1. What is Operations Research (OR) 2 What is Operations Research? - - PDF document
Unit 4 Systems Engineering Tools Deterministic Operations Research, Linear Programming Source: Introduction to Operations Research, 9th edition, Frederick S. Hillier, McGraw-Hill 1 1. What is Operations Research (OR) 2 What is Operations
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Source: Introduction to Operations Research, 9th edition, Frederick S. Hillier, McGraw-Hill
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Operations The activities carried out in an organization. Research The process of observation and testing characterized by the scientific method including situation, problem statement, model construction, validation, experimentation, candidate solutions. Model An abstract representation of reality. Mathematical, physical, narrative, set of rules in computer program.
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It is a Systems Approach Include broad implications of decisions for the organization at each stage in analysis. Both quantitative and qualitative factors are considered. It finds Optimal Solution A solution to the model that optimizes (maximizes or minimizes) some measure of merit over all feasible solutions. It involves a Team A group of individuals bringing various skills and viewpoints to a problem. It includes many different OR Techniques A collection of general mathematical models, analytical procedures, and algorithms.
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OR started just before World War II in Britain with the establishment of teams of scientists to study the strategic and tactical problems involved in military operations. The objective was to find the most effective utilization of limited military resources by the use of quantitative techniques.
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Although scientists had (plainly) been involved in the hardware side of warfare (designing better planes, bombs, tanks, etc) scientific analysis
the
use
military resources had never taken place in a systematic fashion before the Second World War. Military personnel were simply not trained to undertake such analysis.
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These early OR workers came from many different disciplines,
mathematical physicists and a surveyor. What such people brought to their work were "scientifically trained" minds, used to querying assumptions, logic, exploring hypotheses, devising experiments, collecting data, analysing numbers, etc. Many too were of high intellectual calibre (at least four wartime OR personnel were later to win Nobel prizes when they returned to their peacetime disciplines).
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Following the end of the war, OR took a different course in the UK as opposed to in the USA. In the UK (as mentioned above) many of the distinguished OR workers returned to their original peacetime disciplines. As such OR did not spread particularly well, except for a few isolated industries (iron/steel and coal). In the USA OR spread to the universities so that systematic training in OR began.
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You should be clear that the growth of OR since it began (and especially in the last 30 years) is, to a large extent, the result
large number of numeric calculations. Without computers this would simply not be possible.
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Manufacturers used operations research to make products more efficiently, schedule equipment maintenance, and control inventory and distribution. And success in these areas led to expansion into strategic and financial planning … and into such diverse areas as criminal justice, education, meteorology, and communications.
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A number of major social and economic trends are increasing the need for operations researchers. In today’s global marketplace, enterprizes must compete more effectively for their share of profits than ever before. And public and non-profit agencies must compete for ever-scarcer funding dollars.
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This means that all of us must become more productive. Volume must be increased. Consumers’ demands for better products and services must be met. Manufacturing and distribution must be faster. Products and people must be available just in time.
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for complex organizational systems.
techniques such as linear and nonlinear programming to derive values for system variables that will optimize performance.
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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– 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
Problem Statement Situation
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Example: Maximize individual nurse preferences subject to demand requirements, or minimize nurse dissatisfaction costs.
Data
Situation
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Example: Gather information about nurse profiles, work schedule, pay structure, overhead, demand requirement, supply, etc.
Construct a Model
Model
Formulate the Problem
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.
phenomenon that accounts for its known or inferred properties and may be used for further study of its characteristics.
– 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 OR By structure Physical model Analogue model Mathematical model By nature of environment Deterministic Model Probabilistic Model
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– Typically, a single objective function, representing either a profit to be maximized
decision variables. – Software has been developed that is capable of solving problems containing millions of variables and tens of thousands of constraints.
– 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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– Some of the variables are required to take on discrete values.
– 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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Construct a Model
Model
Formulate the Problem
Problem Statement
Data
Situation
Solution
Find a Solution Tools 25
Example: Apply algorithm; post-process results to get monthly schedules.
– Decision Variables, Constraints, Objective Function, Parameters and Data
min or max f(x1; : : : ; xn) ≥ s.t. gi(x1; : : : ; xn) = bi ≤ x X
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constraints are satisfied.
value is less than or equal to that of all other feasible solutions.
value is greater than or equal to that of all other feasible solutions.
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Solution
Establish a Procedure
Solution
Find a Solution Tools Construct a Model
Model
Formulate the Problem
Problem Statement
Data
Situation
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Example: A computer-based scheduling system with graphical user interface.
Solution
Establish a Procedure
Solution
Find a Solution Tools Construct a Model
Model
Formulate the Problem
Problem Statement
Data
Situation
Implement the Solution
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Example: Implement nurse scheduling system in one unit at a time. Integrate with existing HR and T&A
workday.
Data
Solution
Find a Solution Tools
Situation
Formulate the Problem
Problem Statement
Test the Model and the Solution
Solution
Establish a Procedure Implement the Solution Construct a Model
Model
Implement a Solution
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Advantages :
Limitations:
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Set-up cost Unit cost Automatic $50 $0.4 Semi-Automatic $20 $0.6
(in $1,000)
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Objective: Minimize production cost per batch X:
Objective function: Minimize Prod. Cost = Set-up cost + unit cost where unit cost = 0.4 x (Automatic) = 0.6 x (Semiautomatic)
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the batch size < 150
batch size >150
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50+0.4x = 20+0.6x => 30 = 0.2 x => x = 150 20 + 0.6 x 50 + 0.4 x x
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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batch size < 120
is between 120 and 200
size > 200
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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 variables.
* “programming” means “planning of activities”
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returns to scale*.
(the additivity assumption).
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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them in terms of algebraic symbols.
equations or inequalities of the decision variables.
decision variable, which is to be minimized or maximized.
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– Total production cannot exceed 700 dozens. – Number of dozens of Space Rays cannot exceed number of dozens of Zappers by more than 350.
– 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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– 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.
Space Rays = 450 dozen Zapper = 100 dozen Profit = $4100 per week
8(450) + 5(100)
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Space Rays = 450 dozen x 2 lb = 900 lb Zapper = 100 dozen x 1 lb = 100 lb Total = 1,000 lb
Space Rays = 450 dozen x 3 min = 1,350 min Zapper = 100 dozen x 4 min = 400 min Total = 1,750 min = 29.17 hr
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– X1 = Weekly production level of Space Rays (in dozens) – X2 = Weekly production level of Zappers (in dozens).
– Weekly profit, to be maximized
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the non-negativity constraints
X2 X1
the Feasible Region defined by
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1000 500
Feasible
X2
Infeasible
Production Time 3X1+4X2 2400 Total production constraint: X1+X2 700 (redundant)
500 700
The Plastic constraint 2X1+X2 1000
X1 700
800 600
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1000 500
Feasible
X2
Infeasible
Production Time 3X1+4X22400 Total production constraint: X1+X2 700 (redundant)
600 700
Production mix constraint: X1-X2 350 The Plastic constraint 2X1+X2 1000
X1 700
Interior points. Boundary points. Extreme points.
800 350
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Start 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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Space Rays = 320 dozen Zappers = 360 dozen Profit = $4360 – This solution utilizes all the plastic and all the production hours.
– Total production is only 680 (not 700). – Space Rays production is 40 dozens less than Zappers production.
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– 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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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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– Keeping all other factors the same, how much would the
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?
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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 time constraint Maximum profit = $4360 Maximum profit = $4363.4
x1 = 320.8, x2 = 359.4
Shadow price = 4363.40 – 4360.00 = 3.40
The Plastic constraint
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– 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:
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X1=400, X2=300
1000 500
X2 X1
600
Increasing the amount of plastic is only effective until a new constraint becomes active.
The Plastic constraint
This is an infeasible solution
Production time constraint Production mix constraint X1 + X2 700
A new active constraint
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Profit increase from $4,360 to $4,700 If plastic increase to 1100
x1=400, x2=300
1000 500
X2 X1
600
The Plastic constraint
Production time constraint
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 active constraint
Infeasible solution
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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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screen in Excel, open Galaxy.xls
dialog box.
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GALAXY INDUSTRIES
Space Rays Zappers Dozens 320 360 Total Limit Profit 8 5 4360 Plastic 2 1 1000 <= 1000
3 4 2400 <= 2400 Total 1 1 680 <= 700 Mix 1
<= 350
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Solver is ready to provide reports to analyze the
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https://www.youtube.com/watch?v=U_XaHQce--8
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2 3
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– Plant 1: makes aluminum frames and hardware – Plant 2: makes wood frames – Plant 3: produces glass and makes assembly
– Product 1: 8’ glass door with aluminum siding – Product 2: 4’ x 6’ wood framed glass window
a combination of the two products
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– 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
– 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 4 2 2 12 3 3 2 18 Profit per batch ($1,000) $3 $5
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To formulate the LP model, let
Constraints
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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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Wyndor Glass Co. Hrs per unit Product 1 Product 2 Totals plant 1 1 0<= 4 plant 2 2 0<= 12 plant 3 3 2 0<= 18 unit profit 3 5 solution
Class Exercise
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Wyndor Glass Co. Hrs per unit Product 1 Product 2 Totals plant 1 1 2<= 4 plant 2 2 12<= 12 plant 3 3 2 18<= 18 unit profit 3 5 36 solution 2 6
Class Exercise
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The Wyndor Glass Co. will have the maximized profit
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kitchen appliance that requires two resources – labor and material.
– 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 - $3/unit
– Raw material – 200 lbs/day – Labor – 150 hrs/day
models to maximize profits
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To formulate the LP model, let
Constraints
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LP model Maximize Z = 4 x1 + 2 x2 + 3 x3
7 x1 + 3 x2 + 6 x3 ≤ 150 4 x1 + 4 x2 + 5 x3 ≤ 200 x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
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Product-Mix Problem Model A B C Total Labor (hr/unit) 7 3 6 <= 150 Material (lb/unit) 4 4 5 <= 200 Profit ($/unit) 4 2 3 Solution (unit)
Class Exercise
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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) 50
Class Exercise
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The Handy-Dandy Co. will have the maximized profit
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– The objective function criterion (Max or Min). – The type of each constraint: ≤, , ≥. – The actual coefficients for the problem.
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– 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 bond 2. Losing adjacent tiles once the first is lost 3. Burn-through 4. 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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– 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
– 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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– 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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– 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:
– ADANS also developed tools for:
system
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– 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:
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– 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
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– OR/ED Laboratories and the Johnston County schools created a planning system, Integrated Planning for School and Community, to:
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boundaries
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– 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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– 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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