# American Institute of Mathematical Sciences

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

 Department of Management, University of Isfahan, Isfahan, Iran

* Corresponding author: Dariush Mohamadi Zanjirani

Received  October 2017 Revised  June 2018 Published  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.

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:

show all references

##### References:
The flowchart of the proposed process planningscheduling techniques
Structure of the Material Flow FIS
Structure of the Production Ease FIS
The Flowchart of SA Algorithm
Pseudo-Code of SA Algorithm
Parts Operations Sequence in SA solution
Machines Operations Sequence in SA solution
Machines operations sequence in CP
Parts Operations Sequence in CP
Inputs of fuzzy inference systems
Output of fuzzy inference systems
The Goal Programming Formulation of the IPPS Problem
 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: $Min\ D =\displaystyle \left( {{w_1} \times {\rm{ }}\frac{{{d_1}^ - }}{{{G_{TWC}}}}} \right) + \left( {{w_2} \times {\rm{ }}\frac{{{d_2}^ - }}{{{G_{MakeSpan}}}}} \right) + \left( {{w_3} \times {\rm{ }}\frac{{{d_3}^ + }}{{{G_U}}}} \right)~~~~~~~~~~~(1)$ $Subject$ $to:$ $TWC = \sum\limits_{i \in {O^p}}^{} {\sum\limits_{m \in Machin{e_i}} {\sum\limits_{t \in Too{l_i}} {\sum\limits_{d \in TA{D_i}} {operation_{_{imtd}}^p.(M{C_{O{M_i}}} + T{C_{O{T_i}}}} } )} } ~~~~(2)$ $MakeSpan = \mathop {\max }\limits_{\scriptstyle\, \, \, \, \, \, \, i \in {O^p}\atop \scriptstyle p = 1, ..., PNo} (time_{_i}^p)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(3)$ $U = \sum\limits_{p = 1}^{PNo} {\sum\limits_{i, j \in {O^p}/part_{_{ij}}^p = 1}^{} {(ME{F_{O{M_i}, O{M_j}}} + P{E_{O{M_i}, O{M_j}}})} }~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(4)$ $TWC - d_1^ - \le {G_{TWC}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(5)$ $MakeSpan - d_2^ - \le {G_{MakeSpan}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(6)$ $U + d_3^ + \ge {G_U}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(7)$ $d_1^ -, d_2^ -, d_3^ + \ge 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(8)$
Sources Specifications
 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
 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
Technical Specifications of Part 1
 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
 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
Technical Specifications of Part 2
 Features Index operation TAD candidate Machine candidate Tool candidate Machining time for each candidate machine (s) F1 oper1 Milling -Y M1, M2 T14, T15, T16, T17 30, 20 F2 oper2 milling +Z M1, M2 T14, T15, T16, T17 35, 29 F3 oper3 milling -Y, -Y M1, M2 T14, T15, T16, T17 29, 24 F4 oper4 drilling +Z, -Z M3, M4, M5 T4 57, 66, 51 F5 oper5 reaming +X M3, M4, M5 T5 41, 59, 38 oper6 drilling +X M3, M4, M5 T13 52, 71, 41 F6 oper7 reaming -Y, +Y M3, M4, M5 t6 30, 41, 28 F7 oper8 ribbing -Z, +Z, -X M1, M2 T14, T15, T16, T17 47, 49 F8 oper9 milling A M1, M2 T14, T15, T16, T17 40, 41 F9 oper10 boring +X, -X, +Y, -Y m6 T10 40
 Features Index operation TAD candidate Machine candidate Tool candidate Machining time for each candidate machine (s) F1 oper1 Milling -Y M1, M2 T14, T15, T16, T17 30, 20 F2 oper2 milling +Z M1, M2 T14, T15, T16, T17 35, 29 F3 oper3 milling -Y, -Y M1, M2 T14, T15, T16, T17 29, 24 F4 oper4 drilling +Z, -Z M3, M4, M5 T4 57, 66, 51 F5 oper5 reaming +X M3, M4, M5 T5 41, 59, 38 oper6 drilling +X M3, M4, M5 T13 52, 71, 41 F6 oper7 reaming -Y, +Y M3, M4, M5 t6 30, 41, 28 F7 oper8 ribbing -Z, +Z, -X M1, M2 T14, T15, T16, T17 47, 49 F8 oper9 milling A M1, M2 T14, T15, T16, T17 40, 41 F9 oper10 boring +X, -X, +Y, -Y m6 T10 40
Technical Specifications of Part 3
 Features Index operation TAD candidate Machine candidate Tool candidate Machining time for each candidate machine (s) F1 oper1 Milling -Y, -X M1, M2 T14, T15, T16, T17 30, 35 F2 oper2 Milling +Z, +Z M1, M2 T14, T15, T16, T17 28, 30 F3 oper3 Milling +Z, +Y M1, M2 T14, T15, T16, T17 33, 30 F4 oper4 Milling +X, -X M1, M2 T14, T15, T16, T17 29, 25 F5 oper5 Milling +X, -X M1, M2 T14, T15, T16, T17 37, 31 F6 oper6 Drilling -Z, +Z M3, M4, M5 T9 50, 60, 40 F7 oper7 Milling -Z M1, M2 T14, T15, T16, T17 40, 39 F8 ooper8 Drilling -Z M3, M4, M5 T8 39, 50, 38
 Features Index operation TAD candidate Machine candidate Tool candidate Machining time for each candidate machine (s) F1 oper1 Milling -Y, -X M1, M2 T14, T15, T16, T17 30, 35 F2 oper2 Milling +Z, +Z M1, M2 T14, T15, T16, T17 28, 30 F3 oper3 Milling +Z, +Y M1, M2 T14, T15, T16, T17 33, 30 F4 oper4 Milling +X, -X M1, M2 T14, T15, T16, T17 29, 25 F5 oper5 Milling +X, -X M1, M2 T14, T15, T16, T17 37, 31 F6 oper6 Drilling -Z, +Z M3, M4, M5 T9 50, 60, 40 F7 oper7 Milling -Z M1, M2 T14, T15, T16, T17 40, 39 F8 ooper8 Drilling -Z M3, M4, M5 T8 39, 50, 38
Precedence Relations of Part 1
 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.
Precedence Relations of Part 2
 Precedence relations Oper1 is first operation Oper2 is prior to Oper7, Oper9. Oper3 is prior to Oper5, Oper9. Oper4 is prior to Oper5, Oper6. Oper5 is prior to Oper6. Oper7 is prior to Oper8, Oper10.Oper8 is prior to Oper10.
 Precedence relations Oper1 is first operation Oper2 is prior to Oper7, Oper9. Oper3 is prior to Oper5, Oper9. Oper4 is prior to Oper5, Oper6. Oper5 is prior to Oper6. Oper7 is prior to Oper8, Oper10.Oper8 is prior to Oper10.
Precedence Relations of Part 3
 Precedence relations Oper1 is first operation. Oper2 is prior to Oper4, Oper5. Oper3 is prior to Oper4, Oper5.Oper4 is prior to Oper5, Oper6.Oper5 is prior to Oper6. Oper7 is prior to Oper8.
 Precedence relations Oper1 is first operation. Oper2 is prior to Oper4, Oper5. Oper3 is prior to Oper4, Oper5.Oper4 is prior to Oper5, Oper6.Oper5 is prior to Oper6. Oper7 is prior to Oper8.
Rules of Material Flow FIS on Distance and Trans-portation Ease Parameters
 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 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
Score of Pair Machines Based on Material Flow FIS
 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
 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
Rules of Production Ease FIS on Setup Ease and Compatibility Parameters
 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
 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
Score of Each Pair Machines Based on ease in production FIS
 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
 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
Goals and Weights of Objectives
 Objective Function Goal Weight TWC $1484$ 1 Make Span $468$ 2 Utility $288.24$ 3
 Objective Function Goal Weight TWC $1484$ 1 Make Span $468$ 2 Utility $288.24$ 3
The Parameters of SA Algorithm
 Parameters The number of initial population $= 20$ The iterations time $= 300 s$ The number of neighborhood $= 10$ Alpha $= 0.99$
 Parameters The number of initial population $= 20$ The iterations time $= 300 s$ The number of neighborhood $= 10$ Alpha $= 0.99$
The optional list of Positions of Operations of Parts
 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
 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
Solution of SA algorithm
 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
 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
Description of the $Tasks$
 $ID$ 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
 $ID$ 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
Description of the $Modes$
 $task$ 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 $Machine_{Machine.No}$
 $task$ 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 $Machine_{Machine.No}$
he Best Solutions of SA algorithm
 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
 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
The Best Solution obtained through CP
 Part Operation Machine TAD Tool Start Time Machining Time Finish Time 1 1 2 -3 16 0 38 38 1 2 2 3 16 38 37 75 1 3 1 -3 16 75 41 116 1 8 1 -3 16 116 52 168 1 10 3 1 3 168 40 208 1 6 5 -3 1 208 30 238 1 9 5 -1 2 238 20 258 1 11 5 2 4 258 30 288 1 7 5 -3 11 288 30 318 1 13 6 4 10 318 50 368 1 5 6 2 10 368 40 408 1 12 5 2 12 408 30 438 1 4 1 -3 16 438 30 468 2 1 1 -2 16 0 20 20 2 3 1 -3 16 20 24 44 2 2 1 3 16 44 29 73 2 4 5 3 4 73 51 124 2 5 5 1 5 124 38 162 2 6 5 1 13 162 41 203 2 7 5 2 6 318 28 346 2 8 1 -3 16 351 47 398 2 9 1 4 16 398 40 438 2 10 6 1 10 438 40 478 3 1 1 -2 16 168 30 198 3 7 1 -3 16 198 39 237 3 2 1 1 16 237 28 265 3 3 1 3 16 265 30 295 3 4 1 1 16 295 25 320 3 5 1 -1 16 320 31 351 3 6 5 -3 9 351 40 391 3 8 5 -3 8 438 38 476
 Part Operation Machine TAD Tool Start Time Machining Time Finish Time 1 1 2 -3 16 0 38 38 1 2 2 3 16 38 37 75 1 3 1 -3 16 75 41 116 1 8 1 -3 16 116 52 168 1 10 3 1 3 168 40 208 1 6 5 -3 1 208 30 238 1 9 5 -1 2 238 20 258 1 11 5 2 4 258 30 288 1 7 5 -3 11 288 30 318 1 13 6 4 10 318 50 368 1 5 6 2 10 368 40 408 1 12 5 2 12 408 30 438 1 4 1 -3 16 438 30 468 2 1 1 -2 16 0 20 20 2 3 1 -3 16 20 24 44 2 2 1 3 16 44 29 73 2 4 5 3 4 73 51 124 2 5 5 1 5 124 38 162 2 6 5 1 13 162 41 203 2 7 5 2 6 318 28 346 2 8 1 -3 16 351 47 398 2 9 1 4 16 398 40 438 2 10 6 1 10 438 40 478 3 1 1 -2 16 168 30 198 3 7 1 -3 16 198 39 237 3 2 1 1 16 237 28 265 3 3 1 3 16 265 30 295 3 4 1 1 16 295 25 320 3 5 1 -1 16 320 31 351 3 6 5 -3 9 351 40 391 3 8 5 -3 8 438 38 476
Results of SA and CP Algorithm
 Objective TWC Make Span Utility Time (Sec) Max(Min) Function Min Min Max Goal 1484 468 288 CP 1514 478 258.3761 300 SA 1669 493 87.30202 300
 Objective TWC Make Span Utility Time (Sec) Max(Min) Function Min Min Max Goal 1484 468 288 CP 1514 478 258.3761 300 SA 1669 493 87.30202 300
 [1] Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054 [2] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [3] Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071 [4] Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353 [5] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [6] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [7] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [8] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [9] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273 [10] Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357 [11] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [12] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [13] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [14] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447 [15] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [16] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [17] Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326 [18] Mathew Gluck. Classification of solutions to a system of $n^{\rm th}$ order equations on $\mathbb R^n$. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

2019 Impact Factor: 1.366