Rolling Horizon Approach for Optimal Production-Distribution Coordination of Industrial Gases Supply-chainsst but o Coo d at o o dust a Gases Supp y c a s

Miguel Zamarripa, Pablo A. Marchetti, Ignacio E. GrossmannDepartment of Chemical Engineering

Carnegie Mellon UniversityPittsburgh, PA 15213

Lauren Cook Tong Li Tejinder Singh Jean AndréLauren Cook, Tong Li, Tejinder Singh, Jean AndréDelaware Research and Technology Center

American Air Liquide Inc.Newark, DE 19702

Center for Advanced Process Decision-making Enterprise Wide Optimization Meeting, September 18-19, 2014 1

Background and MotivationgDeterministic models Consider fixed data (based on future projections, forecast data, etc.) Demand, market prices, process reliability, etc. Exact data is unavailable or very expensive Easier to solve

Uncertainty Management Realistic decision making (much more expensive to solve) Require efficient solution method for deterministic problem in order to tackle the problem under uncertainty

Develop a Robust decision making model to improve the optimalMain Goal

Develop a Robust decision making model to improve the optimal production, inventory and distribution coordination under

uncertainty2

Main Goal

Deterministic Production Distribution

f I d t i l G

Reduce the computational effort (A l i L

Production Distribution of Industrial Gases

S l Ch iof Industrial Gases Supply Chains Management

(Analyzing Large-scale optimization

techniques)

Supply Chains Management under

uncertainty

Robust Optimization (worst case) No recourse actions (short term problems)( p ) All decisions are here and now Some parameters are affected by an uncertainty set (simplest case upper and lower bounds)

Two Stage Stochastic programming Special attention to the size of the problemp p Discrete scenarios (based on probability distribution)

Deterministic Production Distribution

Reduce the computational effort

of Industrial Gases Supply Chains Management

(Analyzing Large-scale optimization

techniques)

Rolling Horizon Solves the problem in a sequence of iterations, in each iteration part of Solves the problem in a sequence of iterations, in each iteration part of the planning horizon will be modeled in detail (“detailed time block”) and the rest of the time horizon is represented in an aggregated manner (“the aggregated time block”).

Problem Statement and Main AssumptionspGiven

Plants, Products, Operating Modes and Production LimitsC1 D1 C3

Daily Electricity Prices (off-peak and peak) Customers and their demand/consumption profiles Max/Min inventory at production sites and customer

locations Alternative sources and product availabilities

Plant 1

D1

p Depots, Truck availabilities and capacities,

Distances Fixed Planning Horizon (usually 1-2 weeks)

Decisions in each time period t Modes and production rates at each plant

C2C4

Modes and production rates at each plant Inventory level at customer location and plants How much product to be delivered to each customer

through which routeObjective Function Plant 2

D2C5

Minimize total production and distribution cost over planning horizon

Main Assumptions – Distribution Side Two time periods per day (peak and off-peak) are

consideredAlternative

D3

considered Trucks do not visit more than 4 customers in a single

delivery

5

Source

Rolling Horizon approach ),(min yxTotalCostFull

spaceg pp

Tt,,

TtS,s ..

1

MmSsDyCx

BxAxts

ts

ts

ts

ts

The planning horizon in detail (“the detailed time block”),

Rest of the horizon is represented in an aggregate manner (“the aggregate time block”).

t1 t2 T… Detailed block

A t bl k

t1 t2 T…

Aggregate block Fixed variables

t1 t2 Tt3 …

6

RH Relaxed model The planning horizon in detail (“the detailed time block”), Rest of the horizon is represented in an aggregate manner (“the aggregate time block”).

A) Relaxed model (binary variables are replaced by continuous variables [0,1]

DETAILED BLOCK Customer PlantsDepots

B) Aggregate model (Distribution side constraints are replaced by a tailored distribution model)

Detailed Distribution

PDC model

Detailed P d ti

sets

s3

s2

D1

k

P1truck

tipD ,, kptE

Main binary variables

Production

s4

k1

k2

k3kptL

variables

 pmtB

mode selection s5P2D2

Relaxed variables (option a)Y RH(k source time) whether or not truck is loaded at

7

Production modeSame production modeMaterial Balances

• Y_RH(k,source,time) whether or not truck is loaded at source at time period t

• y_route_RH(k,r,time) whether or not truck k delivers to route r (a customer set) in time period t

Rolling Horizon approachg pp The planning horizon in detail (“the detailed time block”), Rest of the horizon is represented in an aggregate manner (“the aggregate time block”).

A) Relaxed model (binary variables are replaced by continuous variables [0,1]

Aggregate BLOCK Aggregate Distribution model (option b)

B) Aggregate model (Distribution side constraints are replaced by a tailored distribution model)

Customer sets s2

PlantsDepotsDetailed Production

model (option b)

s3

D1P1 tupDist ,,

Main binary variables

mode selection k∞

s5

s4

D2

 pmtB

k∞

Production mode

8

s5

P2D2

Same production modeMaterial Balances

Products are shipped directly from plant to customer (disregarding the route generation)

Rolling Horizon (illustrative example iteration 2)

t3 t5 t14…t4 t6 t7t1 t2Fixed:

Yk,source,time, B_onp,m,time,

DetailedDetailed Distribution constraints

Aggregate time blockDetailed time blockY_routek,r,time

R ap Relaxed variables (continuous 0—1)

Customer sets

s

s3

s2

D1

k1

PlantsDepots

P1truck

tipD ,,

k2

kptEMain binary

variablesmode selection

Detailed Productione

lax

pproa

• Y_RH(k,source,time) whether or not truck is loaded at source at time period t

• y_route_RH(k,r,time) whether or not truck k delivers to route r (a

s5

4

P2D2

k3 pmtB

mode selection

Production modeSame production modeMaterial Balances

kptL

Aggregate Distribution modelA

ed c

h

truck k delivers to route r (a customer set) in time period t

Customer sets

s3

s2

D1

PlantsDepots

P1 tupDist ,,Main binary

variables

Detailed Production

Aggregate Distribution model

k

Aggre

appro

9s5

s4

P2D2

variables

 pmtB

mode selectionk∞

Production modeSame production modeMaterial Balances

gate

oach

Rolling Horizon (illustrative example iteration 2)

t3 t5 t14…t4 t6 t7t1 t2Fixed:

Yk,source,time, B_onp,m,time,

Detailed Distribution constraints Aggregate Distribution model

Aggregate time blockDetailed time blockY_routek,r,time

Ag a

Customer sets

s3

s2

D1

k1

PlantsDepots

P1truck

tipD ,, kptEMain binary

variables

Detailed Production

Customer sets

s3

s2

D1

PlantsDepots

P1 tupDist ,,Main binary

variablesmode selection

Detailed Production

k∞

ggreg

ppro

s5

s4

P2D2

k2

k3

variables

 pmtB

mode selection

Production modeSame production modeMaterial Balances

kptLs5

s4

P2D2

 pmtB

mode selection

Production modeSame production modeMaterial Balances

gate

ach

Aggregate model (original):• Infinite number of trucks• Time window for demand

Aggregate model (v2):• Infinite number of trucks• Demand (fixed one time period)

10

Time window for demand satisfaction

• Travel distance calculations• Distribution cost

Demand (fixed one time period)• Travel distance calculations• Distribution cost• Minimum truck load

Industrial Size Test Case – Results 4 Plants / Depots 2 products (LIN, LOX) 2 3 d i d f h l

19 sources 2-3 production modes for each plant 15 alternative sources 229 customers

10,032 possible routes reduced to 1,900 by the route generation algorithmgeneration algorithm

11

14 time periods (peak and off-peak) 46 trucks (25 for LIN, 12 for LOX) Min/max inventory, distances, electricity prices, truck deliveries, etc.

Industrial Size Test Case – Results 4 Plants / Depots 2 products (LIN, LOX) 2 3 production modes for each plant

19 sources 2-3 production modes for each plant 15 alternative sources 229 customers 14 time periods (peak and off-peak)

10,032 possible routes reduced to 1,900 by the route generation algorithmp (p p )

46 trucks (25 for LIN, 12 for LOX) Min/max inventory, distances, electricity prices, truck deliveries, etc.

g

RH M2: 7 times faster!

3 22 %

RH Aggregate: 21 times faster!

4.73 % worse

Original RH RM2 Aggregate Agg v2

Model Size

Binary variables 26,486 ‐ ‐ ‐Continuous variables 69,902 ‐ ‐ ‐

3.22 % worse

RH Agg_v2:

200 times faster!SizeConstraints 35,233 ‐ ‐ ‐Total cost

(normalized) 100 103.22 104.73 103.35

200 times faster!

3.3 % worse

CPU results

Time 41,511s 1,297 368s 204sNodes 104,880 1,817 936 900

Relative gap 4.7% 1.9% 1% 1% 12

12 hrs PDC vs RH RM2Total Cost Original model vs RH

RH_aggregate model

Total Cost C t ti l ff t l ti Original Model (12 hrs) 100

Original Model (3hr) 102.21 RH agg 104.73 RH RM2 103 22Total Cost Original model vs RH

Computational effort vs solution efficiency: RH RM2 solution is the closest to the PDC full-space solution, while Aggregate models are much more faster

RH RM2 103.22 RH agg_v2 103.35

gRM2

gg g(with higher penalties for sub optimality).

13

Rolling horizon approach Decompose the full-space problem in a series of iterations. Detailed time block Aggregate time block

Full

RH(TH)

t

RH(t2)

space

Tota

l Cos

t

Under estimation = lower bound

RH(t1) RH problems under

estimate the t t l t

14

total cost

Remarks

Proposed framework provides optimal production and distributioncoordination reducing the computational effortcoordination reducing the computational effort

Proposed rolling horizon approach (both relaxed and aggregate models) arecomputationally faster than the simultaneous approach with a small penalty forsub-optimality.

Aggregate model changes the search in the Branch and Bound method, while in theoriginal model symmetric results impact the solution.

Alternative sources have been widely used by the Rolling horizon approach(comp1, comp6 and comp10; in time periods t1-t4, t7, t9-t13). Otherwise, in theoriginal model were narrowly used (comp7, comp10, and comp 13 used in timeperiods t7, t11, and t13).

Work under development:

15

Receding horizon: we consider that the information after iteration 1 could be used as decision making, and then explore the next time period as a new RH problem. Hybrid decomposition approach.