Capacity Planning, Scheduling, Project and Supply Chain Management

Where the previous chapter asked how well a single operation is run, this one asks how much a facility can produce, in what order jobs should pass through it, how a project made of many linked activities gets scheduled, and how the materials feeding all of this are kept moving without excess stock.

1. Capacity planning

Design capacity is the maximum output a facility could achieve under ideal conditions; effective capacity is realistically achievable output once planned factors — maintenance, changeovers, quality checks — are accounted for; actual output is what the facility genuinely produced. These three figures are almost always different from one another, and a candidate should be able to compute the two ratios that connect them.

$\text{Efficiency}=\frac{\text{Actual Output}}{\text{Effective Capacity}},\text{Capacity Utilisation}=\frac{\text{Actual Output}}{\text{Design Capacity}}$

A firm choosing how to add capacity ahead of demand growth faces three broad strategies: a lead strategy adds capacity ahead of expected demand (risking idle capacity if demand does not materialise, but capturing sales that a slower competitor would miss); a lag strategy adds capacity only after demand has clearly materialised (avoiding idle capacity, but risking lost sales during the shortfall).

A match strategy adds capacity in smaller increments that track demand growth as closely as possible, splitting the difference between the two risks.

2. Sequencing — Johnson's rule for two-machine scheduling

When several jobs must each pass through the same two machines in the same order, the sequence in which jobs are released changes total completion time (makespan) even though the total work content is fixed — and Johnson's rule finds the sequence that minimises it.

Johnson's rule, in three steps: (1) list each job's processing time on Machine 1 and Machine 2; (2) find the single smallest processing time across the entire table; (3) if that smallest time belongs to Machine 1, schedule that job as early as possible in the sequence, and if it belongs to Machine 2, schedule that job as late as possible — remove the job and repeat until all jobs are placed.

The intuition worth remembering, beyond the mechanical rule: a job with a short Machine 1 time should go early, so Machine 2 is not left waiting for it, and a job with a short Machine 2 time should go late, so it does not sit finished and waiting while Machine 2 works through the backlog ahead of it.

3. Project management — PERT and CPM

PERT (Programme Evaluation and Review Technique) and CPM (Critical Path Method) both represent a project as a network of activities and dependencies, and both find the critical path — the longest path through the network, which determines the minimum possible project duration.

They differ in one key respect: CPM uses a single, deterministic time estimate per activity (suited to projects with well-known activity durations, such as routine construction), while PERT uses three time estimates per activity — optimistic, most likely and pessimistic — to build a single expected time, suited to projects with genuine uncertainty, such as research and development work.

$\text{PERT Expected Time}=\frac{t_o+4t_m+t_p}{6}$

where $t_o$ is the optimistic time, $t_m$ the most likely time, and $t_p$ the pessimistic time — the weight of 4 on the most likely estimate reflects that it is the single best guess, while the optimistic and pessimistic estimates each contribute one-sixth to widen the estimate towards the possible extremes.

An activity lying on the critical path has zero float (or slack) — any delay to it delays the entire project — while an activity not on the critical path has positive float, meaning it can be delayed by up to that amount without affecting the project's overall completion date.

Crashing a project (paying extra to shorten an activity's duration, typically through overtime or added resources) is only worth doing on a critical-path activity; crashing a non-critical activity wastes money because the project's completion date does not move until the float is fully used up.

4. Supply chain and inventory models

Inventory exists to smooth the mismatch between the rate at which materials are consumed and the rate at which they can practically be procured, and the classical Economic Order Quantity (EOQ) model finds the order size that minimises the total of two opposing costs: ordering cost (which falls as order size rises, since fewer orders are placed) and carrying cost (which rises as order size rises, since more average stock is held).

$EOQ=\sqrt{\frac{2DO}{C}}$

where $D$ is annual demand, $O$ is the ordering cost per order, and $C$ is the carrying cost per unit per year.

Reorder level is the stock level at which a fresh order must be placed so that the new stock arrives before existing stock runs out, accounting for the lead time between placing and receiving an order:

$\text{Reorder Level}=\text{Lead Time}\times \text{Average Usage Rate}(+\text{Safety Stock, if maintained})$

Safety stock is extra stock held above the level lead-time demand alone would require, specifically to absorb unexpected spikes in usage or unexpected delays in delivery, at the cost of additional carrying cost.

Just-In-Time (JIT) is a philosophy of minimising inventory altogether, procuring materials to arrive exactly when needed rather than holding buffer stock, and it depends on highly reliable suppliers and short lead times to work without stockouts.

The bullwhip effect describes how small fluctuations in actual end-customer demand get progressively amplified into much larger swings in order quantities as they travel backward up a multi-tier supply chain, each tier over-reacting to the demand signal it receives from the tier immediately downstream — a well-known supply-chain phenomenon whose remedy is sharing genuine demand information across tiers rather than each tier inferring demand only from the orders it receives.

Worked Examples

Example 1. A plant has a design capacity of 1,000 units/day, effective capacity of 800 units/day, and actual output of 720 units/day. Find its efficiency and capacity utilisation.

Efficiency = 720 ÷ 800 = 90%. Capacity Utilisation = 720 ÷ 1,000 = 72%.

Example 2. Three jobs must pass through Machine 1 then Machine 2 in that order. Processing times (in hours) are: Job A (M1=5, M2=2), Job B (M1=1, M2=6), Job C (M1=9, M2=7). Find the optimal sequence using Johnson's rule.

Smallest time overall is Job B's Machine 1 time (1 hour) → schedule B first. Next smallest is Job A's Machine 2 time (2 hours) → schedule A last. Job C fills the remaining middle slot. Sequence: B → C → A.

Example 3. An activity has optimistic time 4 days, most likely time 6 days, and pessimistic time 14 days. Find its PERT expected time.

Expected Time = (4 + 4×6 + 14) ÷ 6 = (4 + 24 + 14) ÷ 6 = 42 ÷ 6 = 7 days.

Example 4. In a project network, Activity X has an earliest start of day 10 and must finish by day 18 for the project to stay on schedule, but takes only 5 days to complete. Find its float, and state whether it lies on the critical path.

Float = (18 − 10) − 5 = 3 days. Since float is positive (not zero), Activity X does not lie on the critical path.

Example 5. Annual demand for a component is 7,200 units. Ordering cost is ₹100 per order and carrying cost is ₹8 per unit per year. Find the EOQ.

$EOQ=\sqrt{\frac{2\times 7,200\times 100}{8}}=\sqrt{180,000}\approx 424\text{ units}$

Example 6. Lead time for a component is 10 days and average daily usage is 50 units. No safety stock is held. Find the reorder level.

Reorder Level = 10 × 50 = 500 units.

Example 7. A retailer sees a 5% rise in weekly customer demand, but places an order 20% larger than usual to its distributor "to be safe," and the distributor in turn orders 40% more from the manufacturer. Which supply-chain phenomenon does this illustrate, and what is its usual remedy?

The bullwhip effect — small demand fluctuations amplify into much larger order swings further up the supply chain. The usual remedy is sharing actual end-customer demand data across all tiers, rather than each tier inferring demand only from the order it receives from the tier immediately below it.

Summary

Design capacity, effective capacity and actual output are three distinct figures connected by the efficiency and capacity-utilisation ratios, and firms choose between lead, lag and match capacity strategies when planning ahead of demand growth. Johnson's rule sequences jobs across two machines to minimise total completion time, by placing the smallest Machine-1 time early and the smallest Machine-2 time late.

PERT and CPM both find a project's critical path — its longest, zero-float path, which sets the minimum project duration — differing mainly in that PERT uses a three-estimate expected-time formula for uncertain activity durations while CPM uses a single deterministic estimate. Crashing only shortens the project when applied to a critical-path activity.

The EOQ model balances ordering cost against carrying cost to find the order size that minimises total inventory cost, reorder level is set from lead time and usage (plus safety stock where held), and JIT and the bullwhip effect describe, respectively, a philosophy of minimal inventory and a well-known failure mode of poorly coordinated multi-tier supply chains.