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A Hybrid Decomposition Heuristic UsingGenetic Algorithm and

MIP Tools for Scheduling ChemicalProductiona

Pavel A. Borisovsky, Anton V. EremeevOmsk Branch of Sobolev Institute of Mathematics SB RAS, Omsk, Russia,

Josef KallrathBASF SE, Ludwigshafen, Germany

aTo appear in International Journal of Production Research

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Outline of the Talk

• Production process and optimization problem

• Rolling horizon decomposition approach

• The main units scheduling subproblem

• GA for main units sheduling, its convergence, experiments

• Conclusions

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Production Process

Feed Transfer→Feed Storage→Main Unit→Product Storage→Filling Unit

Example of a single production cycle:

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Production Process

Example of a schedule for three products and two main units

Shaik et al. Computers & Chemical Engineering (2008)

Problem features:• Sequence dependent setup times for the main units• Minimum lot-size requirements• Time windows for some changeovers• Weekend ban for �lling of some products• Strict due dates for MTO products 4


Flow Sheet of a Real-Life Plant

A brief scheme of the production processes in the considered plant.

10 main units (extruders), about 100 units in total, more than 100 �nalproducts over one month time horizon.

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Optimization Criterion

The primary objective is the minimization of the non-delivered amountfor ordered products (underproduction).

Secondary objectives (added with small weights):

• minimization of the deviations from the delivery due dates,

• minimization of the overproduced amounts,

• the number of changeovers,

• the duration of changeovers.

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Problem Complexity

One extruder, arbitrary number of demands: the problem containsthe shortest Hamilton path problem as a special case, which is polynomiallyequivalent to the TSP. Therefore:

• The problem is strongly NP-hard (Garey, Johnson, 1979).

• The problem is NPO-complete, and cannot be approximated withany constant or polynomial factor of the optimum in polynomial time,unless P = NP (Orponen, Mannila, 1987).

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Decomposition Scheme

A. Upper-level decisions (rolling horizon):The whole scheduling period is decomposed into short-term horizons(5-days weekday horizons and 2-days weekend horizons).

B. Lower-level MILP optimization model: Construct a sched-ule in the given short-term horizon.

The decomposition scheme and the MILP model are based on

Shaik, M.A., Floudas, C.A., Kallrath, J., Pitz, H.-J., Productionscheduling of a large-scale industrial continuous plant: short-term andmedium-term scheduling. Computers & Chemical Engineering (2008)

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Flowchart of Decomposition Scheme

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Lower-level optimization model: Constraints

• Allocation of tasks to event points:

One event point contains not more than one task on each unit.

If material is transferred from one task to another, then both tasks arelocated in the same event point.

• Material balance constraints:

The produced amount must �t min and max capacities of a unit.

If material is transferred from one task to another, then producedamount is equal to consumed amount.

• Timing constraints:

Duration of a task depends on its amount.

Tasks on the same unit do not overlap.

If material is transferred from one task to another, the tasks must besynchronized in time.

Sequnce dependent changeovers, week-end truck ban, etc.

MIP is de�ned by > 40 blocks of (in)equalities.Real-life instances: 25000 constraints, 1500 binary, 5500 real variables.10


Core Subproblem: Extruders Scheduling

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Core Subproblem: Extruders Scheduling

S = {1, 2, ..., n} is the set of products;U = {1, 2, ...,m} is the set of main units (extruders);I = {1, 2, ..., l} is the set of tasks;

Ds is the demand for product s;

For each task i ∈ I it is given:ui is the suitable unit (one only)si is the output product (one only)ri is the production rate.Tmini is the minimal possible duration of task i

sij is the setup time if i and j are performed on the same unit

Objective: minimize Makespan

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Model Variables for Extruders Scheduling

Let the set of event points for each unit u be denoted as

Ku := {1, 2, ..., |Iu|},

where Is is the set of tasks executable on u.

Variables:

• xik ∈ {0, 1} such that xik = 1 i� task i is assigned to unit ui atevent point k ∈ Ku.

• τi ≥ 0 is the duration of task i;

• αuk ≥ 0 is the duration of a changeover task on unit u between eventpoints k and (k + 1);

• α0u ≥ 0 is the duration of an initial changeover on unit u.

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MILP for Extruders Scheduling

Min Cmax + P1

∑i∈Iτi + P2

∑u∈U

(α0u +

∑k∈Ku

αuk) (1)∑i∈Iu

xik ≤ 1 ∀u, k ∈ Ku (2)∑k∈Ku

xik ≤ 1 ∀u, i ∈ Iu (3)∑i∈Iu

xi,k−1 ≥∑i∈Iu

xik∀u, k ∈ Ku\{1} (4)∑i:si=s

riτi ≥ Ds ∀s (5)

αuk ≥∑i∈Iu

aijxi,k−1 −M(1− xjk) ∀u, j ∈ Iu, k ∈ Ku\{1}

(6)∑i∈Iu

τi + α0u +

∑k∈K

αuk ≤ Cmax α0u ≥

∑i∈Iu

a0ixi1 ∀u (7)

Tmini

∑k∈K

xik ≤ τi ≤ Tmaxi

∑k∈K

xik ∀i ∈ I. (8)14


Genetic Algorithm for Extruders Scheduling

Representation of solutions

π = (i1, i2, ...il) tasks permutation (not enough!);L = (L1, L2, ..., Lm), where Lu ≤ |Iu| is the number of tasks to beplaced on unit u.

Example:Tasks 1,2,3,4 can be performed on unit 1,Tasks 5,6,7,8,9 can be performed on unit 2.

A possible solution:π = (4, 7, 5, 2, 3, 9, 1, 8, 6)L = (2, 3)

corresponds to the following assignment:

Unit 1: 4,2Unit 2: 7,5,9

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LP Model for Fitness Evaluation

Solution (π, L) de�nes the set of tasks to be performed, and the sequenceof tasks. The changeover times can be calculated straightforwardly. Tocomplete the schedule it is enough to determine the durations of the tasks.

Let I(π, L, s) be the set of tasks producing product s,Ch(π, L, u) be the sum of changeover time on unit u.

Recall that ri is a production rate of task i.Ds is a demand on product s.

Positive variables: τi ≥ 0 is the duration of task i.

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LP Model for Fitness Evaluation

Objective: minimize Makespan

minCmax

Subject to:

1. Demand satisfaction constraint:∑i∈I(π,L,s)

riτi ≥ Ds.

2. Estimation of Makespan:∑i∈Tτi + Ch(π, L, u) ≤ Cmax.

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Scheme of the Genetic Algorithm

1. Generate initial population π(0).2. For t := 1 to tmax do

2.1 π(t) = π(t−1).2.2 Apply selection to choose p1, p2 from π(t).2.3 Apply crossover to p1, p2 to generate new solutions c1 and c2.2.4 Apply mutation: c′1 =M(c1) and c′2 =M(c2).2.5 Repair c′1 and c′2, if infeasible.2.6 Remove two worst solutions q1, q2 from π(t) and add c′1 and c′2.

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GA operators

• Tournament selection: Choose randomly s solutions and return thebest among them.

• Crossover:- The partially mapped crossover (PMX) for the permutation string π.

- The uniform crossover for the substring L.

• Mutation:

- Choose randomly k and l, and exchange ik and il in π.

- Each component Lu, u = 1, ...,m, is incremented with probabil-ity pm, or decremented with probability pm, otherwise no change.

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The Repair Heuristic

The decoded solution can be infeasible if some products can not beproduced (no tasks produce these products due to low values of Lu).

In such a case, the permutation is scanned again, and if the currentoperation i produces a missing product, then i is appended to the sequenceof unit ui. And so on.

Consider the same example as above:Tasks 1,2,3,4 can be performed on unit 1,Tasks 5,6,7,8,9 can be performed on unit 2.

But suppose that there is a product produced by tasks 3 and 8. Again:π = (4, 7, 5, 2, 3, 9, 1, 8, 6) → Unit 1: 4,2L = (2, 3) Unit 2: 7,5,9

The decoded solution is not feasible.

Since task 3 is the leftmost among 3 and 8 in π, the repair operator adds

task 3 to the solution, so we get a feasible solution:Unit 1: 4,2,3Unit 2: 7,5,9 20


Self-Adaptative Usage of Repair Heuristic

• The threshold Lu allows to skip the last |Iu|−Lu tasks for machine uin permutation π, if other machines can substitute u in making of thoseproducts.

• All products with no related tasks in positions 1, . . . ,∑

u∈U Lu ofpermutation π will be scheduled by the Repair Heutistic in a single-batch production mode.

Application of Repair Heuristic is adapted via levels of the genotypestring L.

Consider the extream cases:

L = (0, 0, . . . , 0) ⇒ all products are assigned by Repair Heuristic (sosingle batch mode). In π, only the genes π1, . . . , π|S| matter.

L = (|I1|, |I2|, . . . , |Im|) ⇒ the number of batches for each productequals to the number of machines that can produce it.

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Convergence to the Optimum

Proposition 1. Suppose 0 < pm < 1 and tmax = ∞, then the GAde�ned above �nds an optimum to problem (1)�(8) after a �nite numberof iterations a.s.

Proof. Follows straightforwardly from the result of A. Eiben, E. Aarts andK. van Hee (1989).

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Local Improvement by DynamicProgramming (GA+DP)

Consider the task sequencing subproblem on each main unit alone.

For a set of products P ⊂ S and task i that does not produce anyproducts from P let f(i, P ) be a makespan of an optimal scheduleproducing the set of products {si} ∪ P so that task i is allocated at the�rst place.

Held M., Karp R.M. A dynamic programming approach to sequencing problems // Journ. of the Soc.

for Ind. and Appl. Math. V. 10. - 1962.- P. 196-210.

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GA vs CPLEX on Random Instances

Comparison of the GA with CPLEX in extruders optimization: Cmax andtotal changeovers time (hours).

Instance CPLEX GA GA+DP|S| × |U | × |I| CPU Cmax Ch. Cmax Ch. Cmax Ch.

time time time time1. 30×2×80 300 297 76 288 59 281 482. 40×2×150 600 334 143 307 88 291 573. 50×3×180 900 274 192 251 129 234 754. 60×3×200 1200 327 181 304 114 295 925. 150×10×500 3600 311 675 301 539 281 2906. 300×10×1000 7200 � � 603 2130 502 1026

For all instances, Pr{CGAmax < CCPLEX

max } ≥ 0.83 with con�dence 95%

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GA-Based Short-Term Scheduling

1. Start from empty schedule (�x all binary variables to zero).2. Run the GA to �nd a schedule for extruders.3. For all extrusion tasks i from the GA schedule do:4. Un�x the binary variables related to i and the auxilary tasks.5. Solve the MILP subproblem.5. Fix the binary variables to their values.

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Deands of the Real Life Instance

Articles: 120, Demands: 100Units: 110,

Shaik et al. Computers & Chemical Engineering (2008)

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Genetic Solution and Short-Term Schedule

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Whole schedule for extruders

Extrusion tasks in the solution: 1057All tasks in the solution: 4697Event points used: 211

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Experimental Results on a Real-Life Instance

Criteria Shaik et al Hybrid Hybrid(2008) (GA) (GA+DP)

MTS Underproduction, % 0.96 0.2 0.2Number of changeovers 314 115 116

Total changeovers duration 930 490 383Solving time 16h 6h 5.5h

Increasing the fraction of products with strict deadlines (MTO products)by a factor of 10, we observe even greater superiority of Hybrid (GA+DP):

Shaik et al Hybrid HybridCriteria (2008) (GA) (GA+DP)

MTO Underproduction, % 8.7 0.0 0.0MTS Underproduction, % 1.8 0.64 0.28Number of changeovers 302 127 119

Total changeovers duration 988 509 430Solving time 16h 6h 6h

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Conclusions

A hybrid algorithm was developed to improve the solution quality andreduce the CPU time in continuos production scheduling.

A genetic algorithm (GA) is proposed for auxiliary problem of schedulingthe main production tasks and optimizing the changeovers.

A dynamic programming local improvement procedure is introduced in thegenetic algorithm and implemented on GPU.

The number of changeovers and their duration as well as the CPU timewere reduced by ≈ 60%.

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Thank you for attention!

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