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1 ENGM 670 & MECE 758 Modeling and Simulation of Engineering Systems (Advanced Toics) Winter 011 Lecture 9: Extra Material M.G. Lisett University of Alberta htt:// Extra Material for Grad Students This slide deck introduces the concet of utility in systems, how objective functions are formulated, and a simle method to solve for the maximum utility in linear systems MG Lisett, 009 1

2 Preference & Utility In a system, value is exressed as utility Utility is the benefit associated with some system resonse to resources that are available to an element of the system Utility is a loo variable that has a relationshi to the cost or resources allocated to the element of the system Utility itself is difficult to measure directly; instead the overall system utility is judged by an outut (such as rofit) MG Lisett, Utility Theory Utility theory is a mathematical framework for examining the references of individuals It has imortant alications for systems engineering, in the formulation of objective functions for engineering design MG Lisett, 009 4

3 Preference Utility is usually assessed in the context of relative measures, such as reference Preference is a choice between otions, based on the idea that the referred otion has more value to the decisionmaker than an alternative A reference of otion A over otion B is exressed as A > B When the otions are equivalent, the decision-maker is indifferent, and we write A ~ B If B is not referred to A, this is a weak reference, written A B MG Lisett, Utility If A > B, then u A > u B And if A ~ B, then u A = u B Where u i is the utility of otion i. Sometimes this number is given in units of utiles. Utilities used only for comarisons (and thus limited to establishing reference order) are ordinal utilities. If we assign u A = 15 and u B =30, all we can say is that B is referred over A, not that B is referred twice as much. A utility measure that gives strength of reference is a cardinal utility. There is no unique set of numbers for secifying utility for a set of references. Thus, the reference orderings for a utility function exhibit the roerty of scale indifference. MG Lisett,

4 Marginal Utility Utility is always conditional The value of an object or an activity deends on the context A bottle of water is a lot more valuable in the middle of a desert But even a thirsty erson eventually gets no additional benefit from extra water The utility of a decision thus deends on the circumstances, which are described by the state of the system One condition is the abundance of the resource of interest (or the amount of it that is used by the element, or system, or consumer, deending on what we are interested in) MG Lisett, Marginal Utility () The law of diminishing marginal utility states that the marginal utility decreases as the amount consumed increases. Note that this standard form for showing the relationshi between utility and resource use does not have the loo variable on the x-axis. MG Lisett,

5 Marginal Utility (3) The law of equal marginal utilities states that each good is demanded u to the oint where the marginal utility er unit of cash sent on it is exactly equal to the marginal utility er unit of cash sent on any other good. This is illustrated below, subject to a budgetary constraint β : β = P A q A + P B q B + P C q C where P is rice and q is quantity MG Lisett, Measuring Utility For resources between zero amount and saturation quantity, the utility function monotonically increases, and exhibits decreasing marginal utility (becoming zero at the saturation oint) If we know the utility function for one commodity in a set of resources in a budgetarily constrained system, we can write functions for the others, e.g., which can be written in the form MG Lisett,

6 System Inconsistency It is ossible to have references that form logical inconsistencies (intransitive references) A rational erson has transitive (consistent) reference orders This is an artifact of the roblem that any sufficiently interesting system can form a strange loo that violates loo admissibility, even while maintaining consistency at each element in the loo. Physical systems obey laws that revent this violation of loo admissibility Systems created by eole have the otential to violate admissibility, because a grou may not have transitive reference orders, even when the individuals do. This effect is seen in aradoxes, otical illusions, and decisionmaking frameworks, such as a olitical system; it was formalised by Godel in his incomleteness theorems in Any system caable of self-reference is incomlete. MG Lisett, Examle: Arrow s Imossibility Theorem Consider a rocess in which there are three voters: I, II & III Each casts one vote for one of three candidates: A, B & C The voters each have the following rational references: Voter I II III Preferences A > B > C, A > C B > C > A, B > A C > A > B, C > B MG Lisett,

7 Examle: Arrow s Imossibility Theorem () This leads to the following choices if there is an election between two candidates: Voters Elections A versus B B versus C A versus C I A B A II B B C III A C C Result A > B B > C C > A Thus, the grou s reference is A>B>C>A Even though each individual decision (activity in the system) was rational, the overall system exhibits irrational behaviour. MG Lisett, Examle: Arrow s Imossibility Theorem (3) This result tells us that, for a grou, it is unlikely that a single utility function exists. This makes it literally imossible for oliticians to lease all of the eole all of the time. The imlication for engineers, who aly scientific rinciles for societal benefit, is: How to maximise the utility (benefit) to a subset of society by allocating some scarce resources (costs) Often this challenge is easily defined, because the engineer is working for a client. But sometimes it is not, esecially when a design grou has to made decisions, or a client grou is stating system requirements. This exlains the joke: a camel is a horse designed by a committee MG Lisett,

8 Objective Functions In many technological systems, there is some erformance measure or objective that we can exress as a (transitive) cardinal utility function with resect to controllable variables within the system There may be may ossible solutions for the system state vector, meaning that there are many feasible solutions for the roblem The objective function allows us to link the ossible solutions to the roblem to maximising the utility of the system The rocess of maximising (or minimising) a function is called otimisation. MG Lisett, Introduction to Otimisation MG Lisett, (Thanks to Prof J. Doucette for most of the material in this section) In our course, we have solved systems of equations to find the answer for a given system But usually, as soon as we can find one answer, we then want to know what the best answer is Otimisation is the rocess of finding the best decisions to make in solving a roblem or arranging a system In engineering systems, the definition of best is often a quantitative objective function, either a technical erformance secification, or a business goal (net rofit), which is an economic exression of the total utility of the system. We define objective functions (goals) with resect to the constraints and decision variables of the system. When the exressions are algebraic, we use mathematical rogramming methods. 8

9 Mathematical Programming A mathematical rogramming model is a mathematical decision model for lanning (rogramming) decisions that otimize an objective function and satisfy limitations imosed by mathematical constraints. 1 General symbolic model: Maximize (or minimize): Subject to: f ( x x K ) 1, x n (, K n ) {,, = } (, K ) {,, = } g x x x b g x x x b 1 n (, K ) {,, = } g x x x b m 1 n m Objective Constraints where x 1, x K x n are the decision variables. 1 T.W. Knowles, Management Science: Building and Using Models, Irwin, MG Lisett, Mathematical Programming Terminology Decision variables are quantities you can control, which comletely describe the set of decisions to be made. Constraints are limitations on the values of the decision variables. The objective function is a measure that can be used to rank alternative solutions (e.g., NPV, cost, roduction rate, travel time). The goal is to either maximize or minimize its value. A solution is any combination of values for all decision variables. A feasible solution is a solution that satisfies all of the constraints. An infeasible solution does NOT satisfy one or more constraints. An otimal solution is the best feasible solution As defined by the objective function MG Lisett,

10 Tyes of Mathematical Programming Deending on the nature of the system and how it is modeled, different solution methods will be aroriate For a system modeled as a linear system in terms of decision variables, linear rogramming methods can be used: Linear Programs (LP): the objective and constraint functions are linear and the decision variables are continuous. MG Lisett, Tyes of Mathematical Programming () Integer Linear Programs (ILP): one or more of the decision variables are restricted to discrete (integer) values only, and the functions are linear Pure IP: all decision variables are integer. Mixed IP (MIP): some decision variables are integer, others are continuous. 1/0 MIP: some or all decision variables are further restricted to be valued either 1 or 0. Nonlinear Programs Nonlinear Programs: one or more of the functions is not linear. MG Lisett,

11 Solving Mathematical Programming Problems Grahical method Only useful for decision variables (maybe 3 if you can handle 3-D chart) Simlex method Efficient algorithm to solve LP roblems by erforming matrix oerations on the LP Tableau Develoed by George Dantzig (1947) Can be used to solve small LP roblems by hand Software ackages MS Excel Lindo, LPSolve AMPL/CPLEX: modeling language and solver for large and comlex LP/IP roblems Matlab Otimization Toolbox Sub-Otimal Algorithms (Heuristics) Simulated annealing Genetic algorithms Tabu search Constraint rogramming MG Lisett, Linear Programming General symbolic form Maximize: Subject to: a ij are resource requirements b i are available resources c j are cost coefficients c x + c x + Kc x 1 1 n n {,, } {,, } a x + a x + K+ a x = b n n 1 a x + a x + K+ a x = b 1 1 n n { } a 1x1 + a x + K+ a x,, = b m m mn n m 0 x, j = 1, K, n j Objective Constraints Bounds where and x a, b, c are the model arameters 1, ij j x K x n j are the decision variables. MG Lisett,

12 Examle of Formulating An LP Problem Examle A steel comany must decide how to allocate roduction time on a rolling mill. The mill takes unfinished slabs of steel as inut and can roduce either of two roducts: bands and coils. The roducts come off the mill at different rates and have different rofitability: Tons/ Profit/ hour ton Bands 00 $5 Coils 140 $30 The weekly roduction that can be justified based on current and forecast orders are: Maximum tons: Bands 6,000 Coils 4,000 from, R. Fourer, D. Gay, B. Kernighan, AMPL, Boyd & Fraser, 1993, MG Lisett, Formulating LP Problems () The question facing the comany: If 40 hours of roduction time are available, how many tons of bands and coils should be roduced to bring the greatest rofit? Constructing the verbal model: Put the objective and constraints into words. For constraints, use the form: {a verbal descrition of the LHS} {a relationshi} {an RHS constant} MG Lisett,

13 Formulating LP Problems (3) Define the decision variables: x B number of tons of bands roduced. x C number of tons of coils roduced. Construct the symbolic model: The equations that define constraints are based on exressions for elements in the system. For each constraint, there is some function of the decision variables that cannot be exceeded. These functions are generally based on loo and node admissibility relationshis, where some element (or elements) associated with the exression have limits. The equation is then exressed as an inequality to show the feasible sace of solutions for that exression, and the sensitivity of the different variables. MG Lisett, Examle Your comany roduces wrenches and liers out of steel using an injection moulding machine and an assembly machine. Given the following data on raw materials, machine hours, and demand, how many wrenches and/or liers should you roduce? Steel required: 1.5 lbs/wrench, 1.0 lb/liers, lbs available Moulding machine: 1.0 hr/wrench, 1.0 hr/liers, hrs available Assembly machine: 0.4 hrs/wrench, 0.5 hrs/liers, hrs available Demand: wrenches, liers Profit: $0.40 er wrench, $0.30 er liers Adated from: James Orlin, MIT, 003 MG Lisett,

14 In-Class Examle Solution First ste: identify the arameters whose values are under our control aka the decision variables # of wrenches manufactured = w # of liers manufactured = Second ste: determine the goal of the roblem, which gives us the objective function we want to roduce a mix of wrenches and liers to give us the greatest ossible rofit. Third ste: determine the limitations or restrictions on the decision variables we only have lbs of steel, available hours on the moulding machine, available hours on the assembly machine, and demand for only wrenches and liers MG Lisett, In-Class Examle Solution () Objective Function Subject to Steel: Moulding: Assembly: Wrench demand: Pliers demand: Adated from: James Orlin, MIT, 003 MG Lisett,

15 In-Class Examle Alternate Formulation Objective Function max : rofit = 400 w Subject to Steel: 1.5 w Moulding: 1.0 w Assembly: Wrench demand: Pliers demand: 0.4 w w 8 10 Non-negativity:, w 0 Adated from: James Orlin, MIT, 003 MG Lisett, Solving an LP Grahically 14 Start by lotting the two decision variables on the x and y axes w Adated from: James Orlin, MIT, 003 MG Lisett,

16 Solving an LP Grahically () 14 Then begin lotting the constraints, one at a time w Adated from: James Orlin, MIT, 003 MG Lisett, Solving an LP Grahically (3) w Adated from: James Orlin, MIT, 003 MG Lisett,

17 Solving an LP Grahically (4) w Adated from: James Orlin, MIT, 003 MG Lisett, Solving an LP Grahically (5) w Adated from: James Orlin, MIT, 003 MG Lisett,

18 Solving an LP Grahically (6) Having defined (and in this case grahed) the constraints, we have defined the feasible region of solutions. But how do we find the otimal solution for the objective function? w Adated from: James Orlin, MIT, 003 MG Lisett, Solving an LP Grahically (7) 14 1 Is there a feasible solution that gives a rofit of, say, $100? = 400 w The set of all solutions with the same value objective function defines the isorofit line in a maximization roblem and the isocost line in a minimization roblem. So we ve shown the isorofit line for a rofit of $ w Adated from: James Orlin, MIT, 003 MG Lisett,

19 Solving an LP Grahically (8) 14 What about $400? = 400 w w Adated from: James Orlin, MIT, 003 MG Lisett, Solving an LP Grahically (9) 14 Or $3600? = 400 w Notice what is haening. What is the largest feasible rofit going to be? w Adated from: James Orlin, MIT, 003 MG Lisett,

20 Solving an LP Grahically (10) The largest feasible rofit is going to be defined by the highest rofit curve that intersects the feasible region w Adated from: James Orlin, MIT, 003 MG Lisett, Solving an LP Grahically (11) In this case, the otimal solution is: w = 50 / 7 = = 30 / 7 = rofit = $ Why? Otimal solutions occur at corner oints. In two dimensions, this is the intersection of lines: 1.5w + = w = 5 w Adated from: James Orlin, MIT, 003 MG Lisett,

21 Solving an LP Grahically (1) Notice that there are two constraints in articular that intersect with the otimal solution these are called binding constraints. The other constraints are called non- binding constraints. How can you identify a binding constraint without the grah? LHS = RHS at the otimal solution What do binding constraints mean with regard to the formulation? w Adated from: James Orlin, MIT, 003 MG Lisett, LP Otimal Solutions If an otimal solution exists, it occurs at a corner oint of the feasible region. In two dimensions with all inequality constraints lotted, a corner oint is a solution at which two (or more) constraints are binding. If a feasible solution exists, there is always an otimal solution that is a corner oint solution. An otimal solution isn t necessarily unique There may be multile solutions deending on the roblem Unique Otimal Solution Alternate Otimal Solutions Adated from: James Orlin, MIT, 003 MG Lisett, No Feasible Solution Unbounded Otimal Solution 1

22 Oerations Research History 1947: George Dantzig Project Scoo Develoed the simlex method for linear rograms, an algorithmic aroach to incrementally seeking the otimal state 1950's: Lots of excitement, mathematical develoments queuing theory, mathematical rogramming 1960's: More excitement, more develoment & grand lans 1970's: Disaointment, and a settling down NP-comleteness More realistic exectations Adated from: James Orlin, MIT, 003 MG Lisett, Oerations Research History () 1980's: Widesread availability of ersonal comuters Increasingly easy access to data Widesread willingness of managers to use models 1990's: Imroved use of O.R. systems and technology otimisation and simulation add-ins to sreadsheets modeling languages large scale otimisation more intermixing of A.I. and O.R. 1 st century: Lots of oortunities widesread availability of data (sometimes even the right kinds) need for increased coordination and resource efficiency automated decision making: decision-suort systems Adated from: James Orlin, MIT, 003 MG Lisett,

23 Today: Otimisation is Everywhere Otimisation concets are embedded in our language and the way we think about things around us. maximize value to shareholders make the best choices the highest quality at the lowest rice the best strategy when laying games otimize the use of our time, money, resources MG Lisett, Today: Otimisation Otimisation is Everywhere () Mathematic otimisation has become commonlace Product design System design Process and facility design Production Planning Oerations management Logistics (suly chain, transortation) Finance Marketing E-business Telecommunications Games MG Lisett,

24 Some Otimisation Success Stories American Airlines Otimal crew scheduling saves $0 million/yr Yellow Freight Imroved shiment routing saves over $17.3 million/yr Reynolds Metals Imroved truck disatching imroves on-time delivery and reduces freight cost by $7 million/yr GTE Local caacity exansion saves $30 million/yr Digital Equiment Otimising global suly chains saves over $300 million Proctor and Gamble Restructuring North America oerations reduces lants by 0% and saves $00 million/yr Adated from: James Orlin, MIT, 003 and Hillier & Lieberman, McGraw-Hill, 003 MG Lisett, Some Otimisation Otimisation Success Stories () Hanshin Exressway in Osaka Otimal traffic control saves 17 million driver hours/yr A southern ower comany Otimal scheduling of hydro and thermal generating saves $140 million Sadia (Brazil) Imroved roduction lanning saves $50 million over three years Tata Steel (India) Otimised resonse to ower shortages saves $73 million San Francisco Police Deartment Otimising olice atrol officer scheduling saves $11 million/yr Harris Cororation Production otimisation imroves on-time deliveries from 75% to 90% Adated from: James Orlin, MIT, 003 and Hillier & Lieberman, McGraw-Hill, 003 MG Lisett,

25 Some Otimisation Texaco Otimisation Success Stories (3) Gasoline blending results in saving of over $30 million/yr Netherlands Rijkswaterstaat Otimisation of water management olicies saves $15 million/yr Citgo Petroleum Otimising refinery oeration, suly distribution, and marketing saves $70 million/yr IBM Integration of sare arts inventories reduces inventory by $50 million and saves $0 million/yr Otimisation of suly chain saved $750 million in 1 st year Delta Airlines Otimal flight scheduling saves $100 million/yr Adated from: James Orlin, MIT, 003 and Hillier & Lieberman, McGraw-Hill, 003 MG Lisett, Some Otimisation Otimisation Success Stories (4) South African Defence Force Otimising force structure, weaon systems saves $1.1 billion/yr Taco Bell Otimal emloyee scheduling saves $13 million/yr AT&T PC-based system for call centre design saves $750 million/yr New Haven Health Deartment Otimal needle exchange rogram reduces HIV infections 33% China Otimal selection/scheduling of national strategic energy rojects saves $45 million/yr Sears/Roebuck Otimal delivery/service vehicle routing and scheduling saves $4 million/yr Adated from: James Orlin, MIT, 003 and Hillier & Lieberman, McGraw-Hill, 003 MG Lisett,

26 Reference Hazelrigg, G.A. Systems Engineering: An Aroach to Information-Based Design. Prentice-Hall MG Lisett,