1. What is Operations Research (OR) Slides.pdfDeterministic Operations Research, Linear Programming 1 Source: Introduction to Operations Research, 9th edition, Frederick S. Hillier,

Unit 4 – Systems Engineering Tools

Deterministic Operations Research, Linear Programming

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Source: Introduction to Operations Research, 9th edition, Frederick S. Hillier, McGraw-Hill

1. What is Operations Research (OR)

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• It is an application of scientific methods, techniques and tools to the analysis and solution.

• It is a Process

• It assists Decision Makers

• It has a set of Tools

• It is applicable in many Situations

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What is Operations Research?


OperationsThe activities carried out in an organization.

ResearchThe process of observation and testing characterized by the scientific method including situation, problem statement, model construction, validation, experimentation, candidate solutions.

ModelAn abstract representation of reality. Mathematical, physical, narrative, set of rules in computer program.

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It is a Systems ApproachInclude broad implications of decisions for the organization at each stage in analysis. Both quantitative and qualitative factors are considered.

It finds Optimal SolutionA solution to the model that optimizes (maximizes or minimizes) some measure of merit over all feasible solutions.

It involves a TeamA group of individuals bringing various skills and viewpoints to a problem.

It includes many different OR TechniquesA collection of general mathematical models, analytical procedures, and algorithms.


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History of OR

OR started just before World War II in Britain with theestablishment of teams of scientists to study the strategic andtactical problems involved in military operations. The objectivewas to find the most effective utilization of limited militaryresources by the use of quantitative techniques.

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History of OR

Although scientists had (plainly) been involved in the hardwareside of warfare (designing better planes, bombs, tanks, etc)scientific analysis of the operational use of militaryresources had never taken place in a systematic fashionbefore the Second World War. Military personnel were simplynot trained to undertake such analysis.

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History of OR

These early OR workers came from many different disciplines,one group consisted of a physicist, two physiologists, twomathematical physicists and a surveyor. What such peoplebrought to their work were "scientifically trained" minds, usedto querying assumptions, logic, exploring hypotheses, devisingexperiments, collecting data, analysing numbers, etc. Many toowere of high intellectual calibre (at least four wartime ORpersonnel were later to win Nobel prizes when they returned totheir peacetime disciplines).

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History of OR

Following the end of the war, OR took a different course in theUK as opposed to in the USA. In the UK (as mentioned above)many of the distinguished OR workers returned to their originalpeacetime disciplines. As such OR did not spread particularlywell, except for a few isolated industries (iron/steel and coal). Inthe USA OR spread to the universities so that systematictraining in OR began.

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History of OR

You should be clear that the growth of OR since it began (andespecially in the last 30 years) is, to a large extent, the resultof the increasing power and widespread availability ofcomputers. Most (though not all) OR involves carrying out alarge number of numeric calculations. Without computers thiswould simply not be possible.

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History of OR

Manufacturers used operations research to make productsmore efficiently, schedule equipment maintenance, andcontrol inventory and distribution. And success in these areasled to expansion into strategic and financial planning …and into such diverse areas as criminal justice, education,meteorology, and communications.

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Future of OR

A number of major social and economic trends are increasingthe need for operations researchers. In today’s globalmarketplace, enterprizes must compete more effectively fortheir share of profits than ever before. And public and non-profitagencies must compete for ever-scarcer funding dollars.

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Future of OR

This means that all of us must become more productive.Volume must be increased. Consumers’ demands for betterproducts and services must be met. Manufacturing anddistribution must be faster. Products and people must beavailable just in time.

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Objectives of Operations Research

• Improve the quality in decision making.

• Identify the optimum solution.

• Integrating the system.

• Improve the objectivity of analysis.

• Minimize the cost and maximize the profit.

• Improve the productivity.

• Success in competitions and market leadership.

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Applications of Operations Research

• National plans & budgets.

• Defense services operations.

• Government developments & public sector units.

• Industrial establishment & private sector units.

• R & D and engineering division.

• Public works department.

• Business management.

• Agriculture and Irrigation projects.

• Education & Training.

• Transportation and Communication.

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Definitions

• Operations Research (OR) is the study of mathematical models for complex organizational systems.

• Optimization is a branch of OR which uses mathematical techniques such as linear and nonlinear programming to derive values for system variables that will optimize performance.

• OR professionals aim to provide a rational basis for decision making by seeking to understand and structure complex situations and to use this understanding to predict system behavior and improve system performance.

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The Process: Recognize the Situation

• Manufacturing– Planning

– Design

– Scheduling

– Dealing with Defects

– Dealing with Variability

– Dealing with Inventory

– …

Situation

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Example: Internal nursing staff not happy with their schedules; hospital using too many external nurses.

Formulate the Problem

• Define the Objective

• Select measures


Problem Statement

Situation

• Determine variables

• Identify constraints

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Example: Maximize individual nurse preferences subject to demand requirements, or minimize nurse dissatisfaction costs.

Gather Data

• Production volume

• Scheduling

• Fixed cost

• Variable cost

• Time

• Labor

• ….

Data

Situation

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Example: Gather information about nurse profiles, work schedule, pay structure, overhead, demand requirement, supply, etc.

Construct a Model

• Math. Programming Model

• Stochastic Model

• Statistical Model

• Simulation Model

Construct a Model

Model


Problem Statement

Data

Situation

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Example: Define relationships between individual nurse assignments and preference violations; define tradeoffs between the use of internal and external nursing resources.

Definitions

• Model: A schematic description of a system, theory, or phenomenon that accounts for its known or inferred properties and may be used for further study of its characteristics.

• System: A functionally related group of elements, for example:– The human body regarded as a functional physiological unit.

– An organism as a whole, especially with regard to its vital processes or functions.

– A group of interacting mechanical or electrical components.

– A network of structures and channels, as for communication, travel, or distribution.

– A network of related computer software, hardware, and data transmission devices.

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Models of Operations Research

Models of OR

By structure

Physical model

Analogue model

Mathematical model

By nature of environment

Deterministic Model

Probabilistic Model

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Models

• Linear Programming– Typically, a single objective function, representing either a profit to be maximized

or a cost to be minimized, and a set of constraints that circumscribe the decision variables. The objective function and constraints all are linear functions of the decision variables.

– Software has been developed that is capable of solving problems containing millions of variables and tens of thousands of constraints.

• Network Flow Programming– A special case of the more general linear program. Includes such problems as the

transportation problem, the assignment problem, the shortest path problem, the maximum flow problem, and the minimum cost flow problem.

– Very efficient algorithms exist which are many times more efficient than linear programming in the utilization of computer time and space resources.

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Models

• Integer Programming– Some of the variables are required to take on discrete values.

• Nonlinear Programming– The objective and/or any constraint is nonlinear.

– In general, much more difficult to solve than linear.

– Most (if not all) real world applications require a nonlinear model. In order to make the problems tractable, we often approximate using linear functions.

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Find a Solution

• Linear Programming

• Nonlinear Programming

• Regression

• Direct Search

• Stochastic Optimization

• Trial and Error

Construct a Model

Model


Problem Statement

Data

Situation

Solution

Find a Solution

Tools

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Example: Apply algorithm; post-process results to get monthly schedules.

Mathematical Programming

• A mathematical model consists of:– Decision Variables, Constraints, Objective Function, Parameters and Data

• The general form of a math programming model is:

min or max f(x1; : : : ; xn)

≥

s.t. gi(x1; : : : ; xn) = bi

≤

x X

• Linear programming (LP): all functions f and gi are linear and X is continuous.

• Integer programming (IP): X is discrete.

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Mathematical Programming

• A solution is an assignment of values to variables.

• A feasible solution is an assignment of values to variables such that all the constraints are satisfied.

• The objective function value of a solution is obtained by evaluating the objective function at the given solution.

• An optimal solution (assuming minimization) is one whose objective function value is less than or equal to that of all other feasible solutions.

• An optimal solution (assuming maximization) is one whose objective function value is greater than or equal to that of all other feasible solutions.

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Establish a Procedure

• Production software

• Easy to use

• Easy to maintain

• Acceptable to the user

Solution


Solution

Find a Solution

Tools

Construct a Model

Model


Problem Statement

Data

Situation

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Example: A computer-based scheduling system with graphical user interface.

Implement the Solution

• Change for the organization

• Change is difficult

• Establish controls to recognize change in the situation

Solution


Solution

Find a Solution

Tools

Construct a Model

Model


Problem Statement

Data

Situation


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Example: Implement nurse scheduling system in one unit at a time. Integrate with existing HR and T&A systems. Provide training sessions during the workday.

The Goal is to Solve the Problem

• The model must be valid

• The model must be tractable

• The solution must be useful

Data

Solution

Find a Solution

Tools

Situation


Problem Statement

Test the Model and the Solution

Solution



Construct a Model

Model

Implement a Solution

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Advantages and Limitations of OR

Advantages :

• Logic and systematic approach.

• Indicates scope as well as limitations.

• Helps in finding avenues.

• Make the overall structure of problems.

Limitations:

• Quantification

• Information gap

• Money and time costs.

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2. Overview of the OR Modeling Approach

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Phases of OR Modeling

• Problem Definition

• Gather Data

• Model Construction

• Find Solution of the model

• Adding Constraints

• Find Feasible Solutions

• Validation of the model

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A Simple Example

Scenario

A production manager must decide whether to acquire an automatic or a semi-automatic production machine in a production process

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A Simple Example

Define the Problem

Choose one between

A – Automatic machine, and

B - Semiautomatic Machine

to minimize the production cost per batch.

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A Simple Example

Gather Data

Set-up cost Unit cost

Automatic $50 $0.4

Semi-Automatic $20 $0.6

(in $1,000)

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A Simple Example

Construct a Model

Objective: Minimize production cost per batch

X: No. of units produced in a batch

Objective function:

Minimize Prod. Cost = Set-up cost + unit cost

where unit cost = 0.4 x (Automatic)

= 0.6 x (Semiautomatic)37

A Simple Example

Find a Solution

•Buy semi-auto if the batch size < 150

•Buy auto if the batch size >150

3850+0.4x = 20+0.6x => 30 = 0.2 x => x = 150

20 + 0.6 x

50 + 0.4 x

x

A Simple Example

Adding Constraints

The current solution assumes that both machines produce parts at the same rate so that the batch size corresponding to a given production period are equal

Suppose that the hourly production rate is 25 unit/hr for the auto and 15 unit/hr for the semi-auto. If the factory operates 8 hours daily, the maximum batch size for

Automatic - 25 x 8 = 200;

Semi-automatic – 15 x 8 = 120

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A Simple Example

Find Feasible Solution

•Buy semi-auto if the batch size < 120

•Buy auto if the batch size is between 120 and 200

•Choose none if batch size > 200

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3. Introduction to Linear Programming

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OR problems in general concerned with the allocation of scarce resources in the best possible manner so that costs are minimized and profits are maximized. “Linear Programming” is one of the OR tools that meets the following conditions:

• Decision variable involved are nonnegative

• The objective function can be described by a linear function of the decision variables.

• The constraints can be expressed as a set of linear equations.* “programming” means “planning of activities”

What is Linear Programming

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• A Linear Programming model seeks to maximize or minimize a linear function, subject to a set of linear constraints.

• The linear model consists of the followingcomponents:– A set of decision variables.– An objective function.

– A set of constraints.

What is Linear Programming

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• A large variety of problems can be represented or at least approximated as LP models.

• Efficient techniques for solving LP problems are available.• Sensitivity analysis can be handled through LP models w/o

much difficulty.• There are well-known successful applications in:

• Manufacturing• Marketing• Finance (investment)• Advertising• Agriculture

Why use Linear Programming

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Assumptions of the linear programming model

• The parameter values are known with certainty.

• The objective function and constraints exhibit constant returns to scale*.

• There are no interactions between the decision variables (the additivity assumption).

• The Continuity assumption: Variables can take on any value within a given feasible range.

* Constant returns to scale occur when increasing the number of inputs leads to an equivalent increase in the output.

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Three basic steps• Identify the unknown variables (decision variables) and represent

them in terms of algebraic symbols.• Identify all constraints in the problem and express them as linear

equations or inequalities of the decision variables.• Identify the objective and represent it as a linear function of the

decision variable, which is to be minimized or maximized.

Formulation of LP Models

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The Galaxy Industries Production Problem –A Prototype Example

• Galaxy manufactures two toy doll models:

– Space Ray.

– Zapper.

• Resources are limited to

– 1000 pounds of special plastic.

– 40 hours of production time per week.

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• Marketing requirement

– Total production cannot exceed 700 dozens.

– Number of dozens of Space Rays cannot exceed number of

dozens of Zappers by more than 350.

• Technological input

– Space Rays requires 2 pounds of plastic and 3 minutes of labor per dozen.

– Zappers requires 1 pound of plastic and 4 minutes of labor per dozen.


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• The current production plan calls for: – Producing as much as possible of the more profitable product,

Space Ray ($8 profit per dozen).

– Use resources left over to produce Zappers ($5 profit

per dozen), while remaining within the marketing guidelines.

• The current production plan consists of:

Space Rays = 450 dozenZapper = 100 dozenProfit = $4100 per week

8(450) + 5(100)

The Galaxy Industries Production Problem –An Example

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• The current production plan uses

• Special plastic

Space Rays = 450 dozen x 2 lb = 900 lbZapper = 100 dozen x 1 lb = 100 lb

Total = 1,000 lb

• Production time

Space Rays = 450 dozen x 3 min = 1,350 minZapper = 100 dozen x 4 min = 400 min

Total = 1,750 min = 29.17 hr

The Galaxy Industries Production Problem –An Example

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• Management is seeking a production schedule that will increase the company’s profit.

• A linear programming model can provide an insight and an intelligent solution to this problem.


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• Decisions variables:

– X1 = Weekly production level of Space Rays (in dozens)

– X2 = Weekly production level of Zappers (in dozens).

• Objective Function:

– Weekly profit, to be maximized

The Galaxy Linear Programming Model

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Max 8X1 + 5X2 (Weekly profit)

subject to

2X1 + 1X2 1000 (Plastic)

3X1 + 4X2 2400 (Production Time)

X1 + X2 700 (Total production)

X1 - X2 350 (Mix)

Xj > = 0, j = 1,2 (Nonnegativity)

The Galaxy Linear Programming Model

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The Graphical Analysis of LP Model

• The set of all points that satisfy all the constraints of the model is called a feasible region.

• Using a graphical presentation we can represent all the constraints, the objective function, and the feasible points.

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the non-negativity constraints

X2

X1

the Feasible Region defined by


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1000

500

Feasible

X2

InfeasibleProduction Time3X1+4X2 2400

Total production constraint:X1+X2 700 (redundant)

500

700

The Plastic constraint2X1+X2 1000

X1

700


800

600

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1000

500

Feasible

X2

Infeasible

Production Time3X1+4X22400

Total production constraint:X1+X2 700 (redundant)600

700

Production mix constraint:X1-X2 350

The Plastic constraint2X1+X2 1000

X1

700

• There are three types of feasible pointsInterior points. Boundary points. Extreme points.


800350

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search for an optimal solutionStart at some arbitrary profit, say profit = $2,000...

Then increase the profit, if possible...

...and continue until it becomes infeasible

Profit =$4360

600

700

1000

500

X2

X1

Maximize 8X1 + 5X2


350

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the optimal solution

Space Rays = 320 dozen

Zappers = 360 dozen

Profit = $4360

– This solution utilizes all the plastic and all the production hours.

• 320×2+360×1=1000 lbs of plastic, 320×3+360×4=2400 min = 40 hrs

– Total production is only 680 (not 700).

– Space Rays production is 40 dozens less than Zappers

production.

• (X1 – X2) = -40 <= 350


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– If a linear programming problem has an optimal solution, an extreme (corner) point is optimal.

Extreme points and optimal solutionsThe Graphical Analysis of LP Model

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• For multiple optimal solutions to exist, the objective function must be parallel to one of the constraints

Multiple optimal solutions

•Any weighted average of optimal solutions is also an optimal solution.


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The Role of Sensitivity Analysis of the Optimal Solution

• Is the optimal solution sensitive to changes in input parameters?

• Possible reasons for asking this question:– Parameter values used were only best estimates.– Dynamic environment may cause changes.– “What-if” analysis may provide economical and operational

information.

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• Range of Optimality

– The optimal solution will remain unchanged as long as

• An objective function coefficient lies within its range of

optimality

• There are no changes in any other input parameters.

Sensitivity Analysis of Objective Function Coefficients

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600

1000

500 800

X2

X1


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600

1000

400 600 800

X2

X1

Hold the coefficients of x2 at 5, Range of optimality for x1 coefficient: [3.75, 10]


Hold the coefficients of x1 at 8, Range of optimality for x2 coefficient: [4, 10.67]

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• In sensitivity analysis of right-hand sides of constraints we are interested in the following questions:

– Keeping all other factors the same, how much would the optimal value of the objective function (for example, the profit) change if the right-hand side of a constraint changed by one unit?

– For how many additional or fewer units will this per unit change be valid?

Sensitivity Analysis of Right-Hand Side Values

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• Any change to the right hand side of a binding constraint will change the optimal solution.

• Any change to the right-hand side of a non-binding constraint that is less than its slack or surplus, will cause no change in the optimal solution.

Sensitivity Analysis of Right-Hand Side Values

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Shadow Prices

• Assuming there are no other changes to the input parameters, the change to the objective function value per unit increase to a right hand side of a constraint is called the “Shadow Price”

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1000

500

X2

X1

600

When more plastic becomes available (the plastic constraint is relaxed), the right hand side of the plastic constraint increases.

Production timeconstraint

Maximum profit = $4360

Maximum profit = $4363.4 x1 = 320.8, x2 = 359.4

Shadow price = 4363.40 – 4360.00 = 3.40

Shadow Price – graphical demonstrationThe Plastic constraint

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Range of Feasibility

• Assuming there are no other changes to the input parameters, the range of feasibility is– The range of values for a right hand side of a constraint, in

which the shadow prices for the constraints remain unchanged.

– In the range of feasibility the objective function value changes as follows:Change in objective value = [Shadow price][Change in the right hand side value]

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X1=400, X2=300


1000

500

X2

X1

600

Increasing the amount ofplastic is only effective until a new constraint becomes active.

The Plastic constraint

This is an infeasible solutionProduction timeconstraint

Production mix constraintX1 + X2 700

A new activeconstraint

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Profit increase from$4,360 to $4,700

If plastic increase to 1100x1=400, x2=300


1000

500

X2

X1

600

The Plastic constraint

Production timeconstraint

Note how the profit increases as the amount of plastic increases.

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1000

500

X2

X1

600

Less plastic becomes available (the plastic constraint is more restrictive).

The profit decreases to $3,000(X1 =0, X2 =600)

A new activeconstraint

Infeasiblesolution

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Other Post - Optimality Changes

• Addition of a constraint.

• Deletion of a constraint.

• Addition of a variable.

• Deletion of a variable.

• Changes in the left - hand side coefficients.

Load Solver to Excel• The Solver Add-in is a Microsoft Office Excel add-in program that

is available when you install Microsoft Office or Excel. 1. In Excel 2010 and later, go to File > Options

2. Click Add-Ins, and then in the Manage box, select Excel Add-ins.

3. Click Go.

4. In the Add-Ins available box, select the Solver Add-in check box, and then click OK.

5. After you load the Solver Add-in, the Solver command is available in the Analysis group on the Data tab.

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Using Excel Solver to Find an Optimal Solution and Analyze Results

Using Excel Solver to Find an Optimal Solution and Analyze Results

• To see the input screen in Excel, open Galaxy.xls

• Click Data > Solver to obtain the following dialog box.

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Using Excel Solver

• Click Optionsto obtain the following dialog box.

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• Click Solve to obtain the solution.

Using Excel Solver

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Using Excel Solver – Optimal Solution

GALAXY INDUSTRIES

Space Rays Zappers

Dozens 320 360

Total Limit

Profit 8 5 4360

Plastic 2 1 1000 <= 1000

Prod. Time 3 4 2400 <= 2400

Total 1 1 680 <= 700

Mix 1 -1 -40 <= 350

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Using Excel Solver – Optimal Solution

Solver is ready to providereports to analyze theoptimal solution.

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Using Excel Solver –Answer Report

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Interpreting Excel Solver Sensitivity Report

https://www.youtube.com/watch?v=U_XaHQce--8

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• Infeasibility: Occurs when a model has no feasible point.

• Unboundness: Occurs when the objective can become

infinitely large (max), or infinitely small (min).

• Alternate solution: Occurs when more than one point

optimizes the objective function

Models Without Unique Optimal Solutions

841

No point, simultaneously, lies both above line and

below lines and

.

1

2 32

3

Infeasible Model

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Solver – Infeasible Model

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Unbounded solution

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Solver – Unbounded solution

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The Wyndor Glass Co. Problem –An Example

• Wyndor Glass produces glass windows and doors

• They have 3 plants:– Plant 1: makes aluminum frames and hardware

– Plant 2: makes wood frames

– Plant 3: produces glass and makes assembly

• Two products proposed:– Product 1: 8’ glass door with aluminum siding

– Product 2: 4’ x 6’ wood framed glass window

• Some production capacity in the three plants is available to produce a combination of the two products

• Problem is to determine the best product mix to maximize profits

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The Wyndor Glass Co. Problem –Data

• The O.R. team has gathered the following data:– Number of hours of production time available per week in each plant

– Number of hours of production time needed in each plant for each batch of new products

– Estimated profit per batch of each new product

Production time per batch (hr) Production time available per

week (hr)Product

Plant 1 2

1 1 0 4

2 0 2 12

3 3 2 18

Profit per batch ($1,000) $3 $5

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The Wyndor Glass Co. Problem –Model Construction

To formulate the LP model, let

• x1 = the no. of batches of product 1 produced

• x2 = the no. of batches of product 2 produced

• Z = total profit per week (in $1,000) from producing these two products

Constraints

• x1 ≥ 0, x2 ≥ 0.

• 1 x1 ≤ 4

• 2 x2 ≤ 12

• 3 x1 + 2 x2 ≤ 18

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The Wyndor Glass Co. Problem –Model Construction

LP model

Maximize Z = 3 x1 + 5 x2

Subject to x1 ≤ 4

x2 ≤ 12

3 x1 + 2 x2 ≤ 18

x1 ≥ 0, x2 ≥ 0.

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The Wyndor Glass Co. Problem –The Graphical Analysis

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The Wyndor Glass Co. Problem –The Excel Solver Analysis

Wyndor Glass Co.

Hrs per unitProduct 1 Product 2 Totals

plant 1 1 0 0<= 4plant 2 0 2 0<= 12plant 3 3 2 0<= 18unit profit 3 5 0solution 0 0

Class Exercise

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The Wyndor Glass Co. Problem –The Optimum Solution

Wyndor Glass Co.

Hrs per unitProduct 1 Product 2 Totals

plant 1 1 0 2<= 4plant 2 0 2 12<= 12plant 3 3 2 18<= 18unit profit 3 5 36solution 2 6

Class Exercise

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The Wyndor Glass Co. will have the maximized profit

• Z = $36,000 with

• 2 batches of product 1 produced

• 6 batches of product 2 produced

The Wyndor Glass Co. Problem –The Optimum Solution

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The Handy-Dandy Co. Product Mix Problem –An Example

• The Handy-Dandy Co. Wishes to schedule the production of a kitchen appliance that requires two resources – labor and material.

• They have 3 models:– A: Requires Labor – 7 hr, Material – 4 lb, Generates profit - $4/unit

– B: Requires Labor – 3 hr, Material – 4 lb, Generates profit - $2/unit

– C: Requires Labor – 6 hr, Material – 5 lb, Generates profit - $3/unit

• Two daily restrictions:– Raw material – 200 lbs/day

– Labor – 150 hrs/day

• Problem is to determine the daily production rate of the various models to maximize profits

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The Handy-Dandy Co. Product Mix Problem –Model Construction

To formulate the LP model, let

• x1 = the no. of batches of model A produced

• x2 = the no. of batches of model B produced

• x3 = the no. of batches of model C produced

• Z = total profit per day from producing these three models

Constraints

• x1 ≥ 0, x2 ≥ 0 , x3 ≥ 0.

• 7 x1 + 3 x2 + 6 x3 ≤ 150 (labor)

• 4 x1 + 4 x2 + 5 x3 ≤ 200 (material)

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The Handy-Dandy Co. Product Mix Problem –Model Construction

LP model

Maximize Z = 4 x1 + 2 x2 + 3 x3

• Subject to 7 x1 + 3 x2 + 6 x3 ≤ 150

4 x1 + 4 x2 + 5 x3 ≤ 200

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.

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The Handy-Dandy Co. Product Mix Problem –The Excel Solver Analysis

Product-Mix Problem

Model

A B C Total

Labor (hr/unit) 7 3 6 0 <= 150

Material (lb/unit) 4 4 5 0 <= 200

Profit ($/unit) 4 2 3 0

Solution (unit) 0 0 0

Class Exercise

102

The Handy-Dandy Co. Product Mix Problem –The Optimum Solution

Product-Mix Problem

Model

A B C Total

Labor (hr/unit) 7 3 6 150 <= 150

Material (lb/unit) 4 4 5 200 <= 200

Profit ($/unit) 4 2 3 100

Solution (unit) 0 50 0

Class Exercise

103

The Handy-Dandy Co. will have the maximized profit

• Z = $100 with

• 0 of model A and model C produced

• 50 of model B produced

The Handy-Dandy Co. Product Mix Problem –The Optimum Solution

104

• Linear programming software packages solve large linear models.

• Most of the software packages use the algebraic technique called the Simplex algorithm.

• The input to any package includes:– The objective function criterion (Max or Min).– The type of each constraint: ≤, , ≥.– The actual coefficients for the problem.

Computer Solution of Linear Programs With Any Number of Decision Variables

OR SuccessesRepresentative cases from the annual INFORMS

Edelman Competition

1. Forecasting the Shuttle Disaster at NASA2. The Operation Desert Storm Airlift3. A North Carolina School District Improves Planning4. US Postal Service Automates Delivery

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The End!Questions?

Case 1: Forecasting the Shuttle Disaster at NASA

• The problem– After the Challenger shuttle disaster in 1986, NASA

decided to conduct risk analysis on specific systems to identify the greatest threats of a future disaster and prevent them

– Consultants at Stanford University and Carnegie Mellon were called in to assess risk to the shuttle tiles

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NASA (con’t)

• Objectives and requirements– Identify different possible accident scenarios

– Compute the probability of failure

– Show how safety could be increased

– Prioritize recommended safety measures

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NASA (con’t)

• The OR solution– Model was based on a multiple partition of the orbiter's

surface– For the tiles in each zone, the OR team examined data to

determine the probability of: 1. Debonding due to debris hits or a poor bond2. Losing adjacent tiles once the first is lost3. Burn-through4. Failure of a critical subsystem under the skin of the orbiter if a burn-

through occurs

– A risk-criticality scale was designed based on the results of this model

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NASA (con’t)

• The value– Found that 15% of the tiles account for about 85% of the risk– Recommended NASA inspect the bond of the most risk

critical tiles and reinforce insulation of vulnerable external systems

– Computed that such improvements could reduce probability of a shuttle accident from tile failure by 70%

– 1994 study quoted extensively in the press after the Columbia, a second shuttle, exploded on reentry in 2003, apparently due to tile failure

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Case 2: The Operation Desert Storm Airlift

• The problem – In 1991, the Military Air Command (MAC) was charged with

scheduling aircraft, crew, and mission support resources to maximize the on-time delivery of cargo and passengers to the Persian Gulf

– A typical airlift mission carrying troops and cargo to the Gulf required a three-day round trip, visited 7 or more different airfields, burned almost 1 million pounds of fuel, and cost $280,000

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Desert Storm (con’t)

• Objectives and requirements: – Create a scheduling system

– Create a communications system coordinating the schedule among bases in the US and other countries

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• The OR solution – MAC worked with the Oak Ridge National Laboratory to

develop the Airlift Deployment Analysis System (ADANS)– Within three months, ADANS provided a set of decision

support tools to manage:• Information on cargo and passengers• Information on available resources

– ADANS also developed tools for:• Scheduling missions• Analyzing the schedule • Distributing the schedule to the MAC worldwide command and control

system

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• The value– By August 1991, more than 25,000 missions had

moved nearly 1 million passengers and 800,000 tons of cargo to and from the Persian Gulf

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Case 3: A North Carolina School District Improves Planning

• The problem– School planning, like public sector land-use planning, takes

place within a complex environment, including perceptions of public education, public finance, taxation, politics, and the courts

– Johnston County, NC sought to improve school planning while integrating the concerns of participating agencies and community groups

– It worked under two constraints:• Inadequate data to support sensitive decisions• Externally imposed constraints on decision-making

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NC School District (con’t)

• Objectives and requirements – The school board and administration sought to develop a

strong planning culture and a decision-support mechanism that would restore public confidence and win the support of the community’s political leaders

– The OR consulting group wanted to fulfill these requests and while creating models that would be effective and portable to other school districts

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• The OR solution– OR/ED Laboratories and the Johnston County schools

created a planning system, Integrated Planning for School and Community, to:

• Forecast enrollments

• Compare enrollment projections to capacity

• Find the optimal locations for new school building

• Set distance-minimized boundaries for all schools to avoid overcrowding and meet racial balance guidelines

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• The value– Implementing the system has increased the school

district’s success in:• Passing bond issues

• Reducing pupil-transportation costs

• Eliminating frequent adjustments to school-attendance boundaries

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Case 4: US Postal Service Automates Delivery

• The problem

• In 1988, the US Postal Service foresaw three interlocking problems:– An increase from 166 billion to 261 billion pieces of mail

handled a year by the turn of the century

– Increased private sector competition

– A complexity of operations that would have to be modeled if automation were to respond to the challenges

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US Postal Service (con’t)

• Objectives and requirements– Create a decision support tool that could simulate

postal operations and quantify the effects of automation alternatives

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• The OR solution– Working with two OR consulting groups, the Postal Service

developed the Model for Evaluating Technology Alternatives (META)

– A simulation model that quantifies the impacts of changes in mail-processing and delivery operations

– Blended OR and software tools in a decision support system

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• The value– META analysis enabled the Postal Service to

release its Corporate Automation Plan, including a cumulative capital investment of $12 billion and labor savings of $4 billion per year

– META spawned a family of systems for use at headquarters and field levels, accelerating and enhancing the use of OR throughout the organization

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Documents

1. What is Operations Research (OR) Slides.pdfDeterministic Operations Research, Linear Programming 1 Source: Introduction to Operations Research, 9th edition, Frederick S. Hillier,