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Variability and Uncertainty

IntroductionStochastic ProgrammingRobust ProgrammingMultiparametric ProgrammingDynamic Programming

2

Introduction: PSE Chemical Supply Chain

Enterprise management

designplanning

scheduling

Supply Chain

Coordination of resources

EfficiencyProfitability

3

Introduction: Modeling systems in PSE

decision detaillevel

temporal scale

Analytical ITproduction plant

supervisory and local control

distribution scheduling modeling systems

production scheduling modeling systems

(site i)

Operational analysis

Tactical analysis

production planning modeling systems

(site i)

logistics modeling systems

tactical modeling systems

Strategic analysis

strategic modeling systems

variablemarket trends

technologychangesenvironmental

conditions

decisions

Modeling system

uncertainty

reliable

inputinformation

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Introduction: Uncertainty

The model will include a series of uncertain parameters There is not a unique solution… There is not an optimal solution…

In spite of this, a decision should be made (before the uncertainty is revealed)

Solution based on average values (or nominal values) does not need to be the best It could be even unfeasible

Decision making implies a risk Typically, there will be a trade-off among expected efficiency and assumed risk

Prediction is very difficult… specially about the futureNiels Bohr

5

ExecutionExecution

Executed scheduleExecuted schedule

Introduction: Scheduling under uncertainty

Data ambiguousoutdatedincomplete

BREAKDOWN

Dynamic & uncertain environment

Infeasible schedulePoor efficiencyOpportunity losses

Predictive schedulePredictive schedule

Decision makingDecision making

HowWhen

Whereinputinformation

t

6

Introduction: Motivating example

u2

u1

u3

ABDECmk: 101 TU wt: 0 TU

Predictive schedule (nominal)

u2

u1

u3

ABDECmk: 104 TU wt: 19 TU

Executed schedule

u2

u1

u3

Predictive schedule

ADEBCmk: 103 TU wt: 0 TU Is it worth spending effort to obtain a predictive

schedule optimal for nominal conditions that will eventually change at execution time due to disruptions and changes in the operation environment?

reaction

U1

dryingcentrifugation

U3U2

ABCDE

wait times

(Balasubramanian & Grossmann, 2002)

7

Safety measuresSlack time Intermediate storageExtra capacity

Sensitivity analysisProactive scheduling

Proactive approaches

Introduction: Management of the uncertainty

Reactive scheduling Completely reactive scheduling

Predictive - reactive scheduling Modification of operation conditions

Reactive approaches

Ability to achieve robust schedule execution despite the occurrence of unexpected events.

uncertaintyrealization

BREAKDOWN

Predictive schedule

How

When

Where

t

Uncertainty

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Sources of Uncertainty

Unexpected events Ambiguous and/or incomplete information

processing / transport timesyield ratiosresources availabilityresources qualitymodel parametersoperators absenteeismcontrol systemsmisjudgements

Strategic Uncertainty

Tactical Uncertainty

Operational Uncertaintyinformation flowsdue datesmisjudgementsinflexible capacitiesraw materials availabilitymarket demandscancelled / rush orders

environmental conditionstechnology changesmarket parametersregion-specific featuresinternational aspectscompetitorsgovernmental regulationsfinancial issuesclinical trials

temporal scale

decision detaillevel

What will customers order?How many products should remain in stock?Will resources be available during production?Will suppliers deliver the materials on time?

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Formal aspects

The way how the model relates the decission variables with the uncertain information Decoupling these two types of variables makes the

problem much more easy !

The way to express the probabiloity distribution of uncertain varaibles Continuous distributions vs. Discrete distributions.

The size of the problem The characteristics of the mathematical model

(convex, continuous, …) Including the characteristics introduced by the

probability functions to be applied (the probability functions are not convex (nor concave) transformations, disretizations, etc.)

There are different ways to face these problems. The most convenient way depends on aspects like:

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Decision making under Uncertainty

Something has to be deceided “here and now” in spite of the fact that decision results only will be known in the future, once uncertainty is revealed

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Typical decision making with a probabilistic representation of uncertainty

Sampling loop

MODEL (scenario simulation)

Objective function (k)

Scenario (k)Uncertain parameters

STOCHASTIC MODELER

Probabilistic objective function

Decision variables

OPTIMIZER Optimal probabilistic

solution

•Numerical or analytic techniques

Discrete distribution (scenario based)

• Sampling techniques

Continuous probability distribution

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Optimization under uncertainty

Stochastic optimization Objective: Mathematic hope related to an efficiency measurement

Variability is not taken into account

Optimización robusta Compromise between “hope and variability ( Risc)

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1.- Stochastic Programming

Combines the power of mathematical programming with statistical advanced techniques: Convexity analysis, development of dual problems, … problem decomposition Constraints management

The solution procedures are driven by the formalization of the

probability model. For example: When the number of scenarios to be contemplated is discrete

Deterministic equivalent problem (the size of the problem will be basically a function of the number of scenarios to be considered)

General Methods of stochastic programming

Multi-step methods(multistage stochastic programming): Penalization (“recourse problems”): The potential losses caused by uncertainty

are “a posteriori” compensated solving another optimization problem. This second problem depend on the decisions already made (“first step”) and on

the uncertainty finally revealed. The variable associated to this new optimization is the “resource” required to

compensate the uncertainty effects. During the first step decision making, the eventual (uncertain) values of the

second stage variables are explicitly considered Limitations:

Not applicable if the decisions (and states) affect the probability distribution

The costs associated to the “resource decisions” should be known in advance.

Chance-constrained programming: some constraints do not need to be maintained in a limited number of cases.

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Equivalent deterministic problem (finite number of scenarios):

Min c'x + p1 d1'y1 + p2 d2'y2 + p3 d3'y3 s.t. A x = b T x + W y1 = h1

T x + W y2 = h2

T x + W y3 = h3

x, yi ≥ 0 

(x: 1st stage variables(yi: 2nd stage variables(pi: probability of scenario “i”       

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Stochastic Programming: Example

Management of a local gas distribution company (http://wiki.mcs.anl.gov/NEOS/index.php/Stochastic_Programming):

Data: Gas demand for the current period (100 units), buying price (1 €/unit) and selling price for the current period (2 €/unit). Storage cost (0.05 €/unit)

Uncertain data: Gas demand for the next period, and buying/selling prices Uncertainty model: 3 equiprobable scenarios:

Decision: How much gas should be bought “here and now” the the global operators (taking into account that gas can be stored and consumed during the next period)

Solution alternatives: Solve the problem for each scenario Solve the problem for the average scenario Stochastic programming

Scenario Probability Gas cost (€/unit) Demand (units)

Normal 1/3 5.0 100

Cold 1/3 6.0 150

Very cold 1/3 7.5 180

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Equivalent deterministic problem (finite number of scenarios):

Min c'x + p1 d1'y1 + p2 d2'y2 + p3 d3'y3 s.t. A x = b T x + W y1 = h1

T x + W y2 = h2

T x + W y3 = h3

x, yi ≥ 0

x: gas to be bought (and/or store)yi: gas to be bought during the second year (for each scenario)di: price of the gas to be bought during the second year (for each scenario)b: demand (in the first year)T, A y c’: parameters to ensure that the extra quantity bought during the 1st year will be stored

(cost!) and will be used to cover the 2nd year demand (or part of it)hi: demand for the second year (for each scenario)pi: probability of scenario “i”       

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Mathematical formulation

     One unique scenarioMinimize (sum Price[t] * Buy[t]) + (sum Store[t] * StorageCost) subject to: Buy[t] + Use_stored[t] >= Demand[t], t in (1,2) Store[t] = Store[t-1] + Buy[t] - Demand[t], t in (1,2) Use_stored[t] <= Store[t-1], t in (1,2)

“n” scenariosMinimize (Price[1][1] * Buy[1][1] + sum Store[1][1] * StorageCost) + + sum Scen Prob[Scen] * (Price[2][Scen] * Buy[2][Scen] + Store[2][Scen] * StorageCost) First-period constraints Buy[1][1] + Use_stored[1][1] >= Demand[1][1] Store[1] = Store[0] + Buy[1] - Demand[1] Use_stored[1] <= Store[0] Second-period constraints Buy[2][Scen] + Use_stored[2][Scen] >= Demand[2][Scen] Store[2][Scen] = Store[1][1] + Buy[2][Scen] - Demand[1][Scen]

Use_stored[2][Scen] <= Store[1][1]

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Excel solution

     

1000.0

1400.0

1800.0

2200.0

50 100 150 200 250 300

Coste total esperado

coste total mínimo

coste total medio

coste total máximo

Mínimo coste esperado: 1400, con una compra de 200Mínimo "coste mínimo": 1000, con una compra de 100 (+100 como mínimo en el escenario "normal")Mínimo "coste máximo": 1595, con una compra de 270 (+10 como máximo en el escenario "muy frío")Mínimo "coste de escenario medio": 1360, con una compra de 243.33 (100+143.33 = D2)

“Chance constrained” programming

Equivalent to stochastic programming, including a new constraint to limit the probability of not matching a certain constraint to a certain maximum value.

Examples: Same problem including a constraint to limit the probability of having to buy more than 30 units in the second periodAccepting solutions no covering the 10% of the second year demand in a 5% of the eventual situations

Helps to assess the “parameters” probability distribution

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2.- Robust programming

The model includes a series of “i” unknow parameters (si); although their value is unknown, they should be among a specific set of specific values (Si)

Optimization = To look for the best solution which meets all constraints (feasibility) for any parameters value combination

Worst Case Analysis for any eventual decision (The set of scenarions is, actually,i irrelevant

“S”: Set of “s” = uncertain parameters (preferably: continuous and convex)

Variation: To look for the best solution which meets all constraints (feasibility) which a probability less than “alfa” to get a result worst than “zmin” (“downside risk”)

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Robust programming example

Management of a local gas distribution company (http://wiki.mcs.anl.gov/NEOS/index.php/Stochastic_Programming):

Data: Gas demand for the current period (100 units), buying price (1 €/unit) and selling price for the current period (2 €/unit). Storage cost (0.05 €/unit)

Uncertain data: Gas demand for the next period, and buying/selling prices Uncertainty model: 3 equiprobable scenarios:

Decision: How much gas should be bought “here and now” the global operators (taking into account that gas can be stored and consumed during the next period)

Solution alternatives Only one: To determine the optimum gas quantity to be bought in the first period

Objective: maximize retrofit benefit in the worst of the uncertain scenarios ( buy 270 Maximum cost of 1595 in the “very cold” scenario. The expected (average) cost is 1575)

Optimization of the expected cost implies buying 200 units and a expected cost of 1400, but this cost can lead to a cost of 1700 in the very cold scenario

Scenario Probability Cost (€/unit) Demand (units)

Normal 1/3 5.0 100

Cold 1/3 6.0 150

Very cold 1/3 7.5 180

Management of a local gas distribution company (http://wiki.mcs.anl.gov/NEOS/index.php/Stochastic_Programming):

Data: Gas demand for the current period (100 units), buying price (1 €/unit) and selling price for the current period (2 €/unit). Storage cost (0.05 €/unit)

Uncertain data: Gas demand for the next period, and buying/selling prices Uncertainty model: 3 equiprobable scenarios:

Decision: How much gas should be bought “here and now” the the global operators (taking into account that gas can be stored and consumed during the next period)

Solution alternatives: Solve the problem for each scenario Solve the problem for the average scenario Stochastic programming

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3.- Multiparametric programming(Origin: Sensibility analysis)

The model (F.Obj. + Constraints) includes a series of bounded parameters (they can vary among certain limits)

Objective To obtain explicit (exact or approximate) analytical expressions for the F.Obj. and other characteristric variables as a function of the uncertain parameters (and, eventualy, the range of applicability of such expressions)

The determination of the applicability ranges is a major problem

Note 1: Multiparametric programming offers “the” exact mathematic solutionNote 2: Is not usefull for decission making (“previous” to uncertainty is revealed)

Very complex mathematical problem (the numerical solution is not valid….) Multiparametric Linear and Quadratic Programming Multiparametric Nonlinear Programming Multiparametric Mixed-Integer Linear Programming Multiparametric Mixed-Integer Quadratic and Nonlinear Programming Parametric Global Optimization Bilevel and Multilevel Programming

Example (AICHE Journal, 53,12, pg. 3183, 2007 // I&EChR, 37, pg 4341, 1998):Li & Ierapetritou Scheduling with uncertain demand

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Equipo Capacidad Tarea(s) Tiempo Estado Almace -namiento

Cantidad Inicial

Precio

U1 100 Task1 3.0 + 0.0300 B S1 Unlimited Unlimited 0.0

U2 75 Task2 2.0 + 0.0266 B S2 100 0.0 0.0

U3 50 Task3 1.0 + 0.0200 B S3 100 0.0 0.7

S4 Unlimited 0.0 1.0

ParámetroIncierto

Rango

Demand S3 0-50

Demand S4 0-100

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Programa Objetivo Estado

SCH-1 -90.46 CR1

SCH-2 -73.547 + 0.029 D2 CR2

SCH-3 -96.140 - 0.158 D1 CR3 + CR5

SCH-4 88.55 CR4

Ejemplo (AICHE Journal, 53,12, pg. 3183, 2007 // I&EChR, 37, pg 4341, 1998):Li & Ierapetritou Scheduling with uncertain demand

90

80

70

60

50

40

30

20

10

If DS3 < 10 this schedule allows to produce the maximum of S4 (74, if DS3 is 0: reducing the batch size of the second S3 batch allows to advance the second batch of s4, and it is possible to produce more S4 (againts S3)

While DS3 < 36, produce as much as possible of S4 (65)(36 of S3 is the maximum)

While DS4 < 50, produce just a batch of S4 and produce as much as possible of S3 (until the end)

If DS3 >36, this sequence allows to produce the maximum of S3, if DS4 is less than DS3

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4.- Dynamic Stochastic Programming

Transformation of problems under uncertainty in dynamic programming problems

Dynamic programming problem characteristics:The problem can be divided in single decision “steps”Each step has a number of associate states.

Decision in a certain step transforms one step into another in the next step.In a certain state, the optimum decision for the next steps does not depend on the previous decision (the system is like an “state function”)There exist a recursive function to obtain the optimum solution at step j, if step j+1 is already solved.Final step solution is obvious.

Optimality principle (Bellman): “Any subset of decisions sequence from the optimal decision sequence is

an optimum solution of the corresponding subproblem

Knapsack problem: n=3 C=15

(b1,b2,b3)=(38,40,24)

(p1,p2,p3)=(9,6,5)

Strategy: Take first the maximum benefit per unit of weight.

Obtained solution:(x1,x2,x3)=(0,1,1), 64

Optimum solution:(x1,x2,x3)=(1,1,0), 78

(because of the integer nature of the problem)

Dynamic programming example:

Se emplea típicamente para resolver problemas de optimización. Permite resolver problemas mediante una secuencia de decisiones.

Como el esquema voraz

A diferencia del esquema voraz, se producen varias secuencias de decisiones (d1, d2, …, dn) y solo al final se sabe cuál es la mejor de ellas.

Si hay «o» opciones posibles para cada una de las decisiones, una técnica de fuerza bruta exploraría un total de on secuencias posibles de decisiones (explosión combinatoria).

La programación dinámica evita explorar todas las secuencias posibles por medio de la resolución de subproblemas de tamaño creciente y almacenamiento en una tabla de las soluciones óptimas de esos subproblemas para facilitar la solución de los problemas más grandes.

Dynamic programming: Introduction

R. Bellman: Dynamic Programming,

Princeton University Press, 1957.

Dynamic programming methods

Problema: ¿ineficiencia ? Un problema de tamaño N (número de objetos disponibles) se reduce a dos

subproblemas de tamaño (N-1). Cada uno de los dos subproblemas se reduce a otros dos…

Por tanto, se obtiene un algoritmo exponencial.

Sin embargo, el número total de sub-problemas a resolver no es tan grande: Sólo hay NxC problemas diferentes:

Decidir si el objeto n va a la mochila ( N) en un momento determinado El momento (C =peso ya obtenido por decisiones anteriores)

Por tanto, la solución recursiva está generando y resolviendo el mismo problema muchas veces Para evitar la repetición de cálculos, las soluciones de los subproblemas se pueden almacenan en una tabla.

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15p1 9 0 0 0 0 0 0 0 0 0 38 38 38 38 38 38 38p2 6 0 0 0 0 0 0 40 40 40 40 40 40 40 40 40 78p3 5 0 0 0 0 0 24 40 40 40 40 40 64 64 64 64 78

Problema de la mochila: n=3

C=15

(b1,b2,b3)=(38,40,24)

(p1,p2,p3)=(9,6,5)

Dynamic Programming example

TSP Problem

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número devértices

n

tiempofuerza bruta

n!

tiempoprog . dinámica

n2 2n

espacioprog . dinámica

n2n

5 120 800 16010 3628800 102400 1024015 1, 311012 7372800 491520

20 2, 431018 419430400 20971520

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Dynamic stochastic programming

The gas company should now cover the demand in 3 different locations, in competence with other local suppliers. They have bought 300 units of gas previously (at a very good price) They will get a benefit of 2.00 €/unit BUT

If they do not sell it, they should have to sell it to the competence, so the benefit will be reduced to 0.50 €/unit.

If more gas is required, they can buy it to the competence, without benefit. The selling probabilities for the next period are estimated as follows:

Objective: Assign gas to cities to obtain the maximum expected benefit

Location Demand Probability

A 50 / 100 / 150 0.60 / 0.00 / 0.40

B 50 / 100 / 150 0.50 / 0.10 / 0.40

C 50 / 100 / 150 0.40 / 0.30 / 0.30

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Dynamic stochastic programming

Transformation to dynamic programming: The expected benefit to send 100 units to location A is:

0.60 * ( 50*2.0 + 50*0.5 + 00*0.0) + 0.00 * (100*2.0 + 00*0.5 + 00*0.0) + 0.40 * (100*2.0 + 50*0.0 + 50*0.0) = 155 €

In a similar way:

Now we have a deterministic problem, which can be solved by dynamic programming

Location Demand Probability

A 50 / 100 / 150 0.60 / 0.00 / 0.40

B 50 / 100 / 150 0.50 / 0.10 / 0.40

C 50 / 100 / 150 0.40 / 0.30 / 0.30

Location Service Expected value

A 0 / 50 / 100 / 150 0.0 / 100.0 / 155.0 / 210.0

B 0 / 50 / 100 / 150 0.0 / 100.0 / 162.5 / 217.5

C 0 / 50 / 100 / 150 0.0 / 100.0 / 170.0 / 217.5

Decissions to be made (and costs)

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300

150

100

50

0

300

250

200

150

Availability toA+B+C

Availability toB+C

Availability toC

210

0217.5

217.5

0

0

100

170

217.5100

155217.5

217.5

0

100

100162.5

162.5

162.5

Location Service Expected value

A 0 / 50 / 100 / 150 0.0 / 100.0 / 155.0 / 210.0

B 0 / 50 / 100 / 150 0.0 / 100.0 / 162.5 / 217.5

C 0 / 50 / 100 / 150 0.0 / 100.0 / 170.0 / 217.5

300

150

100

50

0

300

250

200

150

Availability forA+B+C

Availability forB+C

Availability forC

150 to A

0 to A

150 to B

150 to B

0

0 to C

50 to C100 to C

150 to C

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Procedure

1 step per location 3rd step: to serve “no more” than 0, 50, 100 or 150 units to C

Depending on how much is served to A and B, availability to C will be limited 2nd step: to serve “no more” than 150, 200, 250 o 300 units to B and C

Depending on how much is served to “A” availability to B and C will be limited 1st step: to serve 300 units to A, B y C.

300

150

100

50

0

300

250

200

150

Availability to A+B+C

Availability to B+C

Availability to C

150 a A

0 a A

150 a B

150 a B

0

0 a C

50 a C100 a C

150 a C

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Procedure

Location Service Expected value

A 0 / 50 / 100 / 150 0.0 / 100.0 / 155.0 / 210.0

B 0 / 50 / 100 / 150 0.0 / 100.0 / 162.5 / 217.5

C 0 / 50 / 100 / 150 0.0 / 100.0 / 170.0 / 217.5

300

150

100

50

0

300

250

200

150

Disponibilidadpara

A+B+CDisponibilidadrestante para

B+C

Disponibilidadrestante para

C

150 a A

0 a A

150 a B

150 a B

0

0 a C

50 a C100 a C

150 a C

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Procedure

UnidB+C

Location Serv ice Expected value

A 0 / 50 / 100 / 150 0.0 / 100.0 / 155.0 / 210.0

B 0 / 50 / 100 / 150 0.0 / 100.0 / 162.5 / 217.5

C 0 / 50 / 100 / 150 0.0 / 100.0 / 170.0 / 217.5

Second step: there are different alternatives take the best for each origin

162.5100

162.5

300

15010050

0

300250200150

Availability to A+B+C

Availability to B+C

Availability to C

210

0217.5

217.5+0 = 217.5

0

0

100

170

217.5100

155217.5

217.5

0+217.5=217.5

100+170=270

162.5+100 = 262.5

300

150

100

50

0

300

250

200

150

Availability toA+B+C

Availability to B+C

Availability toC

150 a A

0 a A

150 a B

150 a B

0

0 a C

50 a C100 a C

150 a C

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Procedure

UnidB+C

UnidC

City Service Expected value

A 0 / 50 / 100 / 150 0.0 / 100.0 / 155.0 / 210.0

B 0 / 50 / 100 / 150 0.0 / 100.0 / 162.5 / 217.5

C 0 / 50 / 100 / 150 0.0 / 100.0 / 170.0 / 217.5

Second step: there are different alternatives take the best for each origin

162.5+170=332.5300

15010050

0

300250200150210

0217.5+217.5=435

0170

217.5

100

155

217.5+170=387.5

100+170=270

300

150

100

50

0

300

250

200

150150 a A

0 a A

150 a B

150 a B

0

0 a C

50 a C100 a C

150 a C

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Procedure

UnidB+CThere are 2 equivalent optima:

50 to A, 150 to B and 100 to C. 487.5 €.100 to A, 100 to B and 100 to C. 487.5 €.

300

150

100

50

0

300

250

200

150210

0217.5

217.5

0

0

100

170

217.5100

155217.5

217.5

0

100

100162.5

162.5

162.5

Location Service Expected value

A 0 / 50 / 100 / 150 0.0 / 100.0 / 155.0 / 210.0

B 0 / 50 / 100 / 150 0.0 / 100.0 / 162.5 / 217.5

C 0 / 50 / 100 / 150 0.0 / 100.0 / 170.0 / 217.5

Thirs step: there are different alternatives take the best for each origin

300

150

100

50

0

300

250

200

150480

435

435

0170

217.5487.5

487.5 332.5

270

387.5

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Limitations: what else ?

Uncertainty modeling Fuzzy sets ? Fuzzy programming ? (principle of “difficulty conservation”?)

Generalization of methodologies (modeling and solution) Multiple sources of uncertainty Multiple objectives ( “trade off”)

Computing efficiency (industrial problem) Uncertainty of uncertainty