An integrated approach based on Fuzzy Inference System for scheduling and process planning through multiple objectives

Dariush Mohamadi Zanjirani; Majid Esmaelian

doi:10.3934/jimo.2018202

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May 2020, 16(3): 1235-1259. doi: 10.3934/jimo.2018202

An integrated approach based on Fuzzy Inference System for scheduling and process planning through multiple objectives

Dariush Mohamadi Zanjirani ^, and Majid Esmaelian

Department of Management, University of Isfahan, Isfahan, Iran

* Corresponding author: Dariush Mohamadi Zanjirani

Received October 2017 Revised June 2018 Published May 2020 Early access December 2018

Integrated process planning and scheduling (IPPS) problems are one of the most important flexible planning functions for a job shop manufacturing. In a manufacturing order to produce n jobs (parts) on m machines in a flexible manufacturing environment, an IPPS system intends to generate the process plans for all n parts and the overall job-shop schedule concurrently, with the objective of optimizing a manufacturing objective such as make-span. The optimization of the process planning and scheduling will be applied through an integrated approach based on Fuzzy Inference System (FIS), to provide for flexibilities of the given components and consider the qualitative parameters. The FIS, Constraint Programming (CP) and Simulated Annealing (SA) algorithms are applied in this design. The objectives of the proposed model consist of maximization of processes utility, minimization of make-span and total production costs including costs of flexible tools, machines, process and TADs. The proposed approach indicates that The CP and SA algorithms are able to resolve the IPPS problem with multiple objective functions. The experiments and related results indicate that the CP method outperforms the SA algorithm.

Keywords: Process planning, job shop production, scheduling, fuzzy inference system, simulated annealing, constraint programming.

Mathematics Subject Classification: Primary: 97R40, 90Bxx.

Citation: Dariush Mohamadi Zanjirani, Majid Esmaelian. An integrated approach based on Fuzzy Inference System for scheduling and process planning through multiple objectives. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1235-1259. doi: 10.3934/jimo.2018202

References:

[1]	A. Akgun, E. A. Sezer, H. A. Nefeslioglu, C. Gokceoglu and B. Pradhan, An easy-to-use MATLAB program (MamLand) for the assessment of landslide susceptibility using a Mamdani fuzzy algorithm, Computers & Geosciences, 38 (2012), 23-34. doi: 10.1016/j.cageo.2011.04.012. Google Scholar
[2]	IBM, IBM ILOG CPLEX Optimization Studio 12.5 User's Manual, 2012. Google Scholar
[3]	S. Kirkpatrick, C. Gelatt and M. Vecchi, Optimization by simulated annealing, Science, New Series, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671. Google Scholar
[4]	X. Li, L. Gao, X Shao and C. Wang, Mathematical modeling and evolutionary algorithm-based approach for integrated process planning and scheduling, Computers & Operations Research, 37 (2010), 656-667. doi: 10.1016/j.cor.2009.06.008. Google Scholar
[5]	W. Li and C. A. McMahon, A simulated annealing-based optimization approach for integrated process planning and scheduling, International Journal of Computer Integrated Manufacturing, 20 (2007), 80-95. doi: 10.1080/09511920600667366. Google Scholar
[6]	K. Lian, Optimization of process planning with various flexibilities using an imperialist competitive algorithm, The International Journal of Advanced Manufacturing Technology, 59 (2012), 815-828. doi: 10.1007/s00170-011-3527-8. Google Scholar
[7]	J. Lin, M. Liu, J. Hao and S. Jiang, A multi-objective optimization approach for integrated production planning under interval uncertainties in the steel industry, Computers & Operations Research, 72 (2016), 189-203. doi: 10.1016/j.cor.2016.03.002. Google Scholar
[8]	T. Majozi and X. Zhu, A combined fuzzy set theory and MILP approach in integration of planning and scheduling of batch plants-Personnel evaluation and allocation, Computers & Chemical Engineering, 29 (2005), 2029-2047. doi: 10.1016/j.compchemeng.2004.07.038. Google Scholar
[9]	X. N. Shen and X. Yao, Mathematical modeling and multi-objective evolutionary algorithms applied to dynamic flexible job shop scheduling problems, Information Science, 298 (2015), 198-224. doi: 10.1016/j.ins.2014.11.036. Google Scholar
[10]	J. M. Usher and K. J. Fernandes, Dynamic process planning-The staticphase, Journal of Materials Processing Technology, 61 (1996), 53-58. Google Scholar
[11]	S.-y. Wan, Integrated Process Planning and Scheduling with Setup Time Consideration by Ant Colony Optimization, HKU Theses Online (HKUTO), 2012. doi: 10.5353/th_b4961807. Google Scholar
[12]	Y. Wang, A PSO-based multi-objective optimization approach to the integration of process planning and scheduling, in Control and Automation (ICCA), 8th IEEE International Conference on IEEE, eds. Y. Zhang and J. Y. H. Fuh, 2010. doi: 10.1109/ICCA.2010.5524365. Google Scholar
[13]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X. Google Scholar
[14]	L. Zhang and T. Wong, Solving integrated process planning and scheduling problem with constructive meta-heuristics, Information Sciences, 340/341 (2016), 1-16. doi: 10.1016/j.ins.2016.01.001. Google Scholar
[15]	L. Zhang and T. Wong, An object-coding genetic algorithm for integrated process planning and scheduling, European Journal of Operational Research, 244 (2015), 434-444. doi: 10.1016/j.ejor.2015.01.032. Google Scholar
[16]	W. Zhanjie, The research about integration of process planning and production scheduling based on genetic algorithm, in Computer Science and Software Engineering, International Conference on IEEE, eds.T. Ju, 2008. doi: 10.1109/CSSE.2008.845. Google Scholar

show all references

References:

[1]	A. Akgun, E. A. Sezer, H. A. Nefeslioglu, C. Gokceoglu and B. Pradhan, An easy-to-use MATLAB program (MamLand) for the assessment of landslide susceptibility using a Mamdani fuzzy algorithm, Computers & Geosciences, 38 (2012), 23-34. doi: 10.1016/j.cageo.2011.04.012. Google Scholar
[2]	IBM, IBM ILOG CPLEX Optimization Studio 12.5 User's Manual, 2012. Google Scholar
[3]	S. Kirkpatrick, C. Gelatt and M. Vecchi, Optimization by simulated annealing, Science, New Series, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671. Google Scholar
[4]	X. Li, L. Gao, X Shao and C. Wang, Mathematical modeling and evolutionary algorithm-based approach for integrated process planning and scheduling, Computers & Operations Research, 37 (2010), 656-667. doi: 10.1016/j.cor.2009.06.008. Google Scholar
[5]	W. Li and C. A. McMahon, A simulated annealing-based optimization approach for integrated process planning and scheduling, International Journal of Computer Integrated Manufacturing, 20 (2007), 80-95. doi: 10.1080/09511920600667366. Google Scholar
[6]	K. Lian, Optimization of process planning with various flexibilities using an imperialist competitive algorithm, The International Journal of Advanced Manufacturing Technology, 59 (2012), 815-828. doi: 10.1007/s00170-011-3527-8. Google Scholar
[7]	J. Lin, M. Liu, J. Hao and S. Jiang, A multi-objective optimization approach for integrated production planning under interval uncertainties in the steel industry, Computers & Operations Research, 72 (2016), 189-203. doi: 10.1016/j.cor.2016.03.002. Google Scholar
[8]	T. Majozi and X. Zhu, A combined fuzzy set theory and MILP approach in integration of planning and scheduling of batch plants-Personnel evaluation and allocation, Computers & Chemical Engineering, 29 (2005), 2029-2047. doi: 10.1016/j.compchemeng.2004.07.038. Google Scholar
[9]	X. N. Shen and X. Yao, Mathematical modeling and multi-objective evolutionary algorithms applied to dynamic flexible job shop scheduling problems, Information Science, 298 (2015), 198-224. doi: 10.1016/j.ins.2014.11.036. Google Scholar
[10]	J. M. Usher and K. J. Fernandes, Dynamic process planning-The staticphase, Journal of Materials Processing Technology, 61 (1996), 53-58. Google Scholar
[11]	S.-y. Wan, Integrated Process Planning and Scheduling with Setup Time Consideration by Ant Colony Optimization, HKU Theses Online (HKUTO), 2012. doi: 10.5353/th_b4961807. Google Scholar
[12]	Y. Wang, A PSO-based multi-objective optimization approach to the integration of process planning and scheduling, in Control and Automation (ICCA), 8th IEEE International Conference on IEEE, eds. Y. Zhang and J. Y. H. Fuh, 2010. doi: 10.1109/ICCA.2010.5524365. Google Scholar
[13]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X. Google Scholar
[14]	L. Zhang and T. Wong, Solving integrated process planning and scheduling problem with constructive meta-heuristics, Information Sciences, 340/341 (2016), 1-16. doi: 10.1016/j.ins.2016.01.001. Google Scholar
[15]	L. Zhang and T. Wong, An object-coding genetic algorithm for integrated process planning and scheduling, European Journal of Operational Research, 244 (2015), 434-444. doi: 10.1016/j.ejor.2015.01.032. Google Scholar
[16]	W. Zhanjie, The research about integration of process planning and production scheduling based on genetic algorithm, in Computer Science and Software Engineering, International Conference on IEEE, eds.T. Ju, 2008. doi: 10.1109/CSSE.2008.845. Google Scholar

Figure 1. The flowchart of the proposed process planningscheduling techniques

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Figure 2. Structure of the Material Flow FIS

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Figure 3. Structure of the Production Ease FIS

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Figure 4. The Flowchart of SA Algorithm

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Figure 5. Pseudo-Code of SA Algorithm

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Figure 6. Parts Operations Sequence in SA solution

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Figure 7. Machines Operations Sequence in SA solution

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Figure 9. Machines operations sequence in CP

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Figure 8. Parts Operations Sequence in CP

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Table 1. Inputs of fuzzy inference systems

The cost peroperation(MC)	Symbol	Source
50	M1	CNC milling machine 1
60	M2	CNC milling machine 2
30	M3	Grinding CNC machine
35	M4	Column drilling equipment
20	M5	Hand drilling equipment
20	M6	Grinding machine
The cost per operation(TC)	Symbol	Resource
6	T1	Drill 1
5	T2	Drill 2
10	T3	Drill 3
15	T4	Drill 4
13	T5	Drill 5
14	T6	Drill 6
8	T7	Drill 7
10	T8	Drill 8
5	T9	Drill 9
10	T10	Polishers
15	T11	Reamer 1
20	T12	Reamer 2
18	T13	Reamer 3
15	T14	Diamond blades
18	T15	Milling plate
13	T16	Spark
24	T17	Magnetic stone

Features	Index	Operation	TAD candidate	Machine candidate	Tool candidate	Machining time for each candidate machine (s)
F₁	Oper₁	Milling	-Z	M₁, M₂	T₁₄, T₁₅, T₁₆, T₁₇	40, 38
F₂	Oper₂	Milling	+Z	M₁, M₂	T₁₄, T₁₅, T₁₆, T₁₇	37, 38
F₃	Oper₃	Milling	-Z, -Y	M₁, M₂	T₁₄, T₁₅, T₁₆, T₁₇	41, 43
F₄	Oper₄	Milling	+Y, -Z	M₁, M₂	T₁₄, T₁₅, T₁₆, T₁₇	31, 30
F₅	Oper₅	Boring	+Y, -Y	M₆	T₁₀	40
F₆	Oper₆	Drilling	-Z	M₃, M₄, M₅	T₁	40, 50, 30
	Oper₇	Reaming	-Z	M₃, M₄, M₅	T₁₁	40, 50, 30
F₇	Oper₈	Ribbing	-Z, -Y	M₁, M₂	T₁₄, T₁₅, T₁₆, T₁₇	54, 52
F₈	Oper₉	Drilling	+X, -X	M₃, M₄, M₅	t₂	20, 30, 21
F₉	Oper₁₀	Drilling	+X, -X	M₃, M₄, M₅	t₃	50, 60, 40
F₁₀	Oper₁₁	Drilling	+Y	M₃, M₄, M₅	t₄	60, 50, 30
	Oper₁₂	reaming	+Y	M₃, M₄, M₅	T₁₂	60, 50, 30
F₁₁	Oper₁₃	Boring	A	M₆	T₁₀	50

		Ease in material displacement in different machines
		VL	L	M	H	VH
Distance between installed machines	VH	X	X	X	U	U
	H	X	U	U	O	O
	M	U	U	O	I	I
	L	U	O	I	E	A
	VL	U	O	I	A	A

Ease in material flow	M1	M2	M3	M4	M5	M6
M1	5.69	2.74	5.69	2.00	1.50	1.50
M2	3.74	5.69	2.56	3.00	4.37	1.50
M3	4.50	4.37	5.69	1.20	0.37	1.50
M4	1.50	1.50	1.20	5.69	2.50	1.20
M5	1.20	2.56	1.50	1.50	5.69	2.56
M6	1.50	2.00	1.50	1.20	3.56	5.69

		Compatibility
		VL	L	M/	H	VH
Setup Ease	VL	X	X	X	U	U
	L	X	U	U	O	O
	M	U	U	O	I	I
	H	U	O	I	E	A
	VH	U	O	I	A	A

Ease in production	M1	M2	M3	M4	M5	M6
M1	5.69	1.50	2.50	4.00	3.00	1.50
M2	4.23	5.69	3.00	5.69	3.00	1.50
M3	3.00	3.00	5.69	3.00	5.63	2.50
M4	2.50	2.50	1.50	5.69	2.50	1.50
M5	2.50	4.00	1.50	1.50	5.69	3.00
M6	3.00	3.00	3.50	3.50	3.00	5.69

Objective Function	Goal	Weight
TWC	$1484$	1
Make Span	$468$	2
Utility	$288.24$	3

Parameters
The number of initial population $= 20$	The iterations time $= 300 s$
The number of neighborhood $= 10$	Alpha $= 0.99$

Position	Parts	Operations	Position	Parts	Operations
1	1	1	17	2	4
2	1	2	18	2	5
3	1	3	19	2	6
4	1	4	20	2	7
5	1	5	21	2	8
6	1	6	22	2	9
7	1	7	23	2	10
8	1	8	24	3	1
9	1	9	25	3	2
10	1	10	26	3	3
11	1	11	27	3	4
12	1	12	28	3	5
13	1	13	29	3	6
14	2	1	30	3	7
15	2	2	31	3	8
16	2	3

Random Position	Considering the Precedence relations	Feasible Position	Parts	Operations	Machines	Tools	TAD
17		14	2	1	2	14	-Z
5		17	2	4	1	15	Z
3		16	2	3	2	16	-Y
10		18	2	5	1	16	-Z
2		19	2	6	6	10	Y
31		15	2	2	4	1	-Z
12		20	2	7	4	11	-Z
19		21	2	8	2	16	-Z
9		23	2	10	4	2	X
27		24	3	1	5	3	-X
21		26	3	3	3	4	Y
18		30	3	7	5	12	Y
14		31	3	8	6	10	A
6		25	3	2	2	15	-Y
13		27	3	4	1	17	Z
4		28	3	5	2	15	-Y
26		29	3	6	3	4	Z
30		22	2	9	4	5	X
7		1	1	1	5	13	X
20		3	1	3	4	6	-Y
16		10	1	10	2	17	-Z
28		2	1	2	1	16	A
8		5	1	5	6	10	-X
25		9	1	9	2	15	-Y
23		6	1	6	2	14	X
29		4	1	4	2	17	z
15		7	1	7	2	17	X
24		8	1	8	2	16	X
22		13	1	13	3	9	z
1		11	1	11	2	14	-Z
11		12	1	12	5	8	-Z

Parameters
$p$: Part indicator, $p=1, \ldots, PNo$
$O^p$: The set indicating operations of $p^{th}$ part
$i$: Operation indicator, $i\in O^p$, $p=1, \ldots, PNo$
d: TAD indicator
t: Tool indicator
m: machine indicator
$Machine_i$ : The set indicating machine candidates of the $i^{th}$ operation
$TAD_i$: The set indicating TAD candidates of the $i^{th}$ operation
$Tool_i$: The set indicating tool candidates of the $i^{th}$ operation
$MC_m$: The cost of using $m^{th}$ machine per operation
$TC_t$: The cost of using $i^{th}$ tool per operation
$MFE_{m_k m_l}$: The rate of Material Flow Ease between machines $m_k$ and $m_l$
$PE_{m_k m_l}$: The rate of Production Ease between machines $m_k$ and $m_l$
Decision Variables
$operation_{imtd}^p = \left\{ \begin{array}{l} 1\;\;{\rm{If the}}\;{i^{th}}\;{\rm{operation}}\;{\rm{of}}\;{\rm{part}}\;p\;{\rm{is}}\;{\rm{done}}\;{\rm{in}}\;{m^{th}}\;{\rm{machine}}\\ \;\;\;{\rm{with}}\;{t^{th}}\;{\rm{tool}}\;{\rm{and}}\;{d^{th}}\;{\rm{TAD}}\\ {\rm{0}}\;\;{\rm{Otherwise}} \end{array} \right.$
$part_{ij}^p = \left\{ \begin{array}{l} 1\;\;{\rm{If}}\;{\rm{the}}\;{j^{th}}\;{\rm{operation}}\;{\rm{of}}\;{\rm{part}}\;p\;{\rm{is}}\;{\rm{done}}\;{\rm{immediately}}\;{\rm{after}}\\ \;\;\;\;{\rm{the}}\;{i^{th}}\;{\rm{operation}}\;{\rm{of}}\;{\rm{part}}\;p\\ 0\;\;{\rm{Otherwise}} \end{array} \right.$
$\begin{array}{l} time_i^p{\rm{ = Cumulative}}\;{\rm{time}}\;{\rm{of}}\;{\rm{operations}}\;{\rm{of}}\;{\rm{part}}\;p\;{\rm{completed}}\;{\rm{until}}\;{\rm{the}}\\ \;\;\;\;\;\;\;\;\;\;{\rm{completion}}\;{\rm{of}}\;{\rm{the}}\;{i^{th}}\;{\rm{operation}} \end{array}$
$O{M_i}{\rm{ = The}}\;{i^{th}}\;{\rm{operation}}\;{\rm{machine, }}\;O{M_i} \in Machin{e_i}$
$O{T_i}{\rm{ = The}}\;{i^{th}}\;{\rm{operation}}\;TAD{\rm{, }}\;O{T_i} \in TA{D_i}$
Proposed Goal Programming Model:
$Min\ D =\displaystyle \left( {{w_1} \times {\rm{ }}\frac{{{d_1}^ - }}{{{G_{TWC}}}}} \right) + \left( {{w_2} \times {\rm{ }}\frac{{{d_2}^ - }}{{{G_{MakeSpan}}}}} \right) + \left( {{w_3} \times {\rm{ }}\frac{{{d_3}^ + }}{{{G_U}}}} \right)~~~~~~~~~~~(1)$
$ Subject$ $to: $
$TWC = \sum\limits_{i \in {O^p}}^{} {\sum\limits_{m \in Machin{e_i}} {\sum\limits_{t \in Too{l_i}} {\sum\limits_{d \in TA{D_i}} {operation_{_{imtd}}^p.(M{C_{O{M_i}}} + T{C_{O{T_i}}}} } )} } ~~~~(2)$
$MakeSpan = \mathop {\max }\limits_{\scriptstyle\, \, \, \, \, \, \, i \in {O^p}\atop \scriptstyle p = 1, ..., PNo} (time_{_i}^p)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(3)$
$U = \sum\limits_{p = 1}^{PNo} {\sum\limits_{i, j \in {O^p}/part_{_{ij}}^p = 1}^{} {(ME{F_{O{M_i}, O{M_j}}} + P{E_{O{M_i}, O{M_j}}})} }~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(4)$
$TWC - d_1^ - \le {G_{TWC}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(5)$
$MakeSpan - d_2^ - \le {G_{MakeSpan}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(6)$
$U + d_3^ + \ge {G_U}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(7)$
$d_1^ -, d_2^ -, d_3^ + \ge 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(8)$

Parameters
$p$: Part indicator, $p=1, \ldots, PNo$
$O^p$: The set indicating operations of $p^{th}$ part
$i$: Operation indicator, $i\in O^p$, $p=1, \ldots, PNo$
d: TAD indicator
t: Tool indicator
m: machine indicator
$Machine_i$ : The set indicating machine candidates of the $i^{th}$ operation
$TAD_i$: The set indicating TAD candidates of the $i^{th}$ operation
$Tool_i$: The set indicating tool candidates of the $i^{th}$ operation
$MC_m$: The cost of using $m^{th}$ machine per operation
$TC_t$: The cost of using $i^{th}$ tool per operation
$MFE_{m_k m_l}$: The rate of Material Flow Ease between machines $m_k$ and $m_l$
$PE_{m_k m_l}$: The rate of Production Ease between machines $m_k$ and $m_l$
Decision Variables
$operation_{imtd}^p = \left\{ \begin{array}{l} 1\;\;{\rm{If the}}\;{i^{th}}\;{\rm{operation}}\;{\rm{of}}\;{\rm{part}}\;p\;{\rm{is}}\;{\rm{done}}\;{\rm{in}}\;{m^{th}}\;{\rm{machine}}\\ \;\;\;{\rm{with}}\;{t^{th}}\;{\rm{tool}}\;{\rm{and}}\;{d^{th}}\;{\rm{TAD}}\\ {\rm{0}}\;\;{\rm{Otherwise}} \end{array} \right.$
$part_{ij}^p = \left\{ \begin{array}{l} 1\;\;{\rm{If}}\;{\rm{the}}\;{j^{th}}\;{\rm{operation}}\;{\rm{of}}\;{\rm{part}}\;p\;{\rm{is}}\;{\rm{done}}\;{\rm{immediately}}\;{\rm{after}}\\ \;\;\;\;{\rm{the}}\;{i^{th}}\;{\rm{operation}}\;{\rm{of}}\;{\rm{part}}\;p\\ 0\;\;{\rm{Otherwise}} \end{array} \right.$
$\begin{array}{l} time_i^p{\rm{ = Cumulative}}\;{\rm{time}}\;{\rm{of}}\;{\rm{operations}}\;{\rm{of}}\;{\rm{part}}\;p\;{\rm{completed}}\;{\rm{until}}\;{\rm{the}}\\ \;\;\;\;\;\;\;\;\;\;{\rm{completion}}\;{\rm{of}}\;{\rm{the}}\;{i^{th}}\;{\rm{operation}} \end{array}$
$O{M_i}{\rm{ = The}}\;{i^{th}}\;{\rm{operation}}\;{\rm{machine, }}\;O{M_i} \in Machin{e_i}$
$O{T_i}{\rm{ = The}}\;{i^{th}}\;{\rm{operation}}\;TAD{\rm{, }}\;O{T_i} \in TA{D_i}$
Proposed Goal Programming Model:
$Min\ D =\displaystyle \left( {{w_1} \times {\rm{ }}\frac{{{d_1}^ - }}{{{G_{TWC}}}}} \right) + \left( {{w_2} \times {\rm{ }}\frac{{{d_2}^ - }}{{{G_{MakeSpan}}}}} \right) + \left( {{w_3} \times {\rm{ }}\frac{{{d_3}^ + }}{{{G_U}}}} \right)~~~~~~~~~~~(1)$
$ Subject$ $to: $
$TWC = \sum\limits_{i \in {O^p}}^{} {\sum\limits_{m \in Machin{e_i}} {\sum\limits_{t \in Too{l_i}} {\sum\limits_{d \in TA{D_i}} {operation_{_{imtd}}^p.(M{C_{O{M_i}}} + T{C_{O{T_i}}}} } )} } ~~~~(2)$
$MakeSpan = \mathop {\max }\limits_{\scriptstyle\, \, \, \, \, \, \, i \in {O^p}\atop \scriptstyle p = 1, ..., PNo} (time_{_i}^p)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(3)$
$U = \sum\limits_{p = 1}^{PNo} {\sum\limits_{i, j \in {O^p}/part_{_{ij}}^p = 1}^{} {(ME{F_{O{M_i}, O{M_j}}} + P{E_{O{M_i}, O{M_j}}})} }~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(4)$
$TWC - d_1^ - \le {G_{TWC}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(5)$
$MakeSpan - d_2^ - \le {G_{MakeSpan}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(6)$
$U + d_3^ + \ge {G_U}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(7)$
$d_1^ -, d_2^ -, d_3^ + \ge 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(8)$

Precedence relations
Oper₁ is first operation. Oper₂ is prior to Oper₄, Oper₅, Oper₁₁ and Oper₁₂. Oper₃ is prior to Oper₄, Oper₅ and Oper₁₀. Oper₆ is prior to Oper₇, Oper₁₁ and Oper₁₂. Oper₈ is prior to Oper₁₃. Oper₉ is prior to Oper₁₃. Oper₁₁ is prior to Oper₁₂.

Precedence relations
Oper₁ is first operation. Oper₂ is prior to Oper₄, Oper₅, Oper₁₁ and Oper₁₂. Oper₃ is prior to Oper₄, Oper₅ and Oper₁₀. Oper₆ is prior to Oper₇, Oper₁₁ and Oper₁₂. Oper₈ is prior to Oper₁₃. Oper₉ is prior to Oper₁₃. Oper₁₁ is prior to Oper₁₂.

$ID$	Exclusive number of each member of the tuples
$PartNo$	The number of the parts
$OperationNo$	The number of operations necessary for part processing
$\{TADNo\}$	The set of TAD candidates
$\{ToolNo\}$	The set of tool candidates
$\{MachineNo\}$	The set of machine candidates
$\{pTime\}$	The set of Machining time for each candidate machine (s)
$\{Succs\}$	The set of successor operations

$task$	A tuple type data represent the Task which the Mode is derived from
$PartNo$	Indicates the number of the parts which equal to $task.Part.No$
$OperationNo$	Indicates the number of the operations in one part which equals to $task.Operation.No$
$TADNo$	Indicates the number of TADs which is a member of $task.\{TAD.No\}$
$ToolNo$	Indicates the number of tools which are the members of $task.\{Tool.No\}$
$MachineNo$	Indicates the number of machines which are a member of $task.\{Machine.No\}$
$pTime$	Indicates the machining time for $Machine_{Machine.No}$

Part	Operation	Machine	TAD	Tool	Start Time	Machining Time	Finish Time
1	1	2	-3	16	0	38	38
1	2	1	3	14	100	37	137
1	3	1	-3	14	59	41	100
1	4	1	2	16	323	31	354
1	5	6	-2	10	394	40	434
1	6	5	-3	1	139	30	169
1	7	3	-3	11	169	40	209
1	8	1	-3	16	269	54	323
1	9	5	1	2	38	21	59
1	10	5	1	3	354	40	394
1	11	5	2	4	209	30	239
1	12	5	2	12	239	30	269
1	13	6	4	10	434	50	484
2	1	2	-2	16	38	20	58
2	2	2	3	15	82	29	111
2	3	2	-3	16	58	24	82
2	4	4	-1	4	269	66	335
2	5	4	1	5	335	59	394
2	6	5	1	13	394	41	435
2	7	5	-2	6	111	28	139
2	8	2	3	14	180	49	229
2	9	2	4	16	139	41	180
2	10	6	1	10	229	40	269
3	1	1	-2	14	0	30	30
3	2	2	3	14	298	30	328
3	3	2	3	14	229	30	259
3	4	1	-1	17	354	29	383
3	5	2	1	14	383	31	414
3	6	5	-3	9	453	40	493
3	7	2	-3	15	259	39	298
3	8	3	-3	8	414	39	453

Objective	TWC	Make Span	Utility	Time (Sec)
Max(Min) Function	Min	Min	Max	Time (Sec)
Goal	1484	468	288
CP	1514	478	258.3761	300
SA	1669	493	87.30202	300

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American Institute of Mathematical Sciences

An integrated approach based on Fuzzy Inference System for scheduling and process planning through multiple objectives

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