Operation Management Online quiz

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Optimization and Simulation Modeling

Chapter 9

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Learning Objectives

Formulate and solve linear programming problems.

Describe the use of computer simulation modeling in operations decision making.

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Linear Programming Helps Kellogg’s Optimize Production, Inventory, and Distribution

Kellogg’s must manage a highly complex production, inventory control, and distribution system.
Kellogg’s employs an enterprise resource planning (ERP) system to coordinate its raw material purchases, production, distribution, and demand.
The many different varieties of products and brands, packaged in many different sizes and produced at several different plants, require the use of an optimization approach known as linear programming.
The innovative use of optimization techniques has allowed the company to develop a system that is estimated to save between $35 million and $40 million annually.

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Discussion Starter

What benefits can we receive from simulations and modeling?

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Operations Research
or Management Science

Operations research or management science: the use of interdisciplinary scientific methods such as mathematical modeling, statistics, and algorithms that aid decision making for complex real-world problems of coordination and execution of the operations in an organization
The goal is to derive the best possible solution to a problem or to optimize the performance of the organization.

Source: © Image Source/Corbis

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Introduction

Many management decisions involve making the most effective use of limited resources
Linear programming (LP)

Widely used mathematical modeling technique

Planning and decision making relative to resource allocation

Broader field of mathematical programming

Here programming refers to modeling and solving a problem mathematically

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Requirements of a
Linear Programming Problem

Four properties in common

Seek to maximize or minimize some quantity (the objective function)

Restrictions or constraints are present

Alternative courses of action are available

Linear equations or inequalities

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LP Properties and Assumptions

TABLE 7.1

PROPERTIES OF LINEAR PROGRAMS

1. One objective function

2. One or more constraints

3. Alternative courses of action

4. Objective function and constraints are linear – proportionality and divisibility

5. Certainty

6. Divisibility

7. Nonnegative variables

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Formulating LP Problems

Developing a mathematical model to represent the managerial problem
Steps in formulating a LP problem

Completely understand the managerial problem being faced

Identify the objective and the constraints

Define the decision variables

Use the decision variables to write mathematical expressions for the objective function and the constraints

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Formulating LP Problems

Common LP application – product mix problem
Two or more products are produced using limited resources
Maximize profit based on the profit contribution per unit of each product
Determine how many units of each product to produce

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Flair Furniture Company

Flair Furniture produces inexpensive tables and chairs
Processes are similar, both require carpentry work and painting and varnishing

Each table takes 4 hours of carpentry and 2 hours of painting and varnishing

Each chair requires 3 of carpentry and 1 hour of painting and varnishing

There are 240 hours of carpentry time available and 100 hours of painting and varnishing

Each table yields a profit of $70 and each chair a profit of $50

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Flair Furniture Company

The company wants to determine the best combination of tables and chairs to produce to reach the maximum profit

TABLE 7.2

HOURS REQUIRED TO PRODUCE 1 UNIT
DEPARTMENT	(T) TABLES	(C) CHAIRS	AVAILABLE HOURS THIS WEEK
Carpentry	4	3	240
Painting and varnishing	2	1	100
Profit per unit	$70	$50

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Flair Furniture Company

The objective is

Maximize profit

The constraints are

The hours of carpentry time used cannot exceed 240 hours per week

The hours of painting and varnishing time used cannot exceed 100 hours per week

The decision variables are

T = number of tables to be produced per week

C = number of chairs to be produced per week

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Flair Furniture Company

Create objective function in terms of T and C

Maximize profit = $70T + $50C

Develop mathematical relationships for the two constraints

For carpentry, total time used is

(4 hours per table)(Number of tables produced)
+ (3 hours per chair)(Number of chairs produced)

First constraint is

Carpentry time used ≤ Carpentry time available

4T + 3C ≤ 240 (hours of carpentry time)

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Flair Furniture Company

Similarly

Painting and varnishing time used
≤ Painting and varnishing time available

2 T + 1C ≤ 100 (hours of painting and varnishing time)

This means that each table produced requires two hours of painting and varnishing time

Both of these constraints restrict production capacity and affect total profit

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Flair Furniture Company

The values for T and C must be nonnegative

T ≥ 0 (number of tables produced is greater than or equal to 0)

C ≥ 0 (number of chairs produced is greater than or equal to 0)

The complete problem stated mathematically

Maximize profit = $70T + $50C

subject to

4T + 3C ≤ 240 (carpentry constraint)

2T + 1C ≤ 100 (painting and varnishing constraint)

T, C ≥ 0 (nonnegativity constraint)

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Graphical Solution to an
LP Problem

Easiest way to solve a small LP problems is graphically
Only works when there are just two decision variables

Not possible to plot a solution for more than two variables

Provides valuable insight into how other approaches work
Nonnegativity constraints mean that we are always working in the first (or northeast) quadrant of a graph

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Graphical Representation
of Constraints

This Axis Represents the Constraint T ≥ 0

This Axis Represents the Constraint C ≥ 0

FIGURE 7.1 – Quadrant Containing All Positive Values

100 –

–

80 –

–

60 –

–

40 –

–

20 –

–

| | | | | | | | | | | |

0 20 40 60 80 100

Number of Chairs

Number of Tables

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Graphical Representation
of Constraints

The first step is to identify a set or region of feasible solutions
Plot each constraint equation on a graph
Graph the equality portion of the constraint equations

4T + 3C = 240

Solve for the axis intercepts and draw the line

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Graphical Representation
of Constraints

When Flair produces no tables, the carpentry constraint is:

4(0) + 3C = 240

3C = 240

C = 80

Similarly for no chairs:

4T + 3(0) = 240

4T = 240

T = 60

This line is shown on the following graph

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Graphical Representation
of Constraints

(T = 0, C = 80)

FIGURE 7.2 – Graph of Carpentry Constraint Equation

(T = 60, C = 0)

100 –

–

80 –

–

60 –

–

40 –

–

20 –

–

| | | | | | | | | | | |

0 20 40 60 80 100

Number of Chairs

Number of Tables

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FIGURE 7.3 – Region that Satisfies the Carpentry Constraint

Graphical Representation
of Constraints

Any point on or below the constraint plot will not violate the restriction
Any point above the plot will violate the restriction

100 –

–

80 –

–

60 –

–

40 –

–

20 –

–

| | | | | | | | | | | |

0 20 40 60 80 100

Number of Chairs

Number of Tables

(30, 40)

(30, 20)

(70, 40)

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Graphical Representation
of Constraints

The point (30, 40) lies on the line and exactly satisfies the constraint

4(30) + 3(40) = 240

The point (30, 20) lies below the line and satisfies the constraint

4(30) + 3(20) = 180

The point (70, 40) lies above the line and does not satisfy the constraint

4(70) + 3(40) = 400

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Graphical Representation
of Constraints

(T = 0, C = 100)

FIGURE 7.4 – Region that Satisfies the Painting and Varnishing Constraint

(T = 50, C = 0)

100 –

–

80 –

–

60 –

–

40 –

–

20 –

–

| | | | | | | | | | | |

0 20 40 60 80 100

Number of Chairs

Number of Tables

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Graphical Representation
of Constraints

To produce tables and chairs, both departments must be used
Find a solution that satisfies both constraints simultaneously
A new graph shows both constraint plots
The feasible region is where all constraints are satisfied

Any point inside this region is a feasible solution

Any point outside the region is an infeasible solution

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Graphical Representation
of Constraints

FIGURE 7.5 – Feasible Solution Region

Painting/Varnishing Constraint

Carpentry Constraint

Feasible Region

100 –

–

80 –

–

60 –

–

40 –

–

20 –

–

| | | | | | | | | | | |

0 20 40 60 80 100

Number of Chairs

Number of Tables

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Graphical Representation
of Constraints

For the point (30, 20)

For the point (70, 40)

Carpentry constraint	4T + 3C ≤ 240 hours available (4)(30) + (3)(20) = 180 hours used
Painting constraint	2T + 1C ≤ 100 hours available (2)(30) + (1)(20) = 80 hours used

Carpentry constraint	4T + 3C ≤ 240 hours available (4)(70) + (3)(40) = 400 hours used
Painting constraint	2T + 1C ≤ 100 hours available (2)(70) + (1)(40) = 180 hours used

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

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Graphical Representation
of Constraints

For the point (50, 5)

Carpentry constraint	4T + 3C ≤ 240 hours available (4)(50) + (3)(5) = 215 hours used
Painting constraint	2T + 1C ≤ 100 hours available (2)(50) + (1)(5) = 105 hours used

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

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Linear Programming

Optimization: arriving at a maximum or minimum point of a mathematical function
Constraints: the necessary conditions that must be met when a mathematical function is being optimized
Linear programming: a special formulation of an optimization problem in which all equations and inequalities are linear

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Formulation of a
Linear Programming Problem

Five main components of a linear programming

problem:

Objective function: A mathematical formulation of the criterion by which all decisions should be evaluated.

Decision variables: The parameters that can be changed by the decision makers to achieve a higher or lower value of the objective function.

Constraints: The necessary conditions that must be met when a mathematical function is being optimized

Linearity: When formulating linear programming problems, all mathematical equations or inequalities are represented as straight lines.

Non-negativity: Each decision variable within a linear programming formulation is assumed to take only nonnegative values, although this is not essential.

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Solution of a Linear Programming Problem by a Graphical Method

A simple linear programming problem can be solved by using a graphical method.
Because all the equations within a linear programming problem are either straight lines or inequalities, the constraints are first plotted to find the region that satisfies all conditions.
Once that region is identified, the researcher evaluates the objective function at each corner of the feasible region.

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Mount Sinai Hospital: Scheduling Operating Rooms by Integer Linear Programming

Mount Sinai Hospital in Toronto, Canada, has 14 operating rooms and five departments using the OR.
To address the problem of scheduling operating rooms effectively, Mount Sinai Hospital now uses a constrained-optimization model know as integer linear programming.
Since implementing this approach, the hospital has seen a reduction in the number of conflicts in scheduling operating room times and saves $20,000 annually.
In the end, a better schedule means more effective care for Mount Sinai Hospital’s patients.

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PLATO Helps Athens Win Gold During 2004 Summer Olympic Games

During the 2004 Olympic Games in Athens, over the course of 16 days, more than 2,000 athletes participated in 300 events in 28 different sports across 36 venues located across the city.
The events were watched by 3.6 million spectators in the stadiums, 22,000 journalists, and 2,500 members of the international committees.

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PLATO Wins Gold!

The organizing team for the Athens Olympics developed PLATO, the Process Logistics Advanced Technical Optimization approach.

The PLATO project:

Developed business process models for the various venues.
Developed computer simulation models that enabled managers to conduct a variety of what-if analyses.
Developed software that guided the Olympic Committee personnel in using the business process and simulation models.

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VOLCANO Saves $187 Million for UPS

UPS carries more than 13 million packages to more than 8 million customers in more than 200 countries daily.
UPS, along with a team of researchers from MIT, developed and implemented Volume, Location and Aircraft Network Optimization (VOLCANO), an optimization-based planning system that is transforming the business process within UPS.
The VOLCANO system is an interactive transportation modeling and optimization approach.
Prior to VOLCANO, it used to take planners up to nine months to develop a single transportation plan for UPS airline operations manually.
VOLCANO is expected to save more than $189 million for UPS within the next decade.

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Simulation Modeling

Computer simulation models are used in decision making because testing proposed new operating procedures in an actual operation is expensive, complicated, and risky.
Computer simulation models allow the user to try out different strategies without actually implementing them in practice.
Simulation models allow managers to evaluate multiple operations designs and perform what-if types of analyses.
Simulation replaces the wasteful and unreliable practice of testing managers’ ideas through trial-and-error methods.

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Numerical Simulation

Numerical Simulation: simulating outcomes that are controlled by chance, but where the state of the system at specific times is not of interest
Numerical simulations can typically be performed in computer-spreadsheet software.

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Discrete-Event Simulation

Discrete-event simulation: a type of simulation that is applicable when the state of a system over time is the major concern; the term discrete event describes the nature of such systems, where the system changes at discrete times when particular events occur
Three major uses:

Validating other models

Process design

Management decision-making games

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Building Simulation Models

Spreadsheet-Based Models
Simulation-Modeling Tools

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Advantages and Disadvantages
of Simulation-Modeling

Advantages:

The ability to create complex discrete-event models without programming knowledge

The speed at which models can be created by experienced modelers

Disadvantages:

A slow speed of execution

The cost of the software

can be in the tens of thousands of dollars per copy

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More Disadvantages?

Simulation, like an quantitative-decision aid, is dependent on accurate data.
Because simulation models are relatively easy to create, simulation is often overused.

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Computer Simulation
of Check Process Operations

One of the largest commercial banks in the U.S. asked one of the authors of this text to develop a simulation model of the upgrade and redesign plan for its check processing operations at its central check processing facility in Chicago.
The objective of the project: Was it worthwhile to spend more than $1 million for new equipment for check process operations?
It was essential that the check processing operation complete its daily work in a timely manner so that customer accounts could be posted and online balance information updated for branch operations.

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Discussion Starter

What can we use process flowcharts for in companies?

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Figure 9.10: Check
Processing Workflow Schematic

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Computer Simulation of
Check Processing Operations

The reject repair and balancing process is used to manually process any check that the automatic sorter is not able to read.
Animated simulation models were developed to understand the old and new reject repair processes.

A model is only an abstraction of reality. Therefore, models should include all essential and relevant elements of the real system and leave the nonessential elements out.

Simulation can be used to model the effects of such new technology.

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Figure 9.11: Old Check Reject
Repair Process Simulation Model

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Figure 9.12: New Check Reject
Repair Process Simulation Model

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Tables 9.9 and 9.10: Summarized Simulation Results for Current and New Process