(a) A shaft of nominal size 60 mm is to be fitted into a hole. The tolerances are : Hole tolerance – 0·030 mm Shaft tolerance –…

(a) Limits and fits for a 60 mm shaft and hole (10 marks)

Data: nominal size 60 mm, hole tolerance T_h = 0.030 mm, shaft tolerance T_s = 0.020 mm, and minimum clearance 0.010 mm. The hole-basis system is used, so the lower limit of the hole is the basic size.

(ii) Limits.

• Hole: lower limit 60.000 mm, upper limit 60.000 + 0.030 = 60.030 mm.
• Shaft: the upper limit is the smallest hole minus the minimum clearance, 60.000 − 0.010 = 59.990 mm, and the lower limit is 59.990 − 0.020 = 59.970 mm.

(i) Maximum clearance. Largest hole − smallest shaft = 60.030 − 59.970 = 0.060 mm. As a check, 0.010 + 0.030 + 0.020 = 0.060 mm.

(iii) Allowance. The allowance is the intentional difference between the smallest hole and the largest shaft, that is the tightest fit: 60.000 − 59.990 = +0.010 mm (positive, a clearance fit).

The maximum clearance of 0.060 mm does not depend on the choice of basis. (In a shaft-basis system, the shaft would be 60.000/59.980 and the hole 60.010/60.040.)

(b) Facility layout for standardised and customised products (20 marks)

(i) Product layout and process layout compared.

• Material flow: in a product layout, machines are arranged in the sequence of operations, so flow is a smooth, short, linear path with little handling and low work-in-process. In a process layout, similar machines are grouped by function, so parts follow long, criss-crossing routes with much handling, queues and backtracking.
• Flexibility: a product layout is rigid, since it is dedicated to one product or a narrow range, and a change of design means re-laying or re-balancing the line. A process layout is flexible in product mix, routing and design, because general-purpose machines and skilled workers can do many jobs.
• Cost: the product layout needs a large investment in special-purpose equipment and material-handling systems, but the unit cost is low at high volume because of low handling, low inventory and low labour skill. The process layout has a lower investment, but a higher unit cost because of handling, set-ups, work-in-process and more supervision.
• Demand variability: a product layout suits stable, high, predictable demand, and it loses money if demand falls or the product mix changes. A process layout copes with variable demand and a varying mix, since load can be shifted between machines.
• Machine utilisation: in a product layout, utilisation is high only if the line is balanced and demand fills it, and a breakdown stops the whole line. In a process layout, each machine can be used for many jobs, but flow inefficiency and set-ups cut real productive time, and utilisation of individual machines is often 40–60%.

Overall, the product layout is the choice for the standardised products at high volume, and the process layout for the customised products at low volume. A firm making both needs a hybrid.

(ii) Cellular layout. In a cellular layout (based on group technology), parts with similar shape, size or processing are grouped into part families, and the machines needed to complete a family are grouped into a cell placed together. Each cell is a small process-focused unit that behaves like a mini product line for one family. It sits between the two layouts above and suits a mix of standard and customised parts, with the high-volume standard families in dedicated cells or lines.

Advantages.

• Reduced set-up time, since parts in a family need similar tooling and fixtures.
• Shorter travel distances, lower work-in-process, shorter throughput time and lead time.
• Simple scheduling and control, and improved quality, since defects are spotted quickly.
• Team responsibility, multi-skilled workers and better morale.
• Reasonable flexibility combined with much of the efficiency of a product layout.

Challenges.

• Forming part families and cells is difficult and takes time (classification and coding, production flow analysis, clustering methods), and some parts do not fit any family (exceptional parts, bottleneck machines shared between cells).
• Machines may need to be duplicated, and some machines are under-used within a cell, which raises investment and lowers utilisation.
• Loss of flexibility if the product mix or demand changes, since the cells are built round today's families and load between cells may be unbalanced.
• Re-layout cost and disruption of production, and workers need cross-training, teamwork skills and a changed pay and supervision system.
• The management must change its ways, including cell-based performance measures and maintenance.

(c) Continuous-review (Q, R) inventory system (20 marks)

Data: D = 2400 units/year (σ_D = 300 units/year), lead time L = 2 weeks, 50 weeks/year, S = ₹800 per order, h = ₹18/unit/year, unit cost ₹240 and stock-out cost ₹90/unit.

(i) EOQ. Q* = √(2DS/h) = √(2 × 2400 × 800/18) = √213 333 = 462 units (461.9).

(ii) Demand during lead time. The weekly mean demand is 2400/50 = 48 units and the weekly standard deviation is 300/√50 = 42.43 units.

• Mean: μ_L = 48 × 2 = 96 units.
• Standard deviation: σ_L = 42.43 × √2 = 300 √(2/50) = 60 units.

(iii) Safety stock and reorder point. For a 95% service level, z = 1.645:

• Safety stock SS = z σ_L = 1.645 × 60 = 98.7 ≈ 99 units.
• Reorder point R = μ_L + SS = 96 + 98.7 = 194.7 ≈ 195 units.

The policy is therefore to order 462 units whenever the inventory position falls to 195 units. (The stock-out cost is not needed here, because the service level is specified, and the purchase cost of 2400 × 240 = ₹5,76,000 does not vary with the policy.)

(iv) Total annual inventory cost.

• Ordering cost = (D/Q) S = (2400/461.9) × 800 = 5.196 × 800 = ₹4157.
• Cycle-stock holding cost = (Q/2) h = 230.9 × 18 = ₹4157.
• Safety-stock holding cost = SS × h = 98.7 × 18 = ₹1777.

Total = 4157 + 4157 + 1777 = ₹10 090 per year.

Effect of safety stock. Without safety stock, the holding cost is ₹4157, and with it, it is 4157 + 1777 = ₹5934. So the safety stock adds ₹1777 a year, which inflates the holding cost by 42.7% (and the total cost of ₹8314 without safety stock by 21.4%). This is the price of protecting the store against stock-outs during the lead time.