FOUNDATIONS OF OPERATIONS RESEARCH

Transcription

1 Master of Science in Computer Engineering FOUNDATIONS OF OPERATIONS RESEARCH Edoardo Amaldi DEI - Politecnico di Milano amaldi@elet.polimi.it Course website:

2 INTRODUCTION Operations Research is an interdisciplinary branch of applied mathematics in which mathematical models and advanced quantitative methods such as optimization methods are used to analyze complex decision-making problems and arrive at optimal or near-optimal solutions. The overall goal is to help make better decisions At the intersection of applied mathematics, computer science, economics and industrial engineering 1

3 Decision-making problems Problems in which we have to choose a (feasible) solution among a large number of alternatives based on one or several criteria Examples: 1) Assignment problem: Given m tasks (jobs) and m engineers (machines), suppose that each task can be executed by any engineer and that we know the time t ij engineer E j needs to execute task T i. E 1 E 2 E 3 T T T matrix of execution times (m = 3) Decide which task to assign to each engineer so as to minimize the total execution time. Each task must be assigned to exactly one engineer, and each engineer must be assigned exactly one task. Number of alternatives? 2

4 Decision-making problems Problems in which we have to choose a (feasible) solution among a large number of alternatives based on one or several criteria Examples: 1) Assignment problem: Given m tasks (jobs) and m engineers (machines), suppose that each task can be executed by any engineer and that we know the time t ij engineer E j needs to execute task T i. E 1 E 2 E 3 T T T matrix of execution times (m = 3) Decide which task to assign to each engineer so as to minimize the total execution time. Each task must be assigned to exactly one engineer, and each engineer must be assigned exactly one task. Number of alternatives? m! possible assignments (= number of permutations of the m tasks) 3

5 2) Network design: Decide how to connect n cities (offices) via a collection of possible links so as to minimze the total link cost. Given the graph G = (V, E) with a node for each city and an edge for each link with its associated cost, select a subset of the edges of minium total cost which guarantees that each pair of nodes are connected Number of feasible solutions? 4

6 2) Network design: Decide how to connect n cities (offices) via a collection of possible links so as to minimze the total link cost. Given the graph G = (V, E) with a node for each city and an edge for each link with its associated cost, select a subset of the edges of minium total cost which guarantees that each pair of nodes are connected Number of feasible solutions? 2 n(n 1) 5

7 2) Network design: Decide how to connect n cities (offices) via a collection of possible links so as to minimze the total link cost. Given the graph G = (V, E) with a node for each city and an edge for each link with its associated cost, select a subset of the edges of minium total cost which guarantees that each pair of nodes are connected Number of feasible solutions? 2 n(n 1) 3) Personnel scheduling: Determine the week schedule for the hospital personnel so as to meet the daily requirements while minimizing the number of people involved 4) Decide how many service counters/desks to open at a given time of the day so that the average customer waiting time does not exceed a certain value (guarantee a given service quality) 5) Decide which laptop to buy considering the price, the weight and the performances Complex decision-making problems that are tackled via a mathematical modelling approach (mathematical models, algorithms and computer implementations) 6

8 Historical sketch Roots: first attempts to use a scientific approach to tackle decision-making problems Origin in World War II: teams of scientists helped improve the way the scarce resources were allocated to various military operations Examples: radar location, manage convoy operations,... Operations Research teams were asked to do research on the most efficient way to conduct the operations. In the decades after the war, the techniques began to be applied more widely to problems in business, industry and society During the industrial boom, the substantial increase in the size of the companies and organizations gave rise to more complex decision-making problems 7

9 Favourable circumstances: Fast progress in Operations Research and Numerical Analysis methodologies Advent and diffusion of computers (available computing power and widespread software) Operations Research = Management Science 8

10 Main features of decision-making problems number of decision makers (who decide?) number of objectives (based on which criteria?) level of uncertainty in the parameters (based on which information?) Several decision makers Game theory Several objectives Multi-objective programming Uncertainty level > 0 Stochastic programming 9

11 Example: Production planning A company produces three types of electronic devices. Main phases of the production process: assembly, refinement and quality control Time in seconds required for each phase and product: D 1 D 2 D 3 Assembly Refinement Quality control Available resources whithin the planning horizion (depend on the workforce) expressed in seconds: Unitary profit expressed in KEuro: Assembly Refinement Quality control D 1 D 2 D Suppose the company can sell whathever it produces Give a mathematical model for determining a production plan which maximizes the total profit. 10

12 Production planning model Decision variables: x j = number of devices D j produced for j = 1, 2, 3 Objective function: max z = 1.6x 1 + 1x 2 + 2x 3 Constraints: production capacity limit for each phase 80x x x x x x x x x (assembly) (refinement) (quality control) Non-negative variables: x 1, x 2, x 3 0 can take fractional (real) values 11

13 Scheme of an O.R. study Main steps: define the problem construct a model devise or select an appropriate method develop or use an efficient implementation analyze the results with feedbacks (the previous steps are reconsidered whenever useful) A model is a simplified representation of reality To define a model we need to identify the fundamental elements of the problem and the main relationships among them 12

14 Example: Portfolio selection problem An insurance company must decide which investements to select out of a given set of possible assets (stocks, bonds, options, warrants, gold certificates, real estate,...) Investements area capital [ c j ] (KEuro) expected retrun [ r j ] A (automotive) Germany % B (automotive) Italy 150 9% C (ICT) U.S.A % D (ICT) Italy % E (real estate) Italy 125 8% F (real estate) France 100 7% G (short term treasury bonds) Italy 50 3% H (long term treasury bonds) UK 80 5% Available capital = 600 KEuro At most 5 investements to avoid excessive fragmentation Geographic diversification: 3 investements in Italy and 3 abroad Give a mathematical model for selecting the investements to be included in the portfolio so as to maximize the expected return while satisfying the constraints aiming at limiting the risk. 13

15 Portfolio selection model Decision variables: x j = 1 if j-th investement is selected and x j = 0 otherwise for j = 1,..., 8 Objective function: Constraints: 8 max z = c j r j x j j=1 8 c j x j 600 (capital) j=1 8 x j 5 (max 5 investements) j=1 x 2 + x 4 + x 5 + x 7 3 x 1 + x 3 + x 6 + x 8 3 (max 3 in Italy) (max 3 abroad) Binary (integer) variables: x j {0, 1} j, 1 j 8 14

16 To limit risk we may require that if an ICT investement is selected also a treasury bond must be selected. Model: 8 max z = c j r j x j j=1 8 c j x j 600 (capital) j=1 8 x j 5 (max 5 investements) j=1 x 2 + x 4 + x 5 + x 7 3 x 1 + x 3 + x 6 + x 8 3 (max 3 in Italy) (max 3 abroad) x 3 + x (x 7 + x 8 ) (invest. treasury bonds) x j {0, 1} j, 1 j 8 15

17 Example: Assignment problem The manager of a consulting group must assign m = 3 tasks to m consultants based on the estimates of the time each consultant needs to execute each task: C 1 C 2 C 3 T T T matrix of execution times t ij, 1 i, j 3 Each task must be assigned to exactly one consultant, and each consultant must be assigned exactly one task. Give a mathematical model for deciding which task to assign to each consultant so as to minimize the total completion time. 16

18 Assignment model Decision variables: x ij = 1 if job i is assigned to the j-th consultant and x ij = 0 otherwise i, j with 1 i, j 3 Objective function: min z = 3 3 t ij x ij Constraints: i=1 j=1 3 x ij = 1 j = 1, 2, 3 (exactly one task for each consultant ) i=1 3 x ij = 1 j=1 i = 1, 2, 3 (each task i to exactly one consultant) Binary (integer) variables: x ij {0, 1} i, j, 1 i, j 3 17

19 Example: Facility location Three oil pits, located in positions A, B and C, from which oil is extracted. Decide where to locate the refinery so as to minimize the total pipeline cost (which is proportional to the square of the length). The refinery must be at least 100 km away from point D = (100, 200), but the oil pipelines can cross the corresponding forbidden zone. 18

20 Facility location model Decision variables: x 1, x 2 cartesian coordinates of the refinery Objective function: min z = [ (x 1 0) 2 + (x 2 0) 2] [ + (x 1 300) 2 + (x 2 0) 2] + [(x 1 240) 2 + (x 2 300) 2] Constraints: (x 1 100) 2 + (x 2 200) Variables: x 1, x 2 R 19

21 Mathematical Programming (MP) Decision-making problems with a single decision maker, a single objective and reliable parameter estimates opt f (x) with x X decision variables x R n : numerical variables whose values identify a solution of the problem feasible region X R n distinguish between feasible and infeasible solutions (via the constraints) objective function f : X R expresses in quatitative terms the value or cost of each feasible solution opt = min or opt = max NB: max{f(x) : x X} = min{ f(x) : x X} 20

22 Global optima Solving a mathematical programming (MP) problem amounts to finding a feasible solution which is globally optimum, i.e., a vector x X such that f (x ) f (x) x X if opt = min f (x ) f (x) x X if opt = max It may happen that: 1. the problem is infeasible (X = ) 2. the problem is unbounded ( c R, x c X such that f (x c ) c or f (x c ) c) 3. the problem has a huge number, even an infinite number, of optimal solutions (with the same optimal value!) 21

23 Local optima When the problem at hand is very hard we must settle for a feasible solution that is a local optimum, i.e., a vector ˆx X such that f (ˆx) f (x) x with x X and x ˆx ǫ if opt = min f (ˆx) f (x) x with x X and x ˆx ǫ if opt = max for an appropriate value ǫ > 0 An optimization problem can have many local optima! 22

24 Special cases of MP Linear Programming (LP) f (x) linear X = {x R n : g i (x) R 0, i = 1,..., m} with R {=,, } and g i (x) linear for each i Example: Production planning Integer Linear Programming (ILP) f (x) linear X = {x R n : g i (x) R 0, i = 1,..., m} Z n with R {=,, } and g i (x) linear for each i Examples: Portfolio selection, task assignment ILP coincide with LP with the additonal integrality constraints on the variables Non Linear Programming (NLP) f (x) smooth X subset of R n defined by the smooth surfaces {x R n : g i (x) = 0} (with smooth g i (x)) Example: Facility location 23

25 Multi-objective programming Multiple objectives can be taken into account in different ways Suppose we wish to minimize f 1 (x) and maximize f 2 (x) (laptop example: f 1 is cost and f 2 is performance) 1. without priority: reduce to the case with a single objective function by expressing the two objectives in terms of the same unit (e.g. monetary unit) for appropriate scalars λ 1 and λ 2. min λ 1 f 1 (x) λ 2 f 2 (x) 2. with priority: optimize the primary objective function and turn the other objective into a constraint max x X f 2 (x) con X = {x X : f 1 (x) c} for an appropriate constant c. 24

26 Mathematical programming versus simulation Both approaches involve constructing a model and designing an algorithm Mathematical programming Simulation Problem can be well formalized (example: assignment) Algorithm yields a(n optimal) solution The results are certain Problem is difficult to formalize (example: service counters) Algorithm simulate the evolution of the real system and allows to evaluate the performance of alternative solutions The results need to be interpreted 25

27 The impact of Operations Research Some successful applications year company sector results 90 Taco Bell personnel scheduling 7.6M$ annual (fast food) savings 92 American Arlines design fare structure, overbooking + 500M$ and flights coordination 92 Harris Corp. production planning 50% 95% orders (semiconductors) on time 95 GM - car rental use of car park +50M$ per year avoided banckruptcy 96 HP - printers modify production line doubled production 97 Bosques Arauco harvesting logistics 5M$ annual and transport savings 99 IBM supply chain re-engineering 750M$ annual savings 26

28 Most of the top managers interviewed by Fortune 500 declared that they used O.R. methodologies Significant impact not only for large companies and organizations R.O. methodologies help improve the way scarse resources are used In rapidly evolving contexts, characterized by high levels of complexity and uncertainty, it is important to identify and implement efficient solutions The huge amount of data available with modern information systems open new avenues... 27