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Analysis of the required mathematical models

DECLARATION 2 PUBLICATIONS

4.3 Analysis of the required mathematical models

𝑑𝑑𝑐𝑐(𝑠𝑠) = π‘‘π‘‘π‘šπ‘š +𝑑𝑑𝑣𝑣𝑖𝑖𝑠𝑠 +𝑑𝑑𝑝𝑝𝑝𝑝 +π‘‘π‘‘π‘Ÿπ‘Ÿ +𝑑𝑑𝑑𝑑 (4-1) Assuming the values as: π‘‘π‘‘π‘šπ‘š = 0.13𝑠𝑠, 𝑑𝑑𝑣𝑣𝑖𝑖𝑠𝑠 = 0.1𝑠𝑠, 𝑑𝑑𝑝𝑝 = 0.12𝑠𝑠, π‘‘π‘‘π‘Ÿπ‘Ÿ = 0.125𝑠𝑠, 𝑑𝑑𝑑𝑑 = 0.15𝑠𝑠.

The notations and parameters presented were used to develop a model that described the behavior of a robot as a disruptive tool during the pick and place scenario in a virtual manufacturing environment.

Parameters

πœ†πœ† = πœ†πœ†β„Žπ‘’π‘’ π‘šπ‘šπ‘’π‘’π‘šπ‘šπ‘›π‘› π‘šπ‘šπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘–π‘–π‘£π‘£π‘šπ‘šπ‘Žπ‘Ž π‘Ÿπ‘Ÿπ‘šπ‘šπ‘‘π‘‘π‘’π‘’ 𝑓𝑓𝑓𝑓 π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘π‘ π‘  π‘“π‘“π‘Ÿπ‘Ÿπ‘“π‘“π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘šβ„Žπ‘–π‘–π‘›π‘›π‘’π‘’π‘ π‘ .

πœ‡πœ‡ = πœ†πœ†β„Žπ‘’π‘’ π‘šπ‘šπ‘’π‘’π‘šπ‘šπ‘›π‘› π‘ π‘ π‘’π‘’π‘Ÿπ‘Ÿπ‘£π‘£π‘–π‘–π‘šπ‘šπ‘’π‘’ π‘Ÿπ‘Ÿπ‘šπ‘šπ‘‘π‘‘π‘’π‘’ 𝑓𝑓𝑓𝑓 π‘π‘π‘Ÿπ‘Ÿπ‘“π‘“π‘π‘π‘π‘π‘šπ‘šπ‘‘π‘‘ π‘šπ‘šπ‘Žπ‘Žπ‘ π‘ π‘“π‘“ π‘šπ‘šπ‘šπ‘šπ‘Žπ‘Žπ‘Žπ‘Žπ‘’π‘’π‘π‘ π‘–π‘–π‘›π‘›π‘‘π‘‘π‘’π‘’π‘Ÿπ‘Ÿ βˆ’ π‘šπ‘šπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘–π‘–π‘£π‘£π‘šπ‘šπ‘Žπ‘Ž π‘‘π‘‘π‘–π‘–π‘šπ‘šπ‘’π‘’.

𝑝𝑝𝑛𝑛 = π‘π‘π‘Ÿπ‘Ÿπ‘“π‘“π‘π‘π‘šπ‘šπ‘π‘π‘–π‘–π‘Žπ‘Žπ‘–π‘–π‘‘π‘‘π‘π‘ 𝑓𝑓𝑓𝑓 β„Žπ‘šπ‘šπ‘£π‘£π‘–π‘–π‘›π‘›π‘Žπ‘Ž 𝑛𝑛 π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘ 𝑖𝑖𝑛𝑛 π‘žπ‘žπ‘π‘π‘’π‘’π‘π‘π‘’π‘’. 𝜌𝜌 = π‘π‘π‘‘π‘‘π‘–π‘–π‘Žπ‘Žπ‘–π‘–π‘’π‘’π‘šπ‘šπ‘‘π‘‘π‘–π‘–π‘“π‘“π‘›π‘› π‘“π‘“π‘Ÿπ‘Ÿ π‘‘π‘‘π‘Ÿπ‘Ÿπ‘šπ‘šπ‘“π‘“π‘“π‘“π‘–π‘–π‘šπ‘š 𝑖𝑖𝑛𝑛𝑑𝑑𝑒𝑒𝑛𝑛𝑠𝑠𝑖𝑖𝑑𝑑𝑝𝑝. πΏπΏπ‘žπ‘ž = Average number of parts in the queue 𝐿𝐿 = π΄π΄π‘£π‘£π‘’π‘’π‘Ÿπ‘Ÿπ‘šπ‘šπ‘Žπ‘Žπ‘’π‘’ π‘›π‘›π‘π‘π‘šπ‘šπ‘π‘π‘’π‘’π‘Ÿπ‘Ÿ 𝑓𝑓𝑓𝑓 π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘π‘ π‘  𝑖𝑖𝑛𝑛 π‘‘π‘‘β„Žπ‘’π‘’ π‘ π‘ π‘π‘π‘ π‘ π‘‘π‘‘π‘’π‘’π‘šπ‘š

𝑀𝑀 π‘žπ‘ž = Average π‘€π‘€π‘šπ‘šπ‘–π‘–π‘‘π‘‘π‘–π‘–π‘›π‘›π‘Žπ‘Ž π‘‘π‘‘π‘–π‘–π‘šπ‘šπ‘’π‘’ 𝑓𝑓𝑓𝑓 π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘π‘ π‘  𝑓𝑓𝑛𝑛 π‘šπ‘š π‘šπ‘šπ‘“π‘“π‘›π‘›π‘£π‘£π‘’π‘’π‘π‘π‘“π‘“π‘Ÿπ‘Ÿ before being served = Average time apart spent in the whole system

π‘‘π‘‘π‘šπ‘š = πœ†πœ†π‘“π‘“π‘‘π‘‘π‘šπ‘šπ‘Žπ‘Ž π‘šπ‘šπ‘π‘π‘šπ‘šπ‘Žπ‘Žπ‘’π‘’ π‘šπ‘šπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘–π‘–π‘£π‘£π‘šπ‘šπ‘Žπ‘Ž π‘‘π‘‘π‘–π‘–π‘šπ‘šπ‘’π‘’

𝑑𝑑 𝑣𝑣𝑖𝑖𝑠𝑠 = π‘‘π‘‘π‘–π‘–π‘šπ‘šπ‘’π‘’ π‘‘π‘‘π‘šπ‘šπ‘‘π‘‘π‘’π‘’π‘›π‘› 𝑝𝑝𝑝𝑝 π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘šπ‘’π‘’π‘Ÿπ‘Ÿπ‘šπ‘š 𝑑𝑑𝑓𝑓 π‘£π‘£π‘–π‘–π‘ π‘ π‘π‘π‘šπ‘šπ‘Žπ‘Žπ‘–π‘–π‘’π‘’π‘’π‘’ π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘ π‘šπ‘šπ‘›π‘›π‘π‘ 𝑠𝑠𝑒𝑒𝑛𝑛𝑝𝑝 π‘ π‘ π‘–π‘–π‘Žπ‘Žπ‘›π‘›π‘šπ‘šπ‘Žπ‘Ž 𝑑𝑑𝑓𝑓 π‘Ÿπ‘Ÿπ‘“π‘“π‘π‘π‘“π‘“π‘‘π‘‘ 𝑑𝑑𝑝𝑝 = π‘‘π‘‘π‘–π‘–π‘šπ‘šπ‘’π‘’ π‘‘π‘‘π‘šπ‘šπ‘‘π‘‘π‘’π‘’π‘›π‘› 𝑝𝑝𝑝𝑝 π‘‘π‘‘β„Žπ‘’π‘’ π‘Ÿπ‘Ÿπ‘“π‘“π‘π‘π‘“π‘“π‘‘π‘‘ 𝑑𝑑𝑓𝑓 𝑓𝑓𝑒𝑒𝑒𝑒𝑝𝑝 π‘šπ‘šπ‘›π‘›π‘π‘ π‘π‘π‘–π‘–π‘šπ‘šπ‘‘π‘‘ π‘šπ‘š π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘

Using the above notations from the queuing mathematical theory, a single server system was considered whereby the arriving parts are served by a single robot that identifies and pick up the required part. The probability functions of a negative exponential distribution with inter- arrival time Β΅, having a mean number of arrivals of parts from available stations.

The probability Pr that one product arrive at a time (t) is given as πœ†πœ†π‘₯π‘₯ = (t, t + x) for x β†’ 0. Similarly, the probability Pr that more than one part arrive is given asπœŒπœŒπ‘›π‘›{(𝑑𝑑+π‘₯π‘₯), xβ†’ 0}. Let π‘₯π‘₯(𝑑𝑑) π‘šπ‘šπ‘‘π‘‘ 𝑑𝑑 β‰₯0 denote the available product present to be served at a given time 𝑑𝑑.

Therefore, the system behavior can be represented mathematically using a probability function.

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𝑓𝑓(π‘₯π‘₯) = π‘’π‘’βˆ’πœ†πœ† π‘₯π‘₯!πœ†πœ†π‘₯π‘₯ (4-2) πœŒπœŒπ‘›π‘› (𝑑𝑑) = [𝑛𝑛 π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘ π‘šπ‘šπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘–π‘–π‘£π‘£π‘šπ‘šπ‘Žπ‘Ž π‘šπ‘šπ‘‘π‘‘ π‘šπ‘š π‘ π‘ π‘π‘π‘’π‘’π‘šπ‘šπ‘–π‘–π‘“π‘“π‘–π‘–π‘šπ‘š π‘‘π‘‘π‘–π‘–π‘šπ‘šπ‘’π‘’ (0,𝑑𝑑)]

With the set of asumption presented, πœŒπœŒπ‘›π‘› (𝑑𝑑+π‘₯π‘₯), x β†’ 0, can be expressed as

πœŒπœŒπ‘›π‘› (𝑑𝑑+π‘₯π‘₯) = πœŒπœŒπ‘›π‘› (𝑑𝑑) [1βˆ’ πœ†πœ†π‘₯π‘₯] + πœŒπœŒπ‘›π‘›βˆ’1 (𝑑𝑑)πœ†πœ†π‘₯π‘₯ (4-3) The probability above represent the poisson arrival condition of parts.

𝜌𝜌0 (𝑑𝑑+π‘₯π‘₯) = 𝜌𝜌0 (𝑑𝑑)[1βˆ’ πœ†πœ†π‘₯π‘₯] (4-4) This can also be expressed using h to denote the inter- arrival time

πœŒπœŒπ‘›π‘› (𝑑𝑑+β„Ž)βˆ’ πœŒπœŒπ‘›π‘› (𝑑𝑑)

β„Ž = βˆ’πœ†πœ†πœŒπœŒπ‘›π‘› (𝑑𝑑)βˆ’ πœ†πœ† πœŒπœŒπ‘›π‘›βˆ’1 (𝑑𝑑) (4-5) Similrly,

𝜌𝜌0 (𝑑𝑑+β„Ž)βˆ’ 𝜌𝜌0 (𝑑𝑑)

β„Ž = βˆ’πœ†πœ†πœŒπœŒ0 (𝑑𝑑) (4-6) Diferentiation of 5 & 6 gives

π‘‘π‘‘πœŒπœŒπ‘›π‘› (𝑑𝑑)

𝑑𝑑𝑑𝑑 =βˆ’πœ†πœ†π‘π‘π‘›π‘› (𝑑𝑑) + πœ†πœ†πœŒπœŒπ‘›π‘›βˆ’1 (𝑑𝑑), (4-7)

π‘‘π‘‘πœŒπœŒ0 (𝑑𝑑)

𝑑𝑑𝑑𝑑 =βˆ’πœ†πœ†πœŒπœŒ0 (𝑑𝑑) (4-8) Equation 7 and 8 gives the set of the differential equation solving with respect to

𝜌𝜌0 (𝑑𝑑) = π‘’π‘’βˆ’Ξ»π‘‘π‘‘ (4-9) To proof the above, the derivative of 𝜌𝜌0(𝑑𝑑) is computed and compared with equation (4-7) &

(4-8) to obtain 𝜌𝜌1(𝑑𝑑),

𝜌𝜌1(𝑑𝑑) = πœ†πœ†π‘‘π‘‘π‘’π‘’βˆ’πœ†πœ†π‘‘π‘‘ (4-10) Equation 4-7 can be written as

πœŒπœŒπ‘›π‘›(𝑑𝑑) = (Ξ»t)𝑛𝑛!𝑛𝑛 Γ— π‘’π‘’βˆ’Ξ»π‘‘π‘‘ ,π‘€π‘€β„Žπ‘’π‘’π‘›π‘› 𝑛𝑛= 0,1,2,3,4 … (4-11)

For an efficient productivity, 𝜌𝜌= π‘π‘π‘Ÿπ‘Ÿπ‘π‘π‘‘π‘‘π‘π‘π‘π‘π‘‘π‘‘ π‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘–π‘–π‘£π‘£π‘Žπ‘Žπ‘Žπ‘Ž π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘‘π‘‘π‘’π‘’ (πœ†πœ†) π‘ π‘ π‘’π‘’π‘Ÿπ‘Ÿπ‘£π‘£π‘–π‘–π‘π‘π‘’π‘’ π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘‘π‘‘π‘’π‘’ (πœ‡πœ‡) < 1 For a discrete time, πœŒπœŒπ‘›π‘› (𝑑𝑑) = π‘šπ‘šπ‘’π‘’π‘Žπ‘Žπ‘›π‘› π‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘–π‘–π‘£π‘£π‘Žπ‘Žπ‘Žπ‘Ž π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘‘π‘‘π‘’π‘’

π‘šπ‘šπ‘’π‘’π‘Žπ‘Žπ‘›π‘› π‘ π‘ π‘’π‘’π‘Ÿπ‘Ÿπ‘£π‘£π‘–π‘–π‘π‘π‘’π‘’ π‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘‘π‘‘π‘’π‘’ = πœ‡πœ‡(𝑑𝑑) πœ†πœ†(𝑑𝑑) (4-12)

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Equation (4-13) was used to obtain the probability that parts have been cleared from the work envelope. Equation (4-13) expressed the relationship between the arrival rate and the server rate which was used to obtain the utility rate of the server.

Similarly, using notations from the queuing system (M/G/1) with single server, the average numbers of parts in the queue, the average number of parts in the system, average time spent before served, and average time spent in the system was expressed in equations (4-14) and (4-15).

The average number of parts in the queue was modeled as:

πΏπΏπ‘žπ‘ž = πœ‡πœ‡(πœ‡πœ‡βˆ’Ξ»)πœ†πœ†2 (4-13) Average number of parts in the system

𝐿𝐿= πΏπΏπ‘žπ‘ž + πœ‡πœ‡ πœ†πœ† (4-14) Average time a part that waits before being served

π‘€π‘€π‘žπ‘ž = πΏπΏΞ»π‘žπ‘ž (4-15) Average time a part spent in the system

𝑀𝑀 =π‘€π‘€π‘žπ‘ž + πœ‡πœ‡1 (4-16) To predict a production optimal throughput, the cost involve during the pick and place task must also be considered by taking into consideration the average time spent and average number of parts in the system.

Given that;

Cost of server (robot) = 𝐢𝐢𝑠𝑠 ,Cost of parts wating before picked up = 𝐢𝐢𝑀𝑀, Cost of busy robot

= 𝐢𝐢𝑏𝑏, Cost of idle robot = 𝐢𝐢𝑖𝑖 ,Cost of part returning to be picked = πΆπΆπ‘Ÿπ‘Ÿ .

The expression given in equation 16 can be considered to arrive at a total cost for the process,

𝐢𝐢𝑝𝑝= 𝐢𝐢𝑀𝑀 Γ— L + 𝐢𝐢𝑏𝑏× w + πΆπΆπ‘Ÿπ‘Ÿ Γ—π‘€π‘€π‘žπ‘ž + 𝐢𝐢𝑖𝑖 Γ—πΏπΏπ‘žπ‘ž + 𝐢𝐢𝑠𝑠 𝑠𝑠 (4-17) During the pick and place task, the robot does not pick up all arriving part as the conveyor move past the vision camera. Other parts parts were recycled to join the next arriving parts.

During the travelling period of parts, there exists a continuous movement of part whereby a

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minimum distance/gap𝛼𝛼, exist between the work envelope and the clear part boundaries. For effective throughput computation, the distance between two arriving part must not be less than 𝛼𝛼 for the robotic arm gripper to easily pick-up parts. Assuming the number of visible parts fed along the work envelope is 𝑁𝑁𝑓𝑓, the center point of the arriving part must be known.

This can be obtained by considering the fact that the height and width of work envelope is the same. The diameter of the part fed in this model was assumed to be less than 20mm.

Assuming both belts has the same velocity, the throughput equation can be easily obtained with the set of parameters and assumption made.

Assumptions

οƒ˜ System is not saturated or starved.

οƒ˜ There exist continious motion of buffer belt

οƒ˜ The width and height of work envelope is assumed to be equal

οƒ˜ All part are well secured to avoid falling off

Using the following notations and working parameters, suitable equations were derived.

Notation/parmeters

𝑁𝑁𝑓𝑓 =π‘›π‘›π‘π‘π‘šπ‘šπ‘π‘π‘’π‘’π‘Ÿπ‘Ÿ 𝑓𝑓𝑓𝑓 π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘ 𝑓𝑓𝑒𝑒𝑝𝑝 π‘‘π‘‘β„Žπ‘Ÿπ‘Ÿπ‘“π‘“π‘π‘π‘Žπ‘Žβ„Ž π‘‘π‘‘β„Žπ‘’π‘’ π‘€π‘€π‘“π‘“π‘Ÿπ‘Ÿπ‘‘π‘‘ π‘’π‘’π‘›π‘›π‘£π‘£π‘’π‘’π‘Žπ‘Žπ‘“π‘“π‘π‘π‘’π‘’ π‘šπ‘š = 𝑒𝑒π‘₯π‘₯π‘π‘π‘’π‘’π‘šπ‘šπ‘‘π‘‘π‘’π‘’π‘π‘ π‘›π‘›π‘π‘π‘šπ‘šπ‘π‘π‘’π‘’π‘Ÿπ‘Ÿ 𝑓𝑓𝑓𝑓 π‘šπ‘šπ‘Žπ‘Žπ‘’π‘’π‘šπ‘šπ‘Ÿπ‘Ÿπ‘’π‘’π‘π‘ π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘π‘ π‘ 

𝑝𝑝 = π‘π‘π‘–π‘–π‘šπ‘šπ‘šπ‘šπ‘’π‘’π‘‘π‘‘π‘’π‘’π‘Ÿπ‘Ÿ 𝑓𝑓𝑓𝑓 𝑓𝑓𝑒𝑒𝑝𝑝 π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘

π‘šπ‘š = π‘šπ‘šπ‘–π‘–π‘›π‘›π‘–π‘–π‘šπ‘šπ‘π‘π‘›π‘› π‘šπ‘šπ‘Žπ‘Žπ‘’π‘’π‘šπ‘šπ‘Ÿπ‘Ÿπ‘šπ‘šπ‘›π‘›π‘šπ‘šπ‘’π‘’ π‘Ÿπ‘Ÿπ‘’π‘’π‘žπ‘žπ‘π‘π‘–π‘–π‘Ÿπ‘Ÿπ‘’π‘’π‘π‘ = 𝑝𝑝/2 𝑀𝑀 = π»π»π‘’π‘’π‘–π‘–π‘Žπ‘Žβ„Žπ‘‘π‘‘ π‘šπ‘šπ‘›π‘›π‘π‘ π‘€π‘€π‘–π‘–π‘π‘π‘‘π‘‘β„Ž 𝑓𝑓𝑓𝑓 π‘€π‘€π‘“π‘“π‘Ÿπ‘Ÿπ‘‘π‘‘π‘π‘π‘–π‘–π‘’π‘’π‘šπ‘šπ‘’π‘’

𝑃𝑃𝑏𝑏 =π‘π‘π‘Ÿπ‘Ÿπ‘“π‘“π‘šπ‘šπ‘π‘π‘–π‘–π‘Žπ‘Žπ‘–π‘–π‘‘π‘‘π‘π‘ π‘‘π‘‘β„Žπ‘šπ‘šπ‘‘π‘‘ π‘šπ‘š π‘€π‘€π‘“π‘“π‘Ÿπ‘Ÿπ‘‘π‘‘ β„Žπ‘šπ‘šπ‘ π‘  𝑝𝑝𝑒𝑒𝑒𝑒𝑛𝑛 π‘šπ‘šπ‘Žπ‘Žπ‘’π‘’π‘šπ‘šπ‘Ÿπ‘Ÿπ‘’π‘’π‘π‘ π‘“π‘“π‘Ÿπ‘Ÿπ‘“π‘“π‘šπ‘š π‘π‘π‘“π‘“π‘π‘π‘›π‘›π‘π‘π‘Ÿπ‘Ÿπ‘π‘ 𝑓𝑓𝑓𝑓 π‘‘π‘‘β„Žπ‘’π‘’ π‘€π‘€π‘“π‘“π‘Ÿπ‘Ÿπ‘‘π‘‘ π‘’π‘’π‘›π‘›π‘£π‘£π‘’π‘’π‘Žπ‘Žπ‘“π‘“π‘π‘π‘’π‘’ 𝑣𝑣= π‘£π‘£π‘’π‘’π‘Žπ‘Žπ‘“π‘“π‘šπ‘šπ‘–π‘–π‘‘π‘‘π‘π‘ 𝑓𝑓𝑓𝑓 π‘π‘π‘’π‘’π‘Žπ‘Žπ‘‘π‘‘

πœ†πœ†π‘Ÿπ‘Ÿ = π‘‘π‘‘β„Žπ‘Ÿπ‘Ÿπ‘“π‘“π‘π‘π‘Žπ‘Žβ„Žπ‘π‘π‘π‘π‘‘π‘‘ π‘Ÿπ‘Ÿπ‘šπ‘šπ‘‘π‘‘π‘’π‘’ 𝑓𝑓𝑓𝑓 π‘Ÿπ‘Ÿπ‘“π‘“π‘π‘π‘“π‘“π‘‘π‘‘

π‘Ÿπ‘Ÿπ‘π‘ = π‘Ÿπ‘Ÿπ‘šπ‘šπ‘‘π‘‘π‘’π‘’ 𝑓𝑓𝑓𝑓 π‘šπ‘šπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘–π‘–π‘£π‘£π‘šπ‘šπ‘Žπ‘Ž 𝑓𝑓𝑓𝑓 π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘π‘ π‘  π‘“π‘“π‘Ÿπ‘Ÿπ‘“π‘“π‘šπ‘š π‘šπ‘šπ‘“π‘“π‘›π‘›π‘£π‘£π‘’π‘’π‘π‘π‘“π‘“π‘Ÿπ‘Ÿ 𝑑𝑑𝑓𝑓 π‘ π‘ π‘’π‘’π‘Ÿπ‘Ÿπ‘£π‘£π‘’π‘’π‘Ÿπ‘Ÿ 𝑖𝑖𝑛𝑛 π‘ƒπ‘ƒπ‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘π‘ π‘ /π‘šπ‘šπ‘šπ‘š2

The renewal theorem of impatient customer is implemented for deriving a suitale equation for optimal throughput. The velocity of the conveyor belt is asuumed to be the same and represented with v, using equation 17, the conveyor speed was determined. Where 𝐷𝐷 is the diameter of the motor pulley, and 𝑛𝑛 represent the number of revolutions per seconds.

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𝑉𝑉=πœ‹πœ‹πœ‹πœ‹π‘›π‘›60 (4-18) 𝑝𝑝𝑏𝑏 =(π‘€π‘€βˆ’π‘‘π‘‘βˆ’2𝛼𝛼)2

(π‘€π‘€βˆ’π‘‘π‘‘)2 (4-19) Given the centre of the work envelope as c, to obtain the centre of work envelope, the width and height of the work envelope was assumed to be equal. The diameter of the conveyor belt with the speed (v) was considered in equation 19.

π‘šπ‘š= π‘€π‘€πœ‹πœ‹(𝛼𝛼+𝑑𝑑)2

(π‘€π‘€βˆ’π‘‘π‘‘)2 𝑣𝑣 (4-20) Similarly, there exist another arrival rate denoted as π‘Ÿπ‘Ÿπ‘π‘ which represents the arrival of parts from coveyor to the server (robot). Equation 20 was used to calculate the value for π‘Ÿπ‘Ÿπ‘π‘.

π‘Ÿπ‘Ÿπ‘π‘ = 1𝑐𝑐 π‘€π‘€πœ‹πœ‹(𝛼𝛼+𝑑𝑑)(π‘€π‘€βˆ’π‘‘π‘‘)2𝑣𝑣2 (4-21)

Also, the number of parts fed through the work envelope can be obtained by considering the width and height of work envelope, the velocity of conveyor, the probability of cleared work from work envelope, centre of work envelope, and the arrival rate of parts from buffer station to the pick up point. This is illustrated in equation 21. Equation 22 can be used to obtain the number of arrivin parts fed through the conveyor to the robotic system.

�𝑁𝑁𝑓𝑓�=�𝑀𝑀𝑣𝑣� π‘Ÿπ‘Ÿπ‘π‘π‘π‘π‘π‘ π‘’π‘’βˆ’π‘Ÿπ‘Ÿπ‘π‘π‘π‘ (4-22)

The throughput rate can be obtained using equation (23).

πœ†πœ†π‘Ÿπ‘Ÿ = 𝑁𝑁𝑓𝑓

[�𝑀𝑀𝑣𝑣�+ 𝑑𝑑𝑝𝑝+�𝑁𝑁𝑓𝑓� 𝑑𝑑𝑐𝑐] (4-23) The Engineering equation solver (EES) was used to solve the equations numerically and the results of the analysis were simulated using MATLAB software to achieve the best set of parameters for the system. With the set of equations and parameters assumed, optimal throughput was derived for a pick and place task.

Assumed values; 𝑝𝑝 = 14.4, π‘Šπ‘Š = 250π‘šπ‘š,𝑉𝑉 = 256 π‘Ÿπ‘Ÿπ‘’π‘’π‘£π‘£ π‘π‘π‘’π‘’π‘Ÿπ‘Ÿ π‘šπ‘šπ‘–π‘–π‘›π‘›π‘π‘π‘‘π‘‘π‘’π‘’π‘ π‘ ,πœ†πœ†π‘π‘ = 0.625,𝑛𝑛= 340 π‘Ÿπ‘Ÿπ‘’π‘’π‘£π‘£ π‘π‘π‘’π‘’π‘Ÿπ‘Ÿ π‘šπ‘šπ‘–π‘–π‘›π‘›π‘π‘π‘‘π‘‘π‘’π‘’π‘ π‘ ,π‘šπ‘š=𝑑𝑑2= 7.2π‘šπ‘šπ‘šπ‘š,

Lambdaπœ†πœ†= 0.5π‘π‘π‘šπ‘šπ‘Ÿπ‘Ÿπ‘‘π‘‘π‘ π‘  π‘π‘π‘’π‘’π‘Ÿπ‘Ÿ π‘ π‘ π‘’π‘’π‘šπ‘š,πœ‡πœ‡= 0.4, π‘šπ‘šπ‘€π‘€ =50, π‘šπ‘šπ‘π‘= 0.1,

π‘šπ‘šπ‘Ÿπ‘Ÿ = 60, π‘šπ‘šπ‘–π‘– = 10, π‘šπ‘šπ‘ π‘  = 0.5, π‘‘π‘‘π‘šπ‘š = 0.13𝑠𝑠,𝑑𝑑𝑣𝑣𝑖𝑖𝑠𝑠= 0.1𝑠𝑠,𝑑𝑑𝑝𝑝= 0.12𝑠𝑠,π‘‘π‘‘π‘Ÿπ‘Ÿ = 0.125𝑠𝑠,𝑑𝑑𝑑𝑑 = 0.15𝑠𝑠. π‘“π‘“π‘Ÿπ‘Ÿπ‘“π‘“π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘Žπ‘Žπ‘šπ‘šπ‘π‘π‘Žπ‘Žπ‘šπ‘šπ‘‘π‘‘π‘–π‘–π‘“π‘“π‘›π‘›π‘ π‘ ,𝑑𝑑𝑐𝑐 = 0.625secs

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