tiền gửi 1 tháng vào đầu tháng i

◦ Bi = số tiền ($) để đầu tư vào CC tiền gửi 3 tháng vào đầu tháng i

◦ Ci = số tiền ($) để đầu tư vào CC tiền gửi 6 tháng vào đầu tháng i

Ví dụ 14.11 tiếp theo

🞂 Mô hình tối ưu hoá

Ví dụ 14.11 tiếp theo

Bảng tính - mô hình của D.A Branch

& Sons

Bảng tính - các công thức của

mô hình D.A Branch & Sons

Mô hình Solver của D.A Branch & Sons

🞂 Solver có cách xử lí các giới hạn dưới đơn giản (VD:

C>=500) và giới hạn trên (VD: D<=1000) khác so với các ràng buộc thường trong báo cáo độ nhạy.

🞂 Giới hạn dưới và giới hạn trên cũng tương tự như loại

ràng buộc không tiêu cực, không xuất hiện rõ ràng dưới dạng các ràng buộc trong mô hình.

🞂 Điều này khiến cho việc giải thích thông tin về độ nhạy

trở nên khó, vì ta không còn giá bóng và mức tăng giảm cho phép được liên kết với các ràng buộc

Mô hình với Biến Ràng Buộc

🞂 J&M Manufacturing sản xuất 4 mẫu bếp nướng gas.

🞂 Xác định số lượng bếp nướng cần sản xuất để tối đa

hóa lợi nhuận.

Ví dụ 14.12: J&M Manufacturing

(300-🞂 Ràng buộc bao gồm sự hạn chế về số giờ sản xuất của

mỗi bộ phận, yêu cầu về doanh số tối thiểu, doanh số tối

đa giới hạn tiềm năng.

◦ Cẩn thận khi xem xét các dimensions.

Ví dụ 14.12 tiếp theo

🞂 Ràng buộc:

◦ Khả năng của bộ phận:

◦ Yêu cầu doanh số tối thiểu và doanh số tối đa tiềm năng

Ví dụ 14.12 tiếp theo

Triển khai bảng tính cho J&M Manufacturing

Bảng tính Công thức mô hình của J&M Manufacturing

Mô hình Solver của J&M

Manufacturing

Báo cáo trả lời của J&M Manufacturing

Báo cáo độ nhạy của J&M

Manufacturing

Chú ý rằng không có giới hạn ràng buộc nào xuất hiện trong mục Constraints.

🞂 Đối với sản phẩm B, giới hạn ràng buộc dưới là B ≥ 0 Cần có lợi nhuận bao nhiêu hơn nữa trên sản phẩm B để có lợi nhuận kinh tế trong việc sản xuất số lượng sản phẩm nhiều hơn số lượng tối

thiểu yêu cầu?"

◦ Đáp án đưa ra bởi: Chi phí giảm Lợi nhuận đơn vị trên sản phẩm B phải giảm ít nhất là -1,905 đô la (tức tăng ít nhất là 1,905 đô la) để có thể sản xuất số lượng sản phẩm nhiều hơn số lượng tối thiểu yêu cầu với lợi nhuận kinh tế.

🞂 Sản phẩm D ở giới hạn trên của nó.

◦ Chi phí giảm 19,29 đô la cho biết lợi nhuận đơn vị phải giảm bao nhiêu trước khi không còn kinh tế để sản xuất số lượng tối đa.

Hiểu biết về Độ giảm chi phí