Optimizing Steel Production Scheduling with Energy Consideration

Integrated scheduling of rolling sector in steel production with consideration of energy consumption under time-of-use electricity prices Shengnan Zhao, Lixin Tang. Northeastern University Ignacio Grossmann. Carnegie Mellon University EWO 2017

Background Steel production involves a long process flow with complicated manufacturing techniques We investigate an integrated scheduling of a three-stage process of steel production with consideration of energy consumption under the time-of-use electricity price

Problem Definition Given Sets of jobs with information of : Processing route Processing time at each unit Sequence-dependent changeover costs A set of units at each stage with information of : Processing mode of each unit: batch process or sequential process For batch process: Setup time between adjacent batches Maximum batch capacity Information of energy consumption Unit-dependent electricity consumption per hour Electricity price of a set of discrete time periods

Problem Definition Determine For batch process: Job assignment to the units Batch composition, batch capacity, number of batches at each unit Electricity consumption of each electricity pricing period For sequential process: Coil sequence and timetable at each unit Completion time of the entire production Objective Minimize the total production costs including: Total changeover costs Total rolling facility costs Total energy consumption costs

Proposed MINLP/GDP Model A time-slot based continuous time MINLP model is proposed with GDP constraints Constraints of batch process Assignment and batching: Timing constraints: = = s k s i ' , ,, s k i , sd y s AC k T 1, y i N s , , s k i ac ' ac s AC k T i N s s s k = = + , ,, s k i , nbatch y s AC k T s k , , tf tb sd s AC k T , s k s s k ' ac i N s t s s k 1, + , \ s k r s k r s AC k T + s , , \{ } t tf setup r tb s AC k T , , s + 1 , s k s s k ' tf s s ,, s k r , s k r nbatch nB s AC k T = ' , ,, s k i , ' s tf y i N AC , , max s k s ac i k s AC k T s

Proposed MINLP/GDP Model Constraints of sequential process Allocation constraints: Immediate precedence constraints: = s 1, \ , = , \ , , \{ } t y s S AC i N y , , i j k z s S AC i N k T , , s k i s , , s k i s s s k T j N s s = s 1, \ , = , \ , , \{ } t y s S AC k T y , , i j k z s S AC j N k T , , s k i s , +1, s k j s s s i N i N s s Timing constraints: s s i k , ,, s k i \ , , tb UB y s S AC k T i N s = , \ , tf tf s S AC k T , s s s , i k s k i N s s , ,, s k i \ , , tf UB y s S AC k T i N s s k 1, \ , \{ }, t tf tb s S AC k T i N , i k s s s + s s s k s i tb s s i = , \ , , cd y s S AC k T i N s i k = \ , tb s S AC i N , , s k i s s , s , i k k T s = + s s i k s ,, i k / , , tf tb cd s S AC k T i N s tf s = \ , tf s S AC i N , , i k s s , i k s i k T s s k s i k = , \ , tb tb s S AC k T ' i tf s s i , s , ', ' , tb s S s L i N i N ' s s s

Proposed MINLP/GDP Model Energy consumption constraints = , ,, s k tp T p pwh tp TP Electricity consumption of each time period: tp s s AC k T s = Redundant constraints: , ,, s k tp T , sd s AC k T , s k s tp TP Possible interactions of a slot with a constant time period

Proposed MINLP/GDP Model Energy consumption constraints GDP constraints: , , R R s AC k T , , s k s k s , , s k tp T = 0 , , s k tp A , , s k tp B C , , s k tp s k s k s k s k =0 tb L tp L tp L tp tb cp tb cp tb cp s =0 tf s s s k U tp sd L tp U tp tf cp tf cp tb cp k k k , , s k tp T = s k s s U tp U tp tf cp tf cp k k s s k , , s k tp T = , , s k tp T = L tp U tp tf cp cp tb k D E , , s k tp F , , s k tp , , s k tp s k s k s k L tp L tp U tp tb cp tb cp tb cp , tp TP s s s U tp L tp U tp tf cp tf cp = tf cp = k = k k , , s k tp T , , s k tp T 0 0 U tp L tp , , s k tp T cp cp , , R R s AC k T , , s k s k s , , s k tp B , , s k tp F , , , R , , s k tp A C D E s AC k T tp TP , , , s k tp , , s k tp , , s k tp s k s , , , , , , { , } R , , s k tp A , , s k tp B C D E , , s k tp F True False , , , s k tp , , s k tp , , s k tp s k

Proposed MINLP/GDP Model Objective function = s s buy tp 1 + 2 m + 3 m obj m F z c s k r ce p , roller tp , , i j k ij s S AC i N j N k T s AC k T tp TP / s s s s Estimation of slot number ac = = sn NB The number of slots is estimated by: least nB max Variable bounding The upper bound of timing variables are calculated by: = + ' s i , ', ' UB UB s S s L ' s s s i N ' s

Specified bound Lower bound of total changeover costs We specify a lower bound of the total sequence-dependent changeover costs in the objective function by the underlying cost structure. The lower bound of the total changeover costs is calculated by: = s i s min j i , \ , mc F s S AC i N , i j s = s s s i s i max i N F z BestC mc mc , , i j k ij s s S AC i N j N k T s S AC i N \ \ s s s s

Reformulation Linearization of nonlinear constraints The nonlinear constraints are linearized by the method in (Glover, 1975), s = s Let , then the following constraints are obtained: , ,, s k i , , ' tf y s AC k T i N , i k s ac k ' tf s = s ,, i k , ' I s i AC i s AC k T s s 0 , i k s UB y , , , s k i i k s s s (1 ) tf UB y , , , s k i i k s k s s tf , i k k Glover, F., 1975. Improved Linear Integer Programming Formulations of Nonlinear Integer Problems. Manage. Sci. 22, 455 460. doi:10.1287/mnsc.22.4.455

Reformulation Hull reformulation of GDP constraints s k A B C D E F + + + = + + , , , tb tb tb tb tb tb tb s AC k T tp TP , , s k tp , , s k tp , , s k tp , , s k tp , , s k tp , , s k tp s s A B C D E F + + + = + + , , , tf tf tf tf tf tf tf s AC k T tp TP s , , s k tp , , s k tp , , s k tp , , s k tp , , s k tp , , s k tp k A L tp U tp , ,, s k tp , , cp a tb cp a s AC k T tp TP , , s k tp , , s k tp s A L tp U tp , , , cp a tf cp a s AC k T tp TP , , s k tp , , s k tp s , , s k tp F U tp , ,, s k tp , , cp f tb UB f s AC k T tp TP , , s k tp , , s k tp s s F U tp , , , cp f tf UB f s AC k T tp TP , , s k tp , , s k tp s s , , s k tp + + + + + = , , , a b c d e f s k r s AC k T tp TP , , s k tp , , s k tp , , s k tp , , s k tp , , s k tp , , s k tp , s B C , , s k tp T = + ) ( + s k cp L tp U tp ( ) sd a tf cp , , s k tp b AC k cp c tb , , s k tp , , s k tp , , s k tp TP , , s k tp d + U tp L tp ( ) , , , cp s T tp , , s k tp s

Computational Results Computational statistics of 8 instances Coil number Model size Integrality gap (%) Cost Instance Period CPUs * (103CNY) Discrete variable Continuous variable C1 C2 C3 C4 C5 Constraints 0.91 0.91 1 10 7 4 6 5 4 860 1,944 1,716 114.14 10.80 1.66 1.39 2 10 7 4 6 5 7 970 2,289 2,151 112.80 9.74 5.51 4.98 3 15 10 10 10 5 4 3,163 5,440 3,381 116.46 4.46 31.33 20.72 4 15 10 10 10 5 7 3,307 5,899 3,960 118.11 5.79 66.75 15.98 5 20 15 15 13 6 4 9,132 13,350 5,984 157.28 5.57 50.72 46.44 6 20 15 15 13 6 7 9,312 13,923 6,707 158.78 6.50 1511.28 1181.98 7 25 20 20 20 10 4 24,484 32,277 10,766 210.18 10.97 6172.25 1983.33 8 25 20 20 20 10 7 24,736 33,078 11,777 208.52 10.70 *: CPU seconds for 10-4and 10-2optimality tolerance, respectively.

Computational Results Gantt chart of the optimal solution of Instance 2

Remarks Integrated scheduling of multi-stage production in rolling sector of steel production Schedule optimizes coordination of production and electricity consumption. Novel continuous time mixed-integer linear programming model. Numerical experiments demonstrate effectiveness and tightness of the model.