SCHEDULING IN FLEXIBLE ROBOTIC MANUFACTURING CELLS HAKAN GÜLTEKİN

HAKAN GULTEKIN Mc 1 Mc 2Mc m... Input Buffer Output Buffer Robot Linear Tracks Parts The System:

HAKAN GULTEKIN Robot never loads an already loaded machine and never unloads an empty machine. General Assumptions: No buffers between the workstations. Robot loads and unloads the machines and transports the parts between the machines.

HAKAN GULTEKIN General Assumptions: Maximize production rate = minimize cycle time Cycle time: Long run average time to produce one part Loading and unloading times,  i or  Robot travel times are  ij or .

HAKAN GULTEKIN Mc 1 Mc 2 Robot move sequence: Alternative 1:

HAKAN GULTEKIN Mc 1 Mc 2 Alternative 1: Robot move sequence:

HAKAN GULTEKIN Mc 1 Mc 2 Alternative 1: Robot move sequence:

HAKAN GULTEKIN Mc 1 Mc 2 Alternative 1: Robot move sequence:

HAKAN GULTEKIN Mc 1 Mc 2 Alternative 2: Robot move sequence:

HAKAN GULTEKIN Mc 1 Mc 2 Alternative 2: Robot move sequence:

HAKAN GULTEKIN Mc 1 Mc 2 Alternative 2: Robot move sequence:

HAKAN GULTEKIN Multiple or identical parts to be processed. Mc 1 Mc 2 Multiple Parts Case: Different sequences of parts result in different production rates. Part input sequence:

HAKAN GULTEKIN Mc 1 Mc 2 Identical Parts Case: Different sequences of parts result in identical production rates. Multiple or identical parts to be processed. Part input sequence:

HAKAN GULTEKIN Robot move cycles: Cyclic production: Production by repeating a fixed sequence of robot moves. Easy to implement and control. n-unit cycle: Sequence of robot moves that load and unload each machine exactly n times in which the initial and final states of the cell are the same. S1:A 0 A 1 A 2 S2:A 0 A 2 A 1 S 12 S 21 :A 0 A 1 A 0 A 2 A 1 A 2

HAKAN GULTEKIN Previous results (identical parts): 1-unit cycles are optimal for two and three machine cells. If P 1 +P 2 2  S2 is optimal. If P 1 +P 2 = 2  both perform equally well. Six 1-unit cycles in three machine cells.

HAKAN GULTEKIN Previous results (identical parts): 1-unit cycles need not be optimal for cells with four or greater number of machines. m! 1-unit cycles Algorithm to find the best 1-unit cycle.

HAKAN GULTEKIN Previous results (multiple parts): 1-unit cycles need not be optimal even in two machine cells Algorithm that simultaneously finds the optimal part input sequence and robot move sequence. Two Machine Case:

HAKAN GULTEKIN Previous results (multiple parts): Fix the robot move cycle to one of the six 1-unit robot move cycles and find the part input sequence. NP-Hard for 2 out of 6 1-unit cycles. Three Machine Case: Suboptimal! Worst case?

HAKAN GULTEKIN Other special cell configurations: Parallel machines More than 1 robots Dual gripper robots Different assumptions on ,  and P i ’s Robotic cells with buffers in front of the machines Different objective functions

HAKAN GULTEKIN Each part has a number of operations to be performed on the machines. Assumptions pertinent to this study: CNC machines. Operational flexibility: ability to interchange the ordering of operations. Process flexibility: ability to perform multiple operations on the same machine.

HAKAN GULTEKIN The problem: Find the allocation of the operations to the machines and the corresponding robot move sequence that jointly minimize the cycle time.

HAKAN GULTEKIN Definitions: k-allocation type : Part operations (o 1,o 2,o 3,o 4,o 5 ) (o 1,o 2 ) k M1M2 (o 3,o 4,o 5 ) (o 2,o 3,o 5 ) (o 1,o 4 ) (o 4 ) (o 1,o 2,o 3,o 5 ) (o 1,o 2 ) k (o 3,o 4,o 5 ) (o 2,o 3,o 5 ) (o 1,o 4 ) (o 4 ) (o 1,o 2,o 3,o 5 )

HAKAN GULTEKIN Example: Let  =1,  =2, o 1 =13, o 2 =17, o 3 =10, o 4 =5, o 5 =5 P = 50 Assume a 2-m/c robotic cell and consider robot move cycle S2. 1-allocation S2: P 1 =o 1 +o 3 =23, P 2 =o 2 +o 4 +o 5 =27 T S2(1) = 4  +4  +max{2  +4 , P 1, P 2 } = 39 2-allocation S2: P 11 =o 1 +o 3 =23, P 12 =o 2 +o 4 +o 5 = 27, P 21 =o 2 +o 4 +o 5 =27, P 22 =o 1 +o 3 = 23. T S2(2) = 4  +4  +1/2max{2  +4 , P 11, P 22 } +1/2max{2  +4 , P 21, P 12 } = 37

HAKAN GULTEKIN 2-machine case: Limited tool magazine capacity In most cases number of required tools exceeds the tool magazine capacity Then for each part we have 3 sets of operations: 1.Operations that can only be processed on the first machine with total processing time P 1. 2.Operations that can only be processed on the second machine with total processing time P 2. 3.Operations that can be processed on both machines with total processing time P.

HAKAN GULTEKIN 2-machine case: The problem: Find the allocation of the operations that are in the third set to the machines and the corresponding robot move sequence that jointly minimize the cycle time.

HAKAN GULTEKIN 1. If P 1 + P 2 ≥ 2  then S2 gives the minimum cycle time, 2. Else, 2.1. If 2P + P 1 + P 2 ≤ 2 , then S1 gives the minimum cycle time 2.2. Else, If 2P 1 + P 2 + P ≤ 2  + 6  then S 12 S 21 gives the minimum cycle time, Else, depending on the allocations of the operations for S2 either S2 or S 12 S 21 gives the minimum cycle time. Theorem: 2-machine case:

HAKAN GULTEKIN P 2 =  P 1 2-machine case:

HAKAN GULTEKIN Note that whenever P = 0, then there is no allocation problem and the result reduces to: S1 is optimal if P 1 +P 2 < 2  and S2 is optimal otherwise. The problem as well as the result becomes identical to two machine identical parts case. 2-machine case:

HAKAN GULTEKIN Duplicate all cutting tools: P 1 = P 2 = 0. 2-machine case:

HAKAN GULTEKIN A new robot move cycle: T proposed = 4  +2(m+1)   +1/m(max{0, P-2(m-1)  -(m-1)(m+2)  })

Lower bound: max{2(m+1)(  +  )+min{P,  }, 4  +4  +P/m} - Load and unload all machines once: 2(m+1)  - Travel from input buffer to output buffer and return back: 2(m+1)  - For all machines either wait or do some other activity: ∑min{P i,  } ≥ min{P,  }. First component: HAKAN GULTEKIN

- Either wait or do some other activity: P i, - Unload M i :  - Transport to M (i+1) :  - Load M (i+1) :  - Transport to M (i-1) : 2  - Unload M (i-1) :  - Transport to M i :  - Load M i :  4  +4  +max i {P i } ≥ 4  +4  +P/m Second component: Lower bound: max{2(m+1)(  +  )+min{P,  }, 4  +4  +P/m} HAKAN GULTEKIN

The new robot move cycle gives the minimum cycle time. Changing the layout from in-line robotic cell to robot- centered cell reduces the cycle time of the new cycle even further. 2-machine case:

HAKAN GULTEKIN Example: o 1 = 40, o 2 = 45, o 3 = 50, o 4 = 60, o 5 = 50, o 6 = 55  = 2,  = 10 Allocation: M1 M2 M3 o 1,o 4 P 1 =100 o 2,o 6 P 2 =100 o 3,o 5 P 3 =100 T S6 = 4  +4  +max{4  +8 , P 1, P 2, P 3 } = 148 T proposed = 1/3(max{12  +24 , P+8  +14  }) = 152

HAKAN GULTEKIN Theorem: The proposed cycle gives the minimum cycle time if: (m-2)  ≤ 2  or P ≤ 2(m 2 -1)  +(m 2 +2m-2)  Theorem: For (m-2)  > 2  and P > 2(m 2 -1)  +(m 2 +2m-2) , using the proposed cycle instead of the optimal robot move cycle has the following worst case performance bound: m= m= m= m= m-machine case:

HAKAN GULTEKIN Optimal number of machines: Theorem: The optimal number of machines, m *, for the proposed cycle is either Theorem: Proposed cycle with optimal number of machines dominates all traditional robot move cycles with optimal number of machines. where.

HAKAN GULTEKIN Bicriteria Robotic Cell Scheduling: No existing studies considering cost objectives in robotic cell scheduling literature. Minimize manufacturing cost vs Minimize cycle time. Manufacturing Cost = Machining Cost+Tooling Cost

HAKAN GULTEKIN Cutting speed, v Bicriteria Robotic Cell Scheduling:

HAKAN GULTEKIN Cost Processing time Total cost Machining cost Tooling cost PuPu PlPl Bicriteria Robotic Cell Scheduling:

HAKAN GULTEKIN Increasing the processing times reduces the cost but may increase the cycle time. Reducing the processing times increases the cost but may reduce the cycle time. Find the set of efficient solutions. Bicriteria Robotic Cell Scheduling:

HAKAN GULTEKIN Cycle timeTlTl TuTu Manufacturing cost ClCl A B C Dominated Solution Nondominated Solutions Bicriteria Robotic Cell Scheduling:

HAKAN GULTEKIN Cycle time Manufacturing cost S1S1 S2S2 Bicriteria Robotic Cell Scheduling: T C

HAKAN GULTEKIN QUESTIONS