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**Unformatted text preview: **1 Introduction to
Linear Programming
Chapter 3 © L.Vanhaverbeke 2 Introduction •Linear programming •Programming means planning
•Model contains linear mathematical functions •An application of linear programming
•Allocating limited resources among competing activities in the best possible way
Applies to wide variety of situations • © L.Vanhaverbeke 20 pc/batch 3 4 Site A Site B Site C alu wood glass +
assembly 1 hour — — 2 hours 3 hours 2 hours 5 Site A Site B Site C alu wood glass +
assembly 4 hours 12 hours 18 hours
profit 1 hour — — 2 hours 3 hours 2 hours 3.000€ 5.000€ 6 3.1 Prototype Example •Wyndor Glass Co. •Produces windows and glass doors
•Plant 1 makes aluminum frames and hardware
Plant 2 makes wood frames
Plant 3 produces glass and assembles
products •
• © L.Vanhaverbeke 7 Prototype Example •Company introducing two new products •Product 1: 8 ft. glass door with aluminum frame
•Product 2: 4 x 6 ft. double-hung, wood-framed
window •Problem: What mix of products would be
most profitable? •Assuming company could sell as much of either
product as could be produced •Products produced in batches of 20
© L.Vanhaverbeke 8 OR/Analytics Cycle Grossman, Mehrotra, and Sidaoui: A Student-Centered Approach to the Business School Management Science Course 47 INFORMS Transactions on Education 16(2), pp. 42–53, © 2016 INFORMS Figure 1 The Analysis Roadmap (Business Analysis Lifecycle) REAL WORLD Downloaded from informs.org by [134.184.69.192] on 20 July 2016, at 01:44 . For personal use only, all rights reserved. Business
situation “COMMUNICATE” MODEL WORLD
“FORMULATE”
Create a model Construct highquality
spreadsheet Recommend,
influence,
persuade,
decide, and
act Managerial
insights Mathematical
model Perform
analysis
on model
Make sense of
and explain
model insights
“INTERPRET” “ENGINEER” Spreadsheet
model “ANALYZE” Model
insights uses his understanding of the business situation to
that ultimately affect the business situation. This
Grossman, Mehrotra, and Sidaou (2016) A Student-Centered Approach to the Business
formulate a mathematical model that captures the School
quan-Management
requires
leadership
and Transactions
managerial
skills to
craft
a
Science
Course, INFORMS
on Education
16(2),
pp. 42–53
titative relationships inherent in the business situamessage that can be understood and will be acted
tion. The act of formulation is the bridge between
upon by people in the organization.
the Real World and the Model World. It is here that
This roadmap establishes several key ideas imme© L.Vanhaverbeke
the analyst makes assumptions about what to include
diately, including:
and exclude from the model and which data will be
• The time and effort we ask students to expend is
necessary.
for the ultimate purpose of improved business underThe analyst must then engineer a spreadsheet model
standing, insight, and action
to implement the mathematical model as a com• Any analytical shortcuts (or cool technical tricks)
puter program. Because the only programming lanexist in service to the business outcome, and are
guage available to business students is the spreadnot valuable in and of themselves. (It is important
sheet, all models are implemented in spreadsheets. As
for research-active management science instructors to
discussed later in the paper, careful attention needs
remember this!)
to be paid to the nuances of spreadsheet design and
• The importance of separating formulation of a
programming.
model from engineering a spreadsheet. For small
The analyst next will analyze the spreadsheet model
problems, especially illustrative textbook examples,
using one or more structured techniques or algoone can write a spreadsheet model directly with9 rithms to generate model insights that summarize the
out a separate formulation step. For larger problems,
analytical results. Model insights are framed in the
this approach breaks down and one must consider
Model World and therefore are numeric, exact, and
the conceptual representation of the business situuse technical terminology.
ation (formulation) separate from the programming
Next, the analyst needs to interpret the model
(spreadsheet engineering).
insights and articulate them as business insights that
• The importance of having in hand a wellsummarize the meaning of the analytical results. Busiengineered spreadsheet model as a prerequisite to
ness insights are framed in the Real World and thereconducting analysis.
fore are verbal, approximate (the model is only an
We keep a copy of the roadmap in a separate file and
approximation to reality), and use business terminolroutinely display it throughout the course, especially
ogy. Interpretation is the bridge between the Model
before embarking on a technical topic that students
World back into the Real World. The analyst must
find challenging. In our experience, this is a very helprevisit any elements of the business situation that
ful mechanism for helping students to understand
were assumed away during formulation.
when it is appropriate to raise a business question,
Finally the analyst communicates the business
which resides in “real world,” and when it is approinsights to influence real-world decisions and actions
priate to raise a technical question, which resides in Prototype Example •Data needed •Number of hours of production time available per week in each plant for new
products
Production time used in each plant for each
batch of each new product
Profit per batch of each new product •
• © L.Vanhaverbeke 10 Prototype Example © L.Vanhaverbeke 11 Prototype Example © L.Vanhaverbeke 12 Prototype Example © L.Vanhaverbeke 13 Prototype Example •Problem can be solved graphically •Two dimensional graph with x and x as the axes
•First step: identify values of x and x permitted by
1 1 2 2 the restrictions
Next step: pick a point in the feasible region that
maximizes value of Z • © L.Vanhaverbeke 14 Prototype Example © L.Vanhaverbeke 15 Prototype Example © L.Vanhaverbeke 16 Prototype Example © L.Vanhaverbeke 17 3.2 The Linear Programming
Model •General problem terminology and examples •Resources: money, particular types of machines, vehicles, or personnel
Activities: investing in particular projects,
advertising in particular media, or shipping from a
particular source • •Problem involves choosing levels of activities to
maximize overall measure of performance © L.Vanhaverbeke 18 The Linear Programming Model © L.Vanhaverbeke 19 The Linear Programming Model •Standard form © L.Vanhaverbeke 20 The Linear Programming Model •Other legitimate forms •Minimizing (rather than maximizing) the objective function
Functional constraints with greater-than-orequal-to inequality
Some functional constraints in equation form
Some decision variables may be negative •
•
• © L.Vanhaverbeke 21 The Linear Programming Model •Feasible solution •Solution for which all constraints are satisfied
•Might not exist for a given problem •Infeasible solution •Solution for which at least one constraint is
violated © L.Vanhaverbeke 22 © L.Vanhaverbeke 23 The Linear Programming Model •Feasible solution •Solution for which all constraints are satisfied
•Might not exist for a given problem •Infeasible solution •Solution for which at least one constraint is • violated Optimal solution •Has most favorable value of objective function
•Might not exist for a given problem
© L.Vanhaverbeke 24 © L.Vanhaverbeke 25 © L.Vanhaverbeke 26 The Linear Programming Model •Corner-point feasible (CPF) solution •Solution that lies at the corner of the feasible
region © L.Vanhaverbeke 27 © L.Vanhaverbeke 28 The Linear Programming Model •Corner-point feasible (CPF) solution •Solution that lies at the corner of the feasible • region Linear programming problem with feasible
solution and bounded feasible region •Must have CPF solutions and optimal solution(s)
Best CPF solution must be an optimal solution
© L.Vanhaverbeke 29 © L.Vanhaverbeke 30 3.3 Assumptions of Linear
Programming •Proportionality assumption •The contribution of each activity to the value of the objective function (or lefthand side of a functional constraint) is
proportional to the level of the activity
If assumption does not hold, one must
use nonlinear programming (Chapter 13) • © L.Vanhaverbeke 31 Assumptions of Linear Programming •Additivity •Every function in a linear programming model is
the sum of the individual contributions of the
activities •Divisibility •Decision variables in a linear programming
model may have any values • Including noninteger values •Assumes activities can be run at fractional values
© L.Vanhaverbeke 32 Assumptions of Linear
Programming •Certainty •Value assigned to each parameter of a linear programming model is assumed to
be a known constant
Seldom satisfied precisely in real
applications • •Sensitivity analysis used © L.Vanhaverbeke 33 3.4 Additional Examples •Example 1: Design of radiation therapy for Mary’s cancer treatment •Goal: select best combination of beams
and their intensities to generate best
possible dose distribution •Dose is measured in kilorads © L.Vanhaverbeke 34 OR/Analytics Cycle Grossman, Mehrotra, and Sidaoui: A Student-Centered Approach to the Business School Management Science Course
INFORMS Transactions on Education 16(2), pp. 42–53, © 2016 INFORMS Figure 1 REAL WORLD
Business
situation Downloaded from informs.org by [134.184.69.192] on 20 July 2016, at 01:44 . For personal use only, all rights reserved. 47 The Analysis Roadmap (Business Analysis Lifecycle) “COMMUNICATE” MODEL WORLD
“FORMULATE”
Create a model Construct highquality
spreadsheet Recommend,
influence,
persuade,
decide, and
act Managerial
insights Mathematical
model Perform
analysis
on model
Make sense of
and explain
model insights
“INTERPRET” “ENGINEER” Spreadsheet
model “ANALYZE” Model
insights uses his understanding of the business situation to
that ultimately affect the business situation. This
Grossman, Mehrotra, and Sidaou (2016) A Student-Centered Approach to the Business
formulate a mathematical model that captures the School
quan-Management
requires
leadership
and Transactions
managerial
skills to
craft
a
Science
Course, INFORMS
on Education
16(2),
pp. 42–53
titative relationships inherent in the business situamessage that can be understood and will be acted
tion. The act of formulation is the bridge between
upon by people in the organization.
the Real World and the Model World. It is here that
This roadmap establishes several key ideas imme© L.Vanhaverbeke
the analyst makes assumptions about what to include
diately, including:
and exclude from the model and which data will be
• The time and effort we ask students to expend is
necessary.
for the ultimate purpose of improved business underThe analyst must then engineer a spreadsheet model
standing, insight, and action
to implement the mathematical model as a com• Any analytical shortcuts (or cool technical tricks)
puter program. Because the only programming lanexist in service to the business outcome, and are
guage available to business students is the spreadnot valuable in and of themselves. (It is important
sheet, all models are implemented in spreadsheets. As
for research-active management science instructors to
discussed later in the paper, careful attention needs
remember this!)
to be paid to the nuances of spreadsheet design and
• The importance of separating formulation of a
programming.
model from engineering a spreadsheet. For small
The analyst next will analyze the spreadsheet model
problems, especially illustrative textbook examples,
using one or more structured techniques or algoone can write a spreadsheet model directly with35rithms to generate model insights that summarize the
out a separate formulation step. For larger problems,
analytical results. Model insights are framed in the
this approach breaks down and one must consider
Model World and therefore are numeric, exact, and
the conceptual representation of the business situuse technical terminology.
ation (formulation) separate from the programming
Next, the analyst needs to interpret the model
(spreadsheet engineering).
insights and articulate them as business insights that
• The importance of having in hand a wellsummarize the meaning of the analytical results. Busiengineered spreadsheet model as a prerequisite to
ness insights are framed in the Real World and thereconducting analysis.
fore are verbal, approximate (the model is only an
We keep a copy of the roadmap in a separate file and
approximation to reality), and use business terminolroutinely display it throughout the course, especially
ogy. Interpretation is the bridge between the Model
before embarking on a technical topic that students
World back into the Real World. The analyst must
find challenging. In our experience, this is a very helprevisit any elements of the business situation that
ful mechanism for helping students to understand
were assumed away during formulation.
when it is appropriate to raise a business question,
Finally the analyst communicates the business
which resides in “real world,” and when it is approinsights to influence real-world decisions and actions
priate to raise a technical question, which resides in Example 1: Radiation Therapy
Design © L.Vanhaverbeke 36 © L.Vanhaverbeke 37 Example 1: Radiation Therapy
Design •Linear programming model
•Using data from Table 3.7 © L.Vanhaverbeke 38 Example 1: Radiation Therapy Design •A type of cost-benefit tradeoff
problem © L.Vanhaverbeke 39 Example 2: Reclaiming Solid
Wastes •SAVE-IT company collects and treats four
types of solid waste materials •Materials amalgamated into salable products
•Three different grades of product possible
•Fixed treatment cost covered by grants
•Objective: maximize the net weekly profit
•Determine amount of each product grade
•Determine mix of materials to be used for each
grade © L.Vanhaverbeke 40 Example 2: Reclaiming Solid
Wastes © L.Vanhaverbeke 41 Example 2: Reclaiming Solid
Wastes © L.Vanhaverbeke 42 Example 2: Reclaiming Solid
Wastes • © L.Vanhaverbeke 43 © L.Vanhaverbeke 44 3.5 Formulating and Solving
Linear Programming Models on
a Spreadsheet •Excel and its Solver add-in •Popular tools for solving small linear
programming problems © L.Vanhaverbeke 45 Formulating and Solving Linear
Programming Models on a
Spreadsheet •The Wyndor example •Data entered into a spreadsheet © L.Vanhaverbeke 46 Formulating and Solving Linear
Programming Models on a Spreadsheet •Changing cells •Cells containing the decisions to be made
C12 and D12 in the Wyndor example
below • © L.Vanhaverbeke 47 Formulating and Solving Linear
Programming Models on a © L.Vanhaverbeke 48 Formulating and Solving Linear
Programming Models on a Spreadsheet © L.Vanhaverbeke 49 3.6 Formulating Very Large
Linear Programming Models •Actual linear programming models
•Can have hundreds or thousands of functional constraints
Number of decision variables may also
be very large • •Modeling language •Used to formulate very large models in
practice
Expedites model management tasks • © L.Vanhaverbeke 50 Formulating Very Large Linear
Programming Models •Modeling language examples •AMPL, MPL, OPL, GAMS, and LINGO •Solvers •CPLEX, Gurobi, … © L.Vanhaverbeke 51 3.7 Conclusions •Linear programming technique
applications •Resource-allocation problems
•Cost-benefit tradeoff problems
•Blending problems •Not all problems can be formulated to
fit a linear programming model •Alternatives: integer programming or
nonlinear programming models © L.Vanhaverbeke ...

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- Spring '08
- Macharis
- Linear Programming, Optimization, analyst