[PPT] - Stochastic optimization for the crude oil procurement problem PowerPoint Presentation

SLIDE 1

Stochastic optimization for the crude oil procurement problem

Thomas Martin, Michel De Lara

Cermics

29/03/2019

SLIDE 2

Overview of crude oil procurement

SLIDE 3

Overview of crude oil procurement

SLIDE 4

Overview of the oil supply chain

SLIDE 5

In this presentation we aim at

◮ providing a mathematical representation

f the crude oil procurement problem

◮ presenting a deterministic approach in a simple setting ◮ introducing a stochastic optimization approach ◮ comparing the outputs of the two methods over a toy problem

SLIDE 6

Outline of the presentation

Mathematical modeling including uncertainties Formulation and resolution of optimization problems Extensions

SLIDE 7

Outline of the presentation

Mathematical modeling including uncertainties Formulation and resolution of optimization problems Extensions

SLIDE 8

Outline of the presentation

Mathematical modeling including uncertainties Oil flows and dynamics Prices, costs and decision making Formulation and resolution of optimization problems Deterministic optimization (single refinery over a single week) Stochastic optimization (single refinery over a single week) Numerical applications Extensions

SLIDE 9

Oil flows inside the refinery

Now we need to include stocks dynamics and delays

blue: controls bold: random variables

SLIDE 10

We work with discrete time steps

Refinery activity is continuous while decisions happen once a week ◮ We choose a discrete representation of time ◮ one step = one week Start w1 week1 w2 week2 w3 week3 w4 ... w54 week54 End w55 The activity of the refinery during week i is summed up at point wi

SLIDE 11

Stocks dynamics during week [w, w + 1[

balance mix

pw

1

(Products) uw (Refinery settings) bw

c = (qw c , qualw c )

(Crude oil bought) sw = (sw

qt, sw ql)

aw = (aw

qt, aw ql)

kw sw+1

qt

= aw

qt − kw

sw+1

ql

= aw

ql

= sw

qt + qw c − kw

pw

P

... (Starting stock) (Stock at the end) (Stocks dynamics) (Consumption)

function function

Controls on the system

SLIDE 12

Chronology of oil purchase, shipping and delivery

Order w − τc fc Shipping w − fc τc Delivery w Time

SLIDE 13

Outline of the presentation

Mathematical modeling including uncertainties Oil flows and dynamics Prices, costs and decision making Formulation and resolution of optimization problems Deterministic optimization (single refinery over a single week) Stochastic optimization (single refinery over a single week) Numerical applications Extensions

SLIDE 14

Chronology of costs appearance

Order Primew−τc

c

w − fc Shipping Shipw−fc

c

w − τc w Delivery Ref w

c

Time The cost of a crude is twofold: ◮ the prime (negotiated with the producer and that we observe) ◮ the price of the reference crude (e.g. Brent) Costw

c = Primew−τc c

+ Ref w

c

SLIDE 15

We introduce prices, costs and earnings

balance mix

pw

1

uw (Settings) bw

c

sw = (sw

qt, sw ql)

aw = (aw

qt, aw ql)

kw pw

P

(Starting stock)

upstream

Shipw−fc

c

P rimew−τc

c

dw−τc

c

Ref w

c

(Shipping cost) (Oil cost) P ricew

1

P ricew

P

... (Earnings) (Running costs) runw (Decision to buy crude c) (Consumption)

function function process sales

green: prices red: costs and earnings

SLIDE 16

We compute the total costs over week [w, w + 1[

Costs = (Oil cost) + (Shipping cost) + (Running cost) − (Earnings) Costsw =

c∈Crudes

dw−τc

c

[qc(Prw−τc

c

+ Rw

c )]

(Oil cost) + dw−τc

c

Sw−fc

c

(Shipping cost) + rw (Running cost) −

i∈Products

pw

i Pw i

(Earnings)

SLIDE 17

We present the full model scheme

balance mix

pw

1

uw (Settings) bw

c

sw = (sw

qt, sw ql)

aw = (aw

qt, aw ql)

kw sw+1

qt

= aw

qt − kw

sw+1

ql

= aw

ql

= sw

qt + qw c − kw

pw

P

(Starting stock) (Stock at the end) (Stocks dynamics)

upstream

Shipw−fc

c

P rimew−τc

c

dw−τc

c

Ref w

c

(Shipping cost) (Oil cost) P ricew

1

P ricew

P

... (Earnings) (Running costs) runw (Decision to buy crude c) (Consumption)

function function process sales

SLIDE 18

Outline of the presentation

Mathematical modeling including uncertainties Formulation and resolution of optimization problems Extensions

SLIDE 19

Outline of the presentation

Mathematical modeling including uncertainties Oil flows and dynamics Prices, costs and decision making Formulation and resolution of optimization problems Deterministic optimization (single refinery over a single week) Stochastic optimization (single refinery over a single week) Numerical applications Extensions

SLIDE 20

We choose one crude among five

SLIDE 21

Deterministic approach

Deterministic approach to the situation arg min

c∈C

Valuationc s.t Valuationc = V(s, bc, Prc, Refc, Sc, {Pi}i∈P) ◮ Only uses a reference scenario (for Pr, Ref , S, P) ◮ V is an all-in-one valuation function ◮ Enumerating all Valuationc naturally builds a ranking

SLIDE 22

V breakdown

V is a black box takes in ◮ price scenario (Pr, Ref , S, P) ◮ starting stock (s = (sqt, sql)) ◮ crude (bc = (qc, qualc)) returns (over a week) ◮ valuation ◮ consumption ◮ running costs ◮ products models crudes mixing → (”mix” function) all stages of oil transformation → (”balance” function) products sale → (quantities × prices) summation of all costs → (total cost) production constraints

SLIDE 23

Outline of the presentation

Mathematical modeling including uncertainties Oil flows and dynamics Prices, costs and decision making Formulation and resolution of optimization problems Deterministic optimization (single refinery over a single week) Stochastic optimization (single refinery over a single week) Numerical applications Extensions

SLIDE 24

Prices are now random processes

The prices are now represented by random processes whose value is observed at some point in time (also valid for products and shipping)

SLIDE 25

We represent the chronology of decisions by a tree

SLIDE 26

Stochastic optimization problem over week [w, w + 1[

min

{d w−τc

c

}c∈C

E

c∈C

d w−τc

c

qc (Pr w−τc

c

+ Ref w

c ) + Sw−fc c

−
i∈P

pw

i

Pw

i + r w

s.t

(kw, {pw

i }i∈P, r w) = balance(aw qt, aw ql, uw)

(aw

ql, aw qt) = mix(sw qt, sw ql, {d w−τc c

, qc, qualc}c∈C)

c∈C

d w−τc

c

= 1 d w−τc

c

∈ {0, 1} σ(d w−τc

c

) ⊂ Fw−τc

SLIDE 27

Outline of the presentation

Mathematical modeling including uncertainties Oil flows and dynamics Prices, costs and decision making Formulation and resolution of optimization problems Deterministic optimization (single refinery over a single week) Stochastic optimization (single refinery over a single week) Numerical applications Extensions

SLIDE 28

We consider stochasticity in {Pw

i }i∈P only

min

{d w−τc

c

}c∈C

E

c∈C

d w−τc

c

qc (Prw−τc

c

+ Ref w

c ) + Sw−fc c

−
i∈P

pw

i

Pw

i + r w

s.t

(kw, {pw

i }i∈P, r w) = balance(aw qt, aw ql, uw)

(aw

ql, aw qt) = mix(sw qt, sw ql, {d w−τc c

, qc, qualc}c∈C)

c∈C

d w−τc

c

= 1 d w−τc

c

∈ {0, 1} σ(d w−τc

c

) ⊂ Fw−τc

SLIDE 29

We consider different price scenarios

Product scenario 0 scenario 1 scenario 2 scenario 3 scenario 4 · · · scenario 9 product1 882,6 732,6 882,6 882,6 882,6 · · · 882,6 product2 424,7 424,7 344,7 424,7 424,7 · · · 424,7 product3 360,8 360,8 360,8 290,8 360,8 · · · 360,8 product4 388,7 388,7 388,7 388,7 338,7 · · · 388,7 product5 431,7 431,7 431,7 431,7 431,7 · · · 431,7 product6 418,6 418,6 418,6 418,6 418,6 · · · 418,6 product7 434,1 434,1 434,1 434,1 434,1 · · · 434,1 product8 440,6 440,6 440,6 440,6 440,6 · · · 440,6 product9 485,5 485,5 485,5 485,5 485,5 · · · 485,5 product10 515 515 515 515 515 · · · 515 product11 487,1 487,1 487,1 487,1 487,1 · · · 487,1 product12 362,7 362,7 362,7 362,7 362,7 · · · 292,7 product13 310,8 310,8 310,8 310,8 310,8 · · · 240,8

◮ One reference scenario ( used in the deterministic approach) ◮ 9 alternative scenarios based on the reference

SLIDE 30

Valuations under different scenarios

crude scenario 0 scenario 1 scenario 2 scenario 3 scenario 4 · · · scenario 9 1 18853 · · · 2 18786 17737 18622 18747 15265 · · · 17700 . . . 17 19769 18674 19621 19703 16087 · · · 18990 18 18994 18842 15387 19 18447 17464 18259 18452 17541 20 20697 19609 20526 20621 17260 · · · 18737 21 18726 17647 18522 18644 14981 · · · 17793 . . .

Each cell is a valuation for the corresponding crude and product prices scenario ◮ 24 crudes ◮ 10 price scenarios (including the reference), costs are fixed ◮ No access to balance and mix

SLIDE 31

The deterministic method considers a single scenario

Deterministic result:

scenario ranking scenario 0 20 17 18 10 3 21 19 6 scenario 1 20 17 3 21 19 11 6 14 scenario 2 20 17 18 3 21 19 6 14 scenario 3 20 17 1 3 21 19 6 14 scenario 4 20 17 10 18 3 21 6 13 scenario 5 20 17 10 3 21 19 6 14 scenario 6 20 17 2 18 10 3 19 21 scenario 7 20 17 3 1 21 13 12 9 scenario 8 20 17 1 3 21 19 6 14 scenario 9 17 20 10 21 3 19 13 14

SLIDE 32

We present variations using Total’s method

Deterministic result

scenario ranking scenario 0 20 17 18 10 3 21 19 6 scenario 1 20 17 3 21 19 11 6 14 scenario 2 20 17 18 3 21 19 6 14 scenario 3 20 17 1 3 21 19 6 14 scenario 4 20 17 10 18 3 21 6 13 scenario 5 20 17 10 3 21 19 6 14 scenario 6 20 17 2 18 10 3 19 21 scenario 7 20 17 3 1 21 13 12 9 scenario 8 20 17 1 3 21 19 6 14 scenario 9 17 20 10 21 3 19 13 14

◮ Changes in the scenario change the ranking

SLIDE 33

Stochastic approach answer

distribution scen0 1 2 3 4 5 6 7 8 9 equal 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 ref centered 0.55 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 balanced 0.2 0.2 0.2 0.05 0.05 0.05 0.05 0.05 0.05 0.1

Probability distribution of scenarios Stochastic approach result: 20 has the best average for the constant distribution

SLIDE 34

Stochastic optimization output

distribution scen0 1 2 3 4 5 6 7 8 9 equal 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 ref centered 0.55 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 balanced 0.2 0.2 0.2 0.05 0.05 0.05 0.05 0.05 0.05 0.1

Probability distribution of scenarios Stochastic approach result: 20 has the best average for every distribution

SLIDE 35

We discuss the stability of both approaches

Stochastic answer vs deterministic ranking:

scenario ranking scenario 0 20 17 18 10 3 21 19 6 scenario 1 20 17 3 21 19 11 6 14 scenario 2 20 17 18 3 21 19 6 14 scenario 3 20 17 1 3 21 19 6 14 scenario 4 20 17 10 18 3 21 6 13 scenario 5 20 17 10 3 21 19 6 14 scenario 6 20 17 2 18 10 3 19 21 scenario 7 20 17 3 1 21 13 12 9 scenario 8 20 17 1 3 21 19 6 14 scenario 9 17 20 10 21 3 19 13 14 distribution equal 20 17 3 21 13 9 12 24 ref centered 20 17 3 21 13 12 9 24 balanced 20 17 3 21 13 9 12 24

The stochastic approach exhibits a better stability to perturbations.

SLIDE 36

We discuss the stability of both approaches

Artificial stochastic ranking vs deterministic ranking

scenario ranking scenario 0 20 17 18 10 3 21 19 6 scenario 1 20 17 3 21 19 11 6 14 scenario 2 20 17 18 3 21 19 6 14 scenario 3 20 17 1 3 21 19 6 14 scenario 4 20 17 10 18 3 21 6 13 scenario 5 20 17 10 3 21 19 6 14 scenario 6 20 17 2 18 10 3 19 21 scenario 7 20 17 3 1 21 13 12 9 scenario 8 20 17 1 3 21 19 6 14 scenario 9 17 20 10 21 3 19 13 14 distribution equal 20 17 3 21 13 9 12 24 ref centered 20 17 3 21 13 12 9 24 balanced 20 17 3 21 13 9 12 24

The stochastic approach exhibits better stability to perturbations.

SLIDE 37

Outline of the presentation

Mathematical modeling including uncertainties Formulation and resolution of optimization problems Extensions

SLIDE 38

We need to consider stochasticity in the shipping and buying costs

min

{d w−τc

c

}c∈C

E

c∈C

d w−τc

c

qc (Pr w−τc

c

+ Ref w

c ) + Sw−fc c

−
i∈P

pw

i

Pw

i + r w

s.t

(kw, {pw

i }i∈P, runw) = balance(aw qt, aw ql, u)

(aw

ql, aw qt) = mix(sw qt, sw ql, {d w−τc c

, qc, qualc}c∈C)

c∈C

d w−τc

c

= 1 d w−τc

c

∈ {0, 1} σ(d w−τc

c

) ⊂ Fw−τc

balance and mix can be first modeled as linear functions

SLIDE 39

We want to run a refinery for two weeks

SLIDE 40

We write the corresponding optimization problem

min

{d t−τc

c

}c∈C,t∈[w,w+1]

E

t∈T c∈C

d t−τc

c

qc (Pr t−τc

c

+ Rt

c) + St−fc c

−
i∈P

pt

i Pt i + runt

s.t

(cw, {pw

i }p∈P, runw) = balance(aw qt, aw ql, uw)

(cw+1, {pw+1

i

}p∈P, runw+1) = balance(aw+1

qt

, aw+1

ql

, uw+1) (aw

ql, aw qt) = mix(sw qt, sw ql, {d w c , qw c , qualw c }c∈C)

(aw+1

ql

, aw+1

qt

) = mix(sw+1

qt

, sw+1

ql

, {d w+1

c

, qw+1

c

, qualw+1

c

}c∈C) sw+1

qt

= aw

qt − cw

sw+1

ql

= aw

ql

c∈C

d t−τc

c

= 1 d w

c ∈ {0, 1}

d w+1

c

∈ {0, 1} σ(d t

c ) ⊂ Ft

SLIDE 41

Conclusion

We have

◮ created a model that encompasses upstream supply chain and the processing of crude oil ◮ identified sources of stochasticity as well as control variables ◮ formulated a stochastic optimization problem ◮ exhibited the superior stability of the stochastic approach