Lot-sizing and scheduling with energy constraints and costs

Journée P2LS "Lot-sizing dans l'industrie"LPI6 Paris20 Juin 2014

Grigori German, Claude Lepape, Chloé Desdouits

• Dealing with energy constraints and costs

• Scheduling versus lot-sizing

• A case study of manufacturing scheduling with energy costs

• Lot-sizing perspectives

Agenda

Energy constraints and costs

Introduction

Test DataTest Data

Test DataTest Data

DayNight Time

Cost

• Determine whether it is worth considering energy costs in the planning and scheduling of a given factory or workshop

• Determine what kinds of tradeoffs are worth considering between energy and:• Intermediate or final product inventory• Work shift organization• Other production costs• Tardiness risks• …

• Determine what kinds of models and techniques can be used to answer the questions above • Process simulation• Scheduling with energy costs• Scheduling with energy (power) constraints, i.e., do not exceed a given

power limit• Lot-sizing• …

• Determine how generic can such models and techniques be?

Objectives

• Production planning and scheduling taking into account given energy tariffs• Reducing energy-intensive production during high-cost days and hours• Can mean different things: producing less, producing less energy-intensive

products, avoiding energy-intensive steps, during the high-cost days and hours• Often impacting indirect CO2 emissions

• Selecting or negotiating a better contract based on the energy-aware planning and scheduling capability• In particular concerning power subscription levels and penalties

• Identifying demand-response opportunities• Maintaining a higher stock level to be able to reduce power consumption under

rather short notice• When demand-response is “likely”

Several questions

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An example: the Sarel plant

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Measuring chain

Energy sensor• Self powered• Wireless communication• Non intrusive installation

Accumulator• Provide the Energy value

Collector transmitter• Send historical data

periodically to the time series repository

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Simulation Optimization

Based on a commercial production flow simulator (Rockwell Arena )

Optimization

Input DayNight Time

Cost

Constraints

Objectives

Scheduling versus lot sizing

• What are the time scales?• Duration for the execution of a recipe or

of its critical activities• Versus the frequency of tariff changes

• What is the relationship between the critical resources time-wise and the critical resources energy-wise?

• Do I have batch sizing flexibility and can it impact energy consumption?• Ovens, etc.• Energy-consuming setups / cleaning steps

Scheduling versus lot-sizing: differentiating questions

Time

Cost

Time

Cost

M1 M2

M1 M2

• To exploit batch sizing flexibility

• As an abstraction of the scheduling problem• Less variables• Easiest constraints • …

• As a tool to decompose the scheduling problem

• Depending on the plant, coupled lot-sizing and scheduling can be the best solution

Three motivations for lot-sizing

A case study of manufacturing scheduling with energy costs

Overview of the scheduling problem

Adding the energydimension

act3

time

Resr

capr

st et

calendar

capacity cost interval

capacitycalendar interval

cmax

cminact1

act2

Optimization

Optimization

Input DayNight Time

Cost

Constraints

Objectives

• Simple, classical formulation

• Branching strategy: Earliest Due Date

• No simple formulation for computing the energy cost• Time-based formulation• Perspective: global constraint

• Generates a good first solution

Method 1: Constraint Programming

• Overlap

• Variables

Method 2: MIPHow to express the energy cost?

Taille du bucket act dépasse à gauche

Durée de actact dépasse à droite

act et le bucket sont disjoints

• Constraints

Method 2: MIPHow to express the energy cost?

• Other constraints and variables• Disjunctive constraints: Applegate and Cook (1991) formulation

• Relaxed MIP• Too many variables and constraints (e.g., 700k+ variables and 1.2M+

constraints with 200 activities and a 400 days horizon)• Energy binary variables continuous in [0,1]• Stills leads CPLEX towards a good solution

• Perspectives• Explore different strategies (e.g., branch on all the variables before the

energy variables)• Other formulations with precomputed intervals

Method 2: MIP

• Algorithm

• Perspectives• Adapted time windows size• Sliding time windows• Intensification

Method 3: Hybrid local search

Constraint Programming

S

Local search• While there is still time

• Find a time window F• Set all the variables outside F

•

• Keep the best between S and S’

Optimize F with MIP

S’

S

• Adapted benchmark instances from the literature

• CP, MIP & LS versus best known results

CP MIP LS

All instances (38)= Best known results 25 30 26

Relative deviation 20% 47% 7%

NCGS (20 instances)= Best known results 14 18 14

Relative deviation 30% 0% 11%

NCOS (18 instances)= Best known results 11 12 12

Relative deviation 8% 99% 3%

CP MIP LS

All instances (38)= Best known results 25 30 26

Relative deviation 20% 4% 7%

NCGS (20 instances)= Best known results 14 18 14

Relative deviation 30% 0% 11%

NCOS (18 instances)= Best known results 11 12 12

Relative deviation 8% 8% 3%

Comparison of the 3 methods without the energy

• MIP versus LS

• Local search with and without energy

And with energy ?

MIP

All instances (34)≤ Local search 9

Relative deviation 14%

NCGS (18 instances)≤ Local search 3

Relative deviation 0%

NCOS (16 instances)≤ Local search 6

Relative deviation 31%

Objectives Savings

All instances (29)

Tardiness 0%

Energy -0,95%

Total cost -0,12%

• Application to the SAREL use case

• Multi-objectives: Pareto-optimal schedules

• Piecewise linear energy costs

Scheduling perspectives

Lot-sizing perspectives

• Makes sense only if recipe or critical activity execution duration is smaller than tariff intervals duration

• Recipe-based model• Quantity of recipe r executed in period p for each period p and recipe r• Linked to the energy consumption in period p and hence to the energy cost

(with a linear or piecewise linear relation between consumption and cost – could be subtle in some cases, e.g., if several resources in parallel consume and there is a penalty for exceeding a given amount of power …)

• Linked to quantity of materials produced (or consumed) in period p• Linked to customer demands in different ways: either (i) no tardiness

authorized with the risk that there is no solution, or (ii) delivering the demand when ready, either early or late, or (iii) delivering either just in time or late …

• As a result linked to an evaluation of storage and tardiness costs• Activity-based model• For relevant activities of given batches, deciding in which period they execute• Variation (relaxation) of the model used in our scheduling study• With subtleties to look at when there are multiple energy-relevant activities or

if the energy-relevant activity is not the bottleneck time-wise …

Models with lot-sizing periods corresponding to tariff intervals (buckets)

• Assuming recipe or critical activity execution duration is smaller than the lot-sizing period • An open question is how to approximate the energy cost• An optimistic viewpoint

assumes that inside each period we will be able to exploit intervals with the lowest tariffs, up to some given maximal power

• Can we use historical data to better evaluate an expected cost?

• Shall we do this through some smart coupling of lot-sizing and scheduling?

Models with lot-sizing periods exceeding or not consistent with tariff intervals

Energy

Cost 1 week periodMax power = 10kW88 hours at 0.05€/kWh80 hours at 0.10€/kWh

(0, 0)

(880, 44)

(1680, 124)

Energy

Cost 1 week periodMax power = 10kW88 hours at 0.05€/kWh80 hours at 0.10€/kWh

(0, 0)

(880, 44)

(1680, 124)

• Energy cost reduction is a growing concern• But usually one among multiple optimization criteria

• Multiple technical approaches and models can be considered

• Lot-sizing is one of them• Depending on time scales, relationships between the critical resources time-

wise and the critical resources energy-wise, and on batch sizing flexibility• Sometimes (often) to be coupled with detailed scheduling

A very open topic at this point

Conclusion

Thank you for your attention!