Question 1 of 5: Aggregate Production Planning and Preventive Maintenance

Question 1: Aggregate Production Planning and Preventive Maintenance (20 marks)

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.

Part (a) — the three controllable variables and the four major costs

Aggregate production planning works at the level of a single common measure of output — equivalent units, standard hours, tonnes — over an intermediate horizon of roughly three to eighteen months, bridging the long-range capacity decision above it and the master production schedule below it. Demand over that horizon is a forecast and is therefore not a decision variable; neither is the installed plant capacity, which was fixed by an earlier long-range decision. Only three quantities are genuinely under the planner's control.

The first is work-force size: the number of production employees on the payroll, altered by hiring and by lay-off. The second is the production rate obtained from a given work force, altered by authorising overtime, by accepting undertime or idle time, and by subcontracting work out. The third is the inventory level carried from period to period, together with its negative counterpart, the backlog of unfilled orders. Every aggregate plan — however it is computed — is nothing more than a time path for these three quantities, and the whole art of the problem is that demand variability must be absorbed by some combination of them.

Four cost categories are relevant to that choice, and a plan is judged on their sum:

• Basic production costs. The fixed and variable costs of producing at the normal rate: direct labour at regular time, materials, and the variable overhead attached to output, plus premium payment for output produced on overtime.
• Costs associated with changes in the production rate. Recruiting, selection and training on the way up; severance, notice and unemployment-insurance experience-rating on the way down; and the productivity loss while a new crew climbs its learning curve. These are the costs a level plan exists to avoid.
• Inventory holding costs. The opportunity cost of the capital tied up in finished goods, plus storage space, handling, insurance, property taxes, shrinkage, spoilage and obsolescence. A holding rate of 20 to 35 per cent of item value per year is the usual industrial range.
• Backordering or stockout costs. Expediting and split shipments, contractual penalties, lost contribution margin on sales that go elsewhere, and the loss of customer goodwill. This is the hardest of the four to measure and is normally represented in the model by an assumed penalty per unit short per period.

Part (b) — pure and mixed strategies

A pure strategy absorbs the whole of the demand fluctuation with a single controllable variable while holding the others fixed. The two textbook extremes are the pure chase plan, in which the work force is hired and laid off period by period so that output tracks demand exactly and no seasonal inventory is carried, and the pure level plan, in which work force and production rate are held constant and inventory build-up and backorders absorb every fluctuation. A pure overtime/undertime plan and a pure subcontracting plan are the other two members of the family. Their virtue is that each is trivially easy to cost and to explain, so the four of them are normally evaluated first and used as bounds on what any better plan can achieve.

A mixed strategy uses two or more of the controllable variables at once: for example, a modest work-force increase in the spring, some overtime at the peak, a seasonal inventory build in the shoulder months, and subcontracting for the last slice of the peak. Because each pure strategy pushes one cost category to an extreme — the chase plan maximises hiring and lay-off cost while the level plan maximises holding and backorder cost — and because the total-cost surface is generally convex in the neighbourhood of the optimum, a mixed plan is almost always cheaper than the best pure plan. The price paid is that the search is combinatorial: with two decision variables per period over a twelve-month horizon there are twenty-four unknowns, so the planner needs a formal procedure (linear programming, the transportation method, the linear decision rule, or a computer search) rather than a spreadsheet trial. Practical constraints usually settle the matter anyway: collective agreements limit lay-offs, overtime is capped by fatigue and by statute, subcontract capacity is finite, and warehouse space is finite — so a feasible plan is almost necessarily a mixed one.

Part (c) — the Search Decision Rule method

The Search Decision Rule (SDR), due to Taubert, treats aggregate planning as an unconstrained numerical-optimisation problem rather than as an analytical one. It proceeds in five stages. First, a cost model is written as a computer subroutine that accepts a complete trial schedule and returns the total cost over the planning horizon; the model may take any mathematical form whatever — non-linear, piecewise, discontinuous, containing step charges for a shift added or a supplier engaged — which is precisely the restriction that the linear decision rule cannot escape, since it requires every cost to be quadratic. Second, the decision vector is defined: typically the work-force level and the production rate in each of the N periods, so the search runs in 2N dimensions. Third, a feasible starting schedule is supplied, usually last year's plan or a level plan. Fourth, a direct-search routine — Hooke and Jeeves pattern search, or a conjugate-direction method — makes exploratory perturbations of each variable in turn, retains any move that lowers total cost, makes a longer pattern move in the direction that worked, and contracts the step size whenever a full exploration produces no improvement. Fifth, the search stops when the step size falls below a tolerance, and only the first period's decisions are implemented; the whole calculation is repeated next period on a rolling horizon as the forecast is updated.

Its strengths are realism and flexibility: the company's actual cost structure can be programmed as it is, constraints can be enforced by penalty terms, and no distortion of the cost function is needed to make the mathematics tractable. Its weaknesses follow from the same freedom. A direct search guarantees only a local optimum, so the answer depends on the starting point and different starts should be tried; the computation grows quickly with the horizon; and the quality of the plan can never exceed the fidelity of the cost model, which must be validated by running it retrospectively against historical decisions before it is trusted.

Part (d) — when preventive maintenance is appropriate

Preventive maintenance means intervening on a schedule — inspection, servicing, or replacement — before failure occurs, and it is worth doing only when a set of conditions hold together. The governing one is that the hazard rate must be increasing with age or usage: the item must be in the wear-out region of the bathtub curve, so that a used unit is genuinely more likely to fail in the next interval than a fresh one. If time to failure is exponential, the hazard rate is constant, a survivor is as good as new, and replacing it on a schedule buys nothing at all — the correct policy there is run-to-failure or condition monitoring.

Beyond that the usual conditions are these. The consequence of failure must be expensive relative to the intervention: lost output on a bottleneck machine, scrap or rework of material in process, secondary damage to the machine itself, a safety or environmental incident, or a warranty claim. Failure must be reasonably predictable — the scatter in time to failure must be modest, or the approach of failure must be detectable in advance by inspection, vibration analysis, oil analysis or thermography, which is what turns a fixed-interval policy into a condition-based one. The equipment must be able to be taken out of service on a planned basis, which means either a natural shutdown window, a weekend, or standby redundancy, since preventive work that itself stops production destroys much of the benefit. The direct cost of the preventive action must be small compared with the cost of the failure it averts, which is the economic form of the whole argument. And the intervention must not itself induce infant mortality: if reassembly errors are common, a policy of frequent teardown can raise the failure rate rather than lower it. In practice preventive maintenance is applied to lubrication, filter and belt changes, bearings, seals, cutting-tool replacement, and to safety-critical items where regulation requires it, while electronic components with constant hazard rates are generally left to run.