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

Stochastic optimization for the crude oil procurement problem Thomas Martin, Michel De Lara Cermics 29/03/2019

Overview of crude oil procurement

Overview of crude oil procurement

Overview of the oil supply chain

In this presentation we aim at ◮ providing a mathematical representation of 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

Outline of the presentation Mathematical modeling including uncertainties Formulation and resolution of optimization problems Extensions

Outline of the presentation Mathematical modeling including uncertainties Formulation and resolution of optimization problems Extensions

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

Oil flows inside the refinery Now we need to include stocks dynamics and delays blue: controls bold : random variables

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 End ... w 1 w 2 w 3 w 4 w 54 w 55 week1 week2 week3 week54 The activity of the refinery during week i is summed up at point w i

Stocks dynamics during week [ w , w + 1[ u w p w (Refinery settings) 1 b w c = ( q w c , qual w ... c ) balance p w function P (Crude oil bought) (Products) k w s w = ( s w a w = ( a w qt , s w qt , a w ql ) ql ) (Consumption) mix (Starting stock) function s w +1 = a w qt − k w qt (Stock at the end) = s w qt + q w c − k w s w +1 = a w ql ql (Stocks dynamics) Controls on the system

Chronology of oil purchase, shipping and delivery Order Shipping Delivery w − τ c w − f c w f c Time τ c

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

Chronology of costs appearance Delivery Order Shipping Prime w − τ c Ship w − f c Ref w c c c w − τ c w − f c w 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) Cost w c = Prime w − τ c + Ref w c c

We introduce prices, costs and earnings (Shipping cost) P rice w 1 (Oil cost) ... Ship w − f c P rice w c P (Earnings) sales P rime w − τ c p w c 1 u w upstream process Ref w (Settings) c p w balance P d w − τ c b w function c c (Decision to run w (Running costs) buy crude c) s w = ( s w a w = ( a w qt , s w qt , a w ql ) ql ) k w (Consumption) mix (Starting stock) function green: prices red: costs and earnings

We compute the total costs over week [ w , w + 1[ Costs = (Oil cost) + (Shipping cost) + (Running cost) − (Earnings) Costs w = � d w − τ c [ q c ( Pr w − τ c + R w c )] (Oil cost) c c c ∈ Crudes + d w − τ c S w − f c (Shipping cost) c c + r w (Running cost) � p w i P w − (Earnings) i i ∈ Products

We present the full model scheme (Shipping cost) P rice w 1 ... (Oil cost) Ship w − f c P rice w c P (Earnings) sales P rime w − τ c p w c 1 u w upstream process Ref w (Settings) c balance p w P d w − τ c b w function c c (Decision to run w (Running costs) buy crude c) s w = ( s w a w = ( a w qt , s w ql ) qt , a w ql ) k w (Consumption) mix (Starting stock) function s w +1 = a w qt − k w (Stock at the end) qt = s w qt + q w c − k w s w +1 = a w ql ql (Stocks dynamics)

Outline of the presentation Mathematical modeling including uncertainties Formulation and resolution of optimization problems Extensions

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

We choose one crude among five

Deterministic approach Deterministic approach to the situation arg min Valuation c c ∈C s . t Valuation c = V ( s , b c , Pr c , Ref c , S c , { P i } i ∈P ) ◮ Only uses a reference scenario (for Pr , Ref , S , P ) ◮ V is an all-in-one valuation function ◮ Enumerating all Valuation c naturally builds a ranking

V breakdown V is a black box takes in returns (over a week) ◮ price scenario ( Pr , Ref , S , P ) ◮ valuation ◮ starting stock ( s = ( s qt , s ql )) ◮ consumption ◮ crude ( b c = ( q c , qual c )) ◮ 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

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

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)

We represent the chronology of decisions by a tree

Stochastic optimization problem over week [ w , w + 1[ � � � d w − τ c q c ( Pr w − τ c + Ref w c ) + S w − f c � p w P w i + r w � � �� min − E c c c i { d w − τ c } c ∈C c c ∈C i ∈P ( k w , { p w i } i ∈P , r w ) = balance ( a w qt , a w ql , u w ) s . t ( a w ql , a w qt ) = mix ( s w qt , s w ql , { d w − τ c , q c , qual c } c ∈C ) c � d w − τ c = 1 c c ∈C d w − τ c ∈ { 0 , 1 } c σ ( d w − τ c ) ⊂ F w − τ c c

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

We consider stochasticity in { P w i } i ∈P only � � � d w − τ c q c ( Pr w − τ c c ) + S w − f c � p w P w i + r w � � + Ref w �� min − E c c c i { d w − τ c } c ∈C c c ∈C i ∈P ( k w , { p w i } i ∈P , r w ) = balance ( a w qt , a w ql , u w ) s . t ( a w ql , a w qt ) = mix ( s w qt , s w ql , { d w − τ c , q c , qual c } c ∈C ) c � d w − τ c = 1 c c ∈C d w − τ c ∈ { 0 , 1 } c σ ( d w − τ c ) ⊂ F w − τ c c

We consider different price scenarios Product scenario 0 scenario 1 scenario 2 scenario 3 scenario 4 · · · scenario 9 product 1 882,6 732,6 882,6 882,6 882,6 · · · 882,6 product 2 424,7 424,7 344,7 424,7 424,7 · · · 424,7 product 3 360,8 360,8 360,8 290,8 360,8 · · · 360,8 388,7 388,7 388,7 388,7 338,7 · · · 388,7 product 4 431,7 431,7 431,7 431,7 431,7 · · · 431,7 product 5 product 6 418,6 418,6 418,6 418,6 418,6 · · · 418,6 product 7 434,1 434,1 434,1 434,1 434,1 · · · 434,1 product 8 440,6 440,6 440,6 440,6 440,6 · · · 440,6 product 9 485,5 485,5 485,5 485,5 485,5 · · · 485,5 product 10 515 515 515 515 515 · · · 515 product 11 487,1 487,1 487,1 487,1 487,1 · · · 487,1 362,7 362,7 362,7 362,7 362,7 · · · 292,7 product 12 310,8 310,8 310,8 310,8 310,8 · · · 240,8 product 13 ◮ One reference scenario ( used in the deterministic approach) ◮ 9 alternative scenarios based on the reference

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

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