22-Mec-B4 Integrated Manufacturing Systems · May 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, May 2015 — 07-Mec-B4 Integrated Manufacturing Systems. Three hours, open book, any non-communicating calculator permitted. Seven questions are printed; any five constitute a complete paper and only the first five appearing in the answer book are marked, each of equal value (20 marks). All seven are solved here, because the complete set is the study resource. Questions 5, 6 and 7 are explicitly essay questions, in which the examiners award marks for clarity and organisation as well as content.
Reference texts. E. S. Buffa and R. K. Sarin, Modern Production / Operations Management, 8th ed. (requirements schedules, economic lot size, economic order interval, part-period balancing, production planning); R. B. Chase and F. R. Jacobs, Operations and Supply Chain Management, 16th ed. (demand components, adaptive forecasting, aggregate planning, statistical quality control); B. W. Niebel and A. Freivalds, Methods, Standards, and Work Design, 13th ed. (time study, performance rating, allowances, wage incentive plans); D. C. Montgomery, Introduction to Statistical Quality Control, 8th ed. (Shewhart charts, process capability); M. P. Groover, Automation, Production Systems, and Computer-Integrated Manufacturing, 5th ed. (process planning, CAPP, machinability data systems, maintenance); S. Nahmias and T. L. Olsen, Production and Operations Analysis, 7th ed. (forecasting, aggregate planning).
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.
Aggregate production planning reconciles a forecast of demand, which the planner cannot control, with the resources of the plant, which the planner can. Three variables are controllable. The first is the production rate — the output per unit time achieved with a given work force, adjusted through overtime, undertime, extra shifts, subcontracting and changes in line speed. The second is the work-force size, adjusted by hiring and by layoff, which changes the plant’s underlying capacity rather than how hard it is being pushed. The third is inventory — and, where the market tolerates it, backlog — which decouples the rate of production from the rate of sale and so allows the first two variables to be held steadier than demand.
These three are the levers; the plan is chosen by costing them. Four cost categories dominate. Basic production (payroll) costs are the regular-time labour and the direct material and overhead of making the product at the planned rate. Costs associated with changing the production rate are the hiring, training, and layoff or severance costs of moving the work force up and down, together with the learning-curve losses and quality problems that accompany a new crew. Overtime, undertime and subcontracting costs are the premium paid to raise output without hiring, the cost of carrying an underused work force rather than laying it off, and the margin surrendered to a subcontractor. Inventory-related costs are the carrying charge on finished stock — capital, storage, insurance, obsolescence — together with the backorder, expediting and lost-sales costs incurred when the plan is short. An aggregate plan is nothing more than a chosen point in the trade-off among these four, and the plan that minimises any one of them in isolation is almost always expensive on the others.
A pure strategy uses only one of the controllable variables to absorb the whole of the demand variation, holding the others fixed. Three pure strategies are recognised. The chase strategy matches production to demand period by period by hiring and laying off, so inventory is held near zero and the entire cost of variation is paid in work-force change; it suits low-skill labour markets, perishable output and make-to-order work, and it is destructive where skills are scarce or a collective agreement restricts layoff. The level strategy holds both the work force and the production rate constant and lets inventory and backlog absorb every fluctuation; it gives a stable, skilled work force and smooth operations, and it pays for them in carrying cost and in the risk of obsolescence. The third pure strategy varies output through overtime and subcontracting alone, holding both work force and inventory constant — capacity is rented rather than owned, at a premium.
A mixed strategy uses two or more of the variables together: a modest work-force change to follow the broad seasonal movement, some pre-built inventory into the peak, overtime for the sharpest weeks, and subcontracting for the overflow. Mixed strategies are almost always cheaper, for the straightforward reason that each cost element is convex — the second month of overtime costs more per unit than the first, and inventory accumulated over four months costs more than inventory accumulated over one — so spreading the adjustment across several levers keeps every one of them in its cheap range. The price is complexity: the pure strategies can be evaluated on a single spreadsheet, whereas the number of feasible mixed plans is effectively unbounded, which is exactly the difficulty the search methods of part (c) were invented to overcome. In practice pure strategies are used as bounding cases — they establish what the extremes cost — and the plan adopted is a mix.
The Search Decision Rule (SDR), due to Taubert, abandons the attempt to solve the aggregate planning problem analytically and instead searches it numerically. It rests on one observation: the obstacle to using a realistic cost model is not that the cost cannot be evaluated, but that it cannot be differentiated or forced into the quadratic form the Linear Decision Rule requires or the linear form transportation and linear-programming methods require. If the model need only be evaluated, the modelling restrictions disappear.
The method has three parts. First, a cost model is written as a computer subroutine that accepts a complete plan — a work-force level $W_{t}$ and a production rate $P_{t}$ for each of $N$ periods, so $2N$ decision variables — and returns the total cost of that plan over the planning horizon. Because it is only ever evaluated, the subroutine may contain anything the planner believes to be true: step changes at shift boundaries, non-linear overtime premiums, hiring costs that differ from layoff costs, capacity ceilings, contractual limits on layoff, penalty terms for backorders. Second, a search routine — Hooke and Jeeves’ pattern search in Taubert’s original work — explores the $2N$-dimensional decision space. It begins from a starting plan, makes small exploratory moves in each variable in turn, keeps those that reduce cost, and then makes an accelerated pattern move in the direction that the successful exploratory moves collectively indicate; when a pattern move fails, the step lengths are reduced and exploration resumes. No gradient is computed. Third, the search terminates when the step lengths fall below a tolerance, and the first period of the resulting plan is implemented; the whole search is then repeated next period on a rolled-forward horizon, which is what makes SDR a decision rule rather than a one-off optimisation.
Its strengths and its weakness both follow from this structure. The strength is realism: SDR imposes no restriction whatever on the form of the cost function, so the model can be the planner’s honest description of the plant, and Taubert’s tests reproduced the Linear Decision Rule’s optimum on the paint-company data to within a fraction of a per cent while handling models the LDR could not express. The weakness is that a pattern search guarantees only a local optimum; on a cost surface with several valleys the answer depends on the starting plan, so the search is run from several starts and the results compared. Computationally the method is cheap by modern standards — a few thousand evaluations of a subroutine — and the practical limit on its use has always been the effort of building an honest cost model, not the search itself.
Preventive maintenance replaces or overhauls on a planned basis, before failure; breakdown maintenance repairs after it. Preventive maintenance is not universally superior, and the conditions that make it the economic choice can be stated precisely.
First, the failure rate must be increasing with age or usage — the wear-out region of the bathtub curve. If failures arrive at a constant rate, an item that has run for a year is exactly as likely to fail in the next hour as a new one, so replacing it early buys nothing and simply consumes life; and in the infant-mortality region planned replacement actively makes things worse. Second, the total cost of a failure must substantially exceed the cost of the planned action. That comparison is rarely about the part: it is about lost production on a bottleneck, consequential damage to the machine, scrapped work in process, emergency labour at premium rates, and expedited freight. Third, the consequences of failure must be severe in a non-financial sense — danger to workers, an environmental release, a regulatory breach, loss of a product batch, or damage to a customer commitment. Where safety or a statutory requirement is engaged, planned maintenance is mandatory regardless of the arithmetic; in Canada this is the practical effect of provincial occupational health and safety regulations and of pressure-equipment and lifting-device inspection requirements.
Fourth, the failure must be reasonably predictable: the distribution of time to failure must have a small enough coefficient of variation that a replacement interval can be chosen which catches most failures without discarding most of the useful life. Where the variance is large, condition monitoring — vibration analysis, oil analysis, thermography — is the better answer, since it triggers on evidence rather than on the calendar. Fifth, the equipment must be critical and lightly buffered: a bottleneck machine with no parallel capacity and no inventory downstream justifies preventive attention that an off-line duplicate does not, and the case is strongest in exactly the environments this subject is about — transfer lines, flexible manufacturing systems and just-in-time cells, where there is no buffer to absorb a stoppage. Sixth, there must be a practicable window: planned work has to be doable during a shutdown, a weekend or a scheduled changeover, and it must not itself introduce failures through poor reassembly.
Where several of these hold, the replacement interval is chosen by balancing the cost of planned replacement per unit time against the expected cost of failures per unit time; the optimum is the interval at which the marginal saving in expected breakdown cost equals the marginal cost of replacing earlier. Where none of them holds — a cheap, non-critical, randomly failing item with ample spares and no safety consequence — run-to-failure is the correct and cheapest policy, and saying so is part of a complete answer.
