IndE 310 Linear Programming UW Industrial Engineering Instructor: Prof. Zelda Zabinsky

Operations Research “The Science of Better”: The discipline of applying advanced analytical methods to help make better decisions Operations – problems that concern how to conduct and coordinate the operations within an organization Research – use of the scientific method to address these problems Interdisciplinary – brings together: mathematics statistics industrial engineering management (“management science”) systems engineering... Highly application oriented

Applications of Operations Research Business Manufacturing Project management Scheduling Facility layout and location Many more Service Transportation and logistics Marketing Queueing Economics & Finance Auctions Portfolio selection Capital investment Many more Health Scheduling Queueing Biotechnology And more Military

Examples of O.R. Applications China Select and schedule projects for future energy needs New Haven Health Department Design an effective needle exchange program to combat HIV Continental Airlines Optimize the reassignment of crews to flights IBM Reengineer supply chain Proctor and Gamble Redesign production and distribution system Many more

(and make recommendations) Operations Research Structuring the real-life situation into a mathematical model, abstracting the essential elements so that a solution relevant to the decision maker's objectives can be sought This involves looking at the problem in the context of the entire system (Understand and) Model (and verify) Exploring the structure of such solutions and developing systematic procedures for obtaining them Developing a solution, including the mathematical theory, if necessary, that yields an optimal value of the system measure of desirability (or possibly comparing alternative courses of action by evaluating their measure of desirability) (Strategize and) Solve (and make recommendations)

Operations Research Modeling Toolset Queueing Theory Markov Chains PERT/ CPM Network Programming Dynamic Programming Simulation Markov Decision Processes Inventory Theory Linear Programming Stochastic Programming Forecasting Integer Programming Decision Analysis Nonlinear Programming Game Theory

Operations Research Modeling Toolset 311 Queueing Theory 310 Markov Chains PERT/ CPM Network Programming Dynamic Programming Simulation Markov Decision Processes Inventory Theory Linear Programming 312 Stochastic Programming Forecasting Integer Programming Decision Analysis Nonlinear Programming Game Theory 312 312

IndE 310 Linear Programming Transportation and Assignment Problems Modeling Solving Simplex method Foundations of the simplex method Duality and sensitivity Transportation and Assignment Problems Network Problems Shortest path Minimum spanning tree Minimum cost, maximum flow PERT/CPM

Interesting Links “Operation Everything”, The Boston Globe, June 27, 2004 http://www.boston.com/globe/search/stories/reprints/ operationeverything062704.html Links available on the course web

O.R. Modeling Approach

O.R. Modeling Approach Define the problem and gather data Formulate a (mathematical) model to represent the problem Develop a computer-based procedure for deriving solutions Test model and refine it Prepare for the ongoing application of the model Implement

Defining the problem and gathering data First order of business: Study the relevant system Develop a well-defined statement of the problem Right answer for the wrong problem? Defining objectives What data are needed? Collect them (easily said)

Formulating a (mathematical) model We need to create a model that can be used for Abstracting the essence of the subject Showing interrelationships Facilitating analysis A model: E=mc2 A mathematical model usually consists of: Decision variables Parameters Objective function Constraints

Deriving solutions A common theme in O.R.: Search for an optimal solution Develop a procedure for deriving solutions to the mathematical model e.g. what is the best speed to obtain the highest possible energy? What kind of a procedure? Usually based on theoretical foundations Exact vs. heuristic (slow vs. fast?) Post-optimality and sensitivity analysis

Testing, preparing and implementing Need to validate the model “Debugging” Plausible results? Prepare to apply as prescribed by management Operating procedures, supporting systems, managerial reports… Implementation Coordination, detailed indoctrination, new courses of action

Peeking into Chapter 3 Wyndor Glass Co. 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

Peeking into Chapter 3 Wyndor Glass Co. 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 4 12 3 18 Profit per batch $3,000 $5,000

Peeking into Chapter 3 Formulate LP Model Identify the parameters (activities/values you cannot control) Identify the decision variables (activities/values that you can control and need to make a decision on) Identify the objective function (function of the decision variables for minimization/maximization) Identify the constraints (limitations you cannot control)

Peeking into Chapter 3 Graphical Solution Setup The model:

Prototype Example Graphical Solution The model:

LP Modeling Wyndor Glass Co. is only one of a vast number of applications that LP can be used to address In general, the most common type of an LP addresses: The allocation of limited resources to competing activities for maximizing the value of these activities

LP Terminology The allocation of limited resources to competing activities for maximizing the value of these activities Activities, n Resources, m Decision variables – or level of activities, x Objective function – or value of activities, Z Constraints functional non-negativity

LP Terminology Feasible region, feasible solution Infeasible solution Optimal solution Extreme-point – or corner-point feasible solutions Parameters

Standard (Canonical) Form of an LP Model Maximize Z = c1x1 + c2x2 + … + cnxn subject to a11x1 + a12x2 + … + a1nxn ≤ b1 a21x1 + a22x2 + … + a2nxn ≤ b2 … am1x1 + am2x2 + … + amnxn ≤ bm x1 ≥ 0, x2 ≥ 0, …, xn ≥ 0

Matrix Form of an LP Model

Other Forms that can be Converted into Standard Form Objective function: Minimize Z=cx (instead of Maximize Z=cx) Functional constraints: Ax ≥ b or Ax = b (instead of Ax ≤ b) Non-negativity constraints: x unrestricted in sign (instead of x ≥ 0)

Assumptions of Linear Programming Proportionality Additivity Divisibility Certainty

LP Modeling Examples

A Transportation Example A company has 2 plants and 3 warehouses Supply at plants 100 units in Plant 1, 200 units in Plant 2 Sales potential at warehouses 150 units, 200 units, and 350 units at Warehouses 1, 2 and 3, respectively Revenue 12 $/unit, 14 $/unit and 15 $/unit at Warehouses 1, 2 and 3, respectively Cost of manufacturing one unit at plant i and shipping to w/h j: How many units to ship from each plant to each w/h to maximize profits? Warehouse 1 2 3 Plant 8 10 12 7 9 11

Personnel Scheduling (p.56) Union Airways is adding more flights and needs to hire additional customer service agents Each agent works an eight-hour shift The five possible shifts are Shift 1: 6:00 am – 2:00 pm Shift 2: 8:00 am – 4:00 pm Shift 3: Noon – 8:00 pm Shift 4: 4:00 pm – Midnight Shift 5: 10:00 pm – 6:00 am

Minimum number of agents needed Personnel Scheduling Minimum number of agents needed per two-hour time periods: Time period Time periods covered Minimum number of agents needed Shift 1 2 3 4 5 6:00 am – 8:00 am  48 8:00 am – 10:00 am 79 10:00 am – noon 65 Noon – 2:00 pm 87 2:00 pm – 4:00 pm 64 4:00 pm – 6:00 pm 73 6:00 pm – 8:00 pm 82 8:00 pm – 10:00 pm 43 10:00 pm – midnight 52 Midnight – 6:00 am 15 Daily cost per agent $170 $160 $175 $180 $195

Reclaiming Solid Wastes (p.52) A recycling center takes four types of material Material 1: Newsprint Material 2: White paper Material 3: Mixed paper Material 4: Cardboard Three products are reclaimed Grades A, B, C The SAVE-IT company wants to determine the amount of each grade to produce and the mix of materials in each grade to maximize profit

Reclaiming Solid Wastes Grade Specification Amalgamation Cost per lb ($) Selling Price per lb ($) A Mat’l 1: Not more than 30% of total Mat’l 2: Not less than 40% of total Mat’l 3: Not more than 50% of total Mat’l 4: Exactly 20% of total 3.00 8.50 B Mat’l 1: Not more than 50% of total Mat’l 2: Not less than 10% of total Mat’l 4: Exactly 10% of total 2.50 7.00 C Mat’l 1: Not more than 70% of total 2.00 5.50 Products Solid waste materials Mat’l lbs per week available Treatment cost per lb ($) 1 3,000 3.00 2 2,000 6.00 3 4,000 4.00 4 1,000 5.00 Use at least half of each material collected Cannot use more than $30,000 per week for treatment of mat’ls