Optimization under Uncertainty. with Applications

Transcription

1 with Applications Professor Alexei A. Gaivoronski Department of Industrial Economics and Technology Management Norwegian University of Science and Technology 1

2 Lecture 2 Other sources of uncertainty Cutting plane method Benders Decomposition Example of stochastic program with recourse: introduction of new service Real options and flexibilities in industrial projects Examples from design of telecom networks 2

3 Stochastic problem with recourse: introduction of new service Basic formulation Profit and pricing Evaluation of flexibility, real options MPL implementation Excel implementation 3

4 Computational experience with deterministic equivalents 16 server locations, 16 user locations, 5 demand scenarios 1632 variables, 272 constraints, 96 integer variables solution time 1 min 21 sec 25 server locations, 25 user locations, 6 demand scenarios 4500 variables, 500 constraints, 175 integer variables solution time 1 hour 5 min 3 sec DELL Latitude 810, Pentium III 1130 Mhz, 512 MB MPL modeling system with CPLEX 7.5 solver 4

5 Cutting plane method 5

6 Benders decomposition Why special algorithms for stochastic programming problems? Description of algorithm for stochastic program with recourse 6

7 Case study:introduction of new service Service: internet access through phone line Features: geographical regions, demand is uncertain, costs are uncertain, fixed costs, variable costs, time, competition and substitution between services, relations between different market actors, e.g. network providers and service providers Decisions: Locations of servers, pricing, number of servers in Phase 1 and Phase 2; Strategies: Phase 1 deployment now, look for user response, Phase 2 deployment later; real options: option to expand, option to abandon 7

8 Decision quality, objectives Short time to market Profit Demand satisfaction Cost Total cost for the network and server operations Network operation patterns and customer behavior patterns which can be learned during phase 1, utilize them for phase 2 8

9 Step 1: Simplest case Features: geographical regions, demand is uncertain, costs are uncertain, fixed costs, variable costs, time, competition and substitution between services, relations between different market actors, e.g. network providers and service providers Decisions: Locations of servers, pricing, number of servers in Phase 1 Strategies: phase 1 deployment now, look for user response, phase 2 deployment later; real options: option to expand, option to abandon 9

10 Step 1: Mathematical model Building blocks: regions locations, server Decisions: whether to put server in location - binary variable, amount of service provided by server in location to region Parameters, data: fixed cost for putting server in location, cost for providing a unit of service from location to region, demand for service from region i, server capacity 10

11 Step 1: Mathematical model Objectives and structural relations: total costs Total served demand for region i Total demand served from server in location j 11

12 Step 1: Mathematical model Decision model: find decisions and from minimization of total costs subject to satisfaction of demand and to structural constraints 12

13 Step 2: Introducing uncertainty Features: geographical regions, demand is uncertain, costs are uncertain, fixed costs, variable costs, time, competition and substitution between services, relations between different market actors, e.g. network providers and service providers Decisions: Locations of servers, pricing, number of servers in Phase 1 Strategies: phase 1 deployment now, look for user response, phase 2 deployment later; real options: option to expand, option to abandon 13

14 Step 2: Introducing uncertainty Description of uncertainty demand: scenarios Dependence of decisions on available information Introduction of time. Two decision periods: now and future Make some decisions now and correct them in the future when more information will be available. Decision which we make now should allow decision flexibility and adaptation in the future 14

15 Description of demand uncertainty: scenarios value of demand under scenario r probability (frequency) of scenario r Time structure of decisions: now: placement of servers future correction: assignment of demand to servers when demand scenario r will be known now future 15

16 Model of two stage decision process Decision to take now: find server placement which minimize current placement cost and average future demand service cost where is cost of servicing customers under demand scenario r given server placement y 16

17 Decision to take in the future: Given demand scenario r and server placement y, find the customer assignment which yields the minimal service costs : subject to satisfaction of demand and to structural constraints 17

18 Combining current and future decisions in the same model Find server placement which minimize current placement cost and average future demand service cost 18

19 Step 3: Enriching decision flexibility: option to expand Features: geographical regions, demand is uncertain, costs are uncertain, fixed costs, variable costs, time, competition and substitution between services, relations between different market actors, e.g. network providers and service providers Decisions: Locations of servers, pricing, number of servers in Phase 1 Strategies: phase 1 deployment now, look for user response, phase 2 deployment later; real options: option to expand, option to abandon 19

20 Model of two stage decision process Decision to take now: find server placement and demand assignment which minimize current placement cost and average future demand service cost Discount coefficient 20

21 Decision to take in the future: Given demand scenario r and Phase 1 server placement y, find expansion program for Phase 2 and new demand assignment which yield the minimal expansion and service costs : subject to satisfaction of demand and to structural constraints Admissible number of servers at j 21

22 Combined model Find Phase 1 server placement and current demand assignment which minimize current placement cost and average Phase 2 expansion and demand service cost 22

23 Computational experience with deterministic equivalents 16 server locations, 16 user locations, 5 demand scenarios 1632 variables, 272 constraints, 96 integer variables solution time 1 min 21 sec 25 server locations, 25 user locations, 6 demand scenarios 4500 variables, 500 constraints, 175 integer variables solution time 1 hour 5 min 3 sec DELL Latitude 810, Pentium III 1130 Mhz, 512 MB MPL modeling system with CPLEX 7.5 solver 23

24 Introduction of new service, Part II Summary Important facility location problem Modeling of uncertainty and dynamic decisions robust decision under uncertainty has a structure which can not be recovered from consideration of individual scenarios Interplay between decisions of different scale: tactical and operational decisions; Profit and pricing 24

25 Case study:introduction of new service Service: internet access through phone line Features: geographical regions, demand is uncertain, costs are uncertain, fixed costs, variable costs, time, competition and substitution between services, relations between different market actors, e.g. network providers and service providers Decisions: Locations of servers, pricing, number of servers in Phase 1 and Phase 2; Strategies: Phase 1 deployment now, look for user response, Phase 2 deployment later; real options: option to expand, option to abandon 25

26 Two Phase service deployment Find Phase 1 server placement and current demand assignment which minimize current deployment cost and average Phase 2 expansion and demand service cost costs first stage constraints second stage constraints for scenario r 26

27 Modeling of price and profit: unique service How demand depend on price? Simplest case: linear dependence - reference price h w - incremental price - demand which corresponds to reference price - demand elasticity 27

28 demand Demand vs price f(h) d h price h 0 28

29 Description of demand uncertainty: scenarios base value of demand under scenario r base price under scenario r demand elasticity under scenario r probability (frequency) of scenario r Time structure of decisions: Phase 1 Phase 2 29

30 Two Phase service deployment Find Phase 1 server placement and current demand assignment which maximize expected profit revenue costs first stage constraints second stage constraints for scenario r 30

31 where revenue first stage second stage averaged among scenarios costs first stage second stage averaged among scenarios 31

32 Features Quadratic with respect to continuous variables Binary variables Straightforward linear MIP is not applicable Solution techniques: Benders decomposition Genetic algorithms Iterative fixed point techniques 32

33 Iterative fixed point techniques Idea: substitute one difficult problem with a sequence of simpler ones a.fix binary location variables y, resulting problem is quadratic programming problem in continuous variables, solve it with XPRESS or other solver b. Fix continuous variables to the values just obtained, resulting problem is IP in binary variables, solve it again with XPRESS c. Repeat steps a-b several times 33

34 Profit Dependence of profit on price Price Blue graph: fixed costs doubled 34

35 Competition with other providers Classical efficient market: just charge market price and improve your own efficiency Our situation: we provide service which is different, but not quite Reference service provided by competitors for reference price h 0 Our decision: amount h by which our price differs from reference price Scenarios about future prices of competitors We are back to monopoly case, BUT, more difficult to get scenario data 35

36 Introduction of new service, Part III Decision flexibility and real options 36

37 Evaluation of real options Financial option: right, but not an obligation to buy/sell a specified amount of financial asset (stock) at specified price at some future date Real option: right (possibility) but not an obligation to undertake a specified action directed towards enhancement of project value (value of real investment) hedge risks, increase profitability 37

38 Real options Possibilities to undertake actions if they lead to favourable outcome Hedge against uncertainty, increase profitability Option to expand Option to postpone, wait and see Option to change/update to different use Option to change/upgrade technology 38

39 What is option value? Difference between expected profit without option and with option Important for investment decisions: project which seems unprofitable without considering real options can become profitable if real options are considered 39

40 Example Case 1. No option to expand, all deployment should be made during phase 1 in order to satisfy demand projections Case 2. Option to expand: two phase deployment in response to changing demand Case 3. Option to upgrade technology for Phase 2 deployment 40

41 No option to expand Find Phase 1 server placement and current demand assignment which maximize expected profit revenue costs first stage constraints second stage constraints for scenario r 41

42 where revenue first stage second stage averaged among scenarios costs first stage second stage averaged among scenarios 42

43 Option to upgrade technology Find Phase 1 server placement and current demand assignment which maximize expected profit revenue costs first stage constraints second stage constraints for scenario r 43

44 where revenue first stage second stage averaged among scenarios costs first stage second stage averaged among scenarios 44

45 45