Question 3 of 7: Machine Space Requirements, and Assembly Line Balancing (RPW Technique)

Question 3: Machine Space Requirements, and Assembly Line Balancing (RPW Technique) (20 marks: i–5, ii–5, iii–10)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(i) Steps to Determine Total Machine Space Requirements for an Entire Facility

Total machine space is built up machine-by-machine and then aggregated, not estimated directly at the facility level: (1) list every machine/equipment item the process design requires, with its model/specification; (2) determine each machine’s static (footprint) space from its manufacturer dimensions — the plan-view area the machine itself occupies; (3) add each machine’s space determination allowances — material-in-process storage immediately at the machine (incoming and outgoing), maintenance/service access clearance, and any auxiliary equipment (controllers, tooling carts) that must sit beside it; (4) add gangway (aisle) space around each machine or machine group for operator and material-handling-equipment access, sized to the handling method chosen; (5) total the individual machine-level space requirements (static + allowances + gangway) into a required area per machine, typically recorded on a standard space-determination template or worksheet; (6) sum every machine’s required area across the department/facility; and (7) apply a building/space-utilization (or “look”) factor to the summed total, since not every square metre of floor area can be perfectly packed with productive space — main aisles, columns, and irregular building shape consume additional area beyond the sum of individual machine requirements. The result is the total machine space requirement that feeds directly into the block-layout and space-relationship-diagram steps that follow.

(ii) Determining the Amount of Space per Machine

Space per machine is determined using a space-determination template (a scale drawing, physical or computerized, of the machine’s footprint together with everything that must be accessible around it), built from four component allowances: (1) static (footprint) space — the machine’s own physical dimensions, taken directly from the manufacturer’s specification sheet; (2) machine allowance — clearance for moving machine parts (e.g., a swinging door, an extending ram, a tool-change envelope) that extend beyond the static footprint during normal operation; (3) gangway (aisle) allowance — space for the operator to work at the machine and for material handling equipment to approach and clear it, sized to the specific handling method serving that machine; and (4) equipment/personnel allowance — space for in-process material staged at the machine, auxiliary equipment, and the operator’s own working area. Summing these four components for one machine (or one machine type, if several identical units are used) gives its required space; where a facility has many identical machines, the same per-machine figure is simply multiplied by the machine count, which is what makes the template approach scale to the total-facility calculation in part (i).

(iii) Line Balancing by the Ranked Positional Weight (RPW) Technique

Given. 14 task elements with times and immediate-predecessor precedence as tabulated above (total content time $\sum t_i=6.0$ min); required output $=65$ units/hr, produced on a single assembly line.

Find. The theoretical minimum and actual number of stations, the station assignment (schematic), and the resulting line efficiency.

Approach. Compute the required cycle time from the output rate, find the theoretical minimum station count, rank every element by its Ranked Positional Weight (own time + the time of every element that must follow it), then assign elements to stations in RPW-rank order — adding the highest-ranked precedence-ready element that still fits the station’s remaining cycle time, and opening a new station only when no ready element fits — finally compute the resulting line efficiency.

1. Cycle time from the required output. $C=\dfrac{60\ \text{min/hr}}{65\ \text{units/hr}}=\boxed{0.9231\ \text{min/unit}}$ (55.4 s/unit).
2. Theoretical minimum number of stations. $N_{min}=\left\lceil\dfrac{\sum t_i}{C}\right\rceil=\left\lceil\dfrac{6.0}{0.9231}\right\rceil=\lceil 6.50\rceil=\boxed{7\ \text{stations}}$.
3. Ranked Positional Weight of every element. $RPW_i=t_i+\sum_{j\in\text{followers}(i)}t_j$ (own time plus every element, direct or indirect, that must come after it). Working from the precedence network:

   Element	1	3	6	5	2	9	11	4	10	13	7	8	12	14
   RPW (min)	6.0	4.6	2.3	2.2	1.8	1.4	1.4	1.4	1.2	0.9	0.9	0.9	0.7	0.6

   This ranking is itself already a valid precedence order (every element’s predecessors carry a strictly higher RPW), so it can be assigned to stations directly.
4. Station assignment (largest-candidate rule). Working down the RPW list, add each precedence-ready element to the open station if it fits the remaining cycle time ($C=0.9231$ min); otherwise leave it for a later station and try the next-ranked ready element:

   Station	Elements (RPW order)	Station time (min)	Idle (min)
   1	1, 3	0.9	0.0231
   2	6, 10	0.9	0.0231
   3	5	0.8	0.1231
   4	2, 11	0.9	0.0231
   5	9	0.8	0.1231
   6	4, 13, 7	0.8	0.1231
   7	8, 12, 14	0.9	0.0231

   Every station time is $\le C=0.9231$ min and every precedence requirement is respected (e.g. element 13 needs 10 and 11, both assigned in earlier stations 2 and 4; element 14 needs 9, 12 and 13, all assigned in earlier stations). This uses exactly $N_{actual}=\boxed{7\ \text{stations}}$ — the theoretical minimum from Step 2 is achieved, so no further re-balancing can reduce the station count.
5. Line efficiency and balance delay. $$\text{Efficiency}=\frac{\sum t_i}{N_{actual}\cdot C}=\frac{6.0}{7(0.9231)}=\frac{6.0}{6.4615}=\boxed{92.9\%}$$ Balance delay $=100\%-92.9\%=\boxed{7.1\%}$ (the idle time built into the line by imperfect divisibility of element times into a common cycle). At $C=0.9231$ min/unit the line’s actual output is $60/0.9231=65.0$ units/hr, meeting the required rate exactly.