# P J P Ltd.’s Stock Control and Lighting Replacement

## Minimizing Stocking Costs at P J P Limited

The givens in this case are that:

• The cost penalty for running out of stock is unbearably high;
• Transaction cost per order is £1,000;
• The firm incurs storage cost of £500 a unit a year.

For purposes of analysis, it is assumed that carrying cost can be averaged out evenly no matter how long the component remains in P J P Ltd. storage. Accordingly, working towards solutions for this case depends on the assumption that storage cost per day is the same and reaches a maximum of £500 if the component stays in storage for 364 days of the year.

Total component volume requirement per year is 4,000 units. Given this quantity, any solution must minimise total storage cost even at the price of incurring greater transaction cost. This is illustrated in the table below where placing a single order to limit transaction cost to just £1,000 in total for the year penalises the firm with holding costs of no less than £1,990,000. Whatever the unit price of this component, the average stocking cost of £497 is plainly excessive.

On the other hand, placing daily orders in order to avoid stocking the component at all (and therefore having absolutely zero holding cost) means 200 separate transactions which the supplier may be inclined to charge for at the usual rate of £1,000 each. In this cases, one sees that average stocking cost per component rises again, though only to £50.

Finding a less costly compromise between these two extremes requires the aforementioned uniformity of average storage cost for whatever length of time a component is stocked and, more crucially, that it is the same for every unit of within one lot, whether that lot is drawn down in just two days or two months. As well, 20 units from each delivery incur no storage cost because they are sent directly to the production area for use the next day.

To illustrate, placing two orders a year immediately halves total holding cost so that average stocking cost per component comes down to £248 (from £498 for a single annual order).

Table A.1:

 Annual Unit Demand Order placement cost Holding costs Total Cost Average stocking cost per component Single order 4,000 £1,000 £1,990,000 £1,991,000 £497.75 Two orders 4,000 £2,000 £990,000 £992,000 £248.00 Four orders 4,000 £4,000 £490,000 £494,000 £123.50 Ten orders 4,000 £10,000 £190,000 £200,000 £50.00 Twenty orders 4,000 £20,000 £90,000 £110,000 £27.50 Thirty-six orders 4,000 £36,000 £45,559 £81,559 £20.39 TWO HUNDRED ORDERS 4,000 £200,000 £0 £200,000 £50.00

Given the number of working days in a month (about 17), the optimal solution is to place three orders each month so that: 1) There is enough to meet at least six workings days production requirement for the component; and, 2) Stocking cost is held as low as possible. Under a 36-order plan, average stocking cost per component will drop to just £20.39.

Why not a 48-order plan then? The answer is that transaction cost for 12 more orders per year will add up to more than total stocking costs (£48,000 and £31,660, respectively). At that, average stocking cost is reduced only marginally, to £19.92.

As to the question posed by item c), producing 100 units of the component daily is inflated because the workforce requires just 20 every day. True, the intended production will reduce overhead and other fixed-cost charges per unit of the component since the annual requirement will be accomplished in just 40 working days or two months. On the other hand, even spreading out internal production more rationally:

• We would have to be sure that the component can be produced in-house for a saving of at least £20.00, the optimal sum of transaction and stocking cost.
• And in-house production cost must generate cost savings versus supplier pricing (both unknown at this time), at least enough to pay for the necessary capex, raw materials, training, extra supervision, spoilage as the fabricators for the component ride the learning curve down, workspace and other overhead costs.

Since the crucial items of workers’ wages, supplier unit pricing and total internal production cost are unknown, it is too speculative and laborious (for a paper of this length) to generate all the reasonable permutations. There is, of course, the cost-free solution of adopting “just-in-time” (JIT) ordering frequency, a manufacturing practice first implemented by Ford Motor Co. and embraced enthusiastically by Toyota (Broyles, Beims, Franko & Bergman, 2005; Lean Deployment, n.d.). If PJP Limited had more than one supplier, it may be feasible to arrange for yearly contracts to supply just two days’ requirement at each delivery. Twenty units of the component would go straight to the production floor to be used immediately and the balance of 20 would go into storage as fallback stock, giving PJP time to order from the other supplier if force majeure were to hinder the preferred one from continuing with deliveries on day 3. That way, the transaction cost is minimised to just £1,000 (as in the last line of Table A1 above) and, since just half of total stock is ever stored and for just one day at a time, the result is an average stocking cost of £1.50. The trade-off, of course, is that PJP would have to manage its supply chain more intimately and this can have many hidden costs.

## Lighting Replacement at the Tricycle Shop Floor

Suppose that the value of lost production was £100 every time a bulb had to be replaced. Embarking on the full-replacement option would cost just £2,500 for no trade-off in production time (see Table below). However, the lesson one applies from fault tree analysis is to account for combined probabilities and to assess multiple failure modes after partial installation or replacement has been done (Kim, Ju, & Gens, 1996; Ficalora, 2010; Reliability Analysis Center, n.d.).

Given known probabilities, even a full replacement of all 1,000 bulbs on the shop floor will be subject to failure rates in subsequent years such that individual replacements would still have to be made and production downtime incurred anyway. Were failures solely to occur at the given probabilities only once across the entire replacement set, the combination of bulb replacement and downtime would still add up to about £106,000 at the end of the fifth year.

The rest of the table below illustrates better than a fault tree what the cumulative impact will be of forsaking the full-replacement option and opting to replace as needed. Even if just a second level of failure probability is accounted for, the total cost is so high (principally because of the high assumed cost for lost production) that it is not even sensible to consider full replacement. The bulbs will fail anyway and total cost to the tricycle manufacturer will be at least a quarter of a million pounds.

Table B.1: Overall Cost of Replacement for Combined Probabilities of Failure

 FULL REPLACEMENT OPTION Actual work Bulbs Cost Production Penalty* Total Year 0 One-time cost £2,500 0 £2,500 Year 1 10% fail 100 £600 £10,000 £10,600 Year 2 20% fail 200 £1,200 £20,000 £21,200 Year 3 20% fail 200 £1,200 £20,000 £21,200 Year 4 30% fail 300 £1,800 £30,000 £31,800 Year 5 20% fail 200 £1,200 £20,000 £21,200 TOTAL 1,000 £6,000 £100,000 £106,000 ONE YEAR LATER Year 2 20% fail 200 £1,200 £20,000 £21,200 Year 3 20% fail 200 £1,200 £20,000 £21,200 Year 4 30% fail 300 £1,800 £30,000 £31,800 Year 5 20% fail 200 £1,200 £20,000 £21,200 ADD: PROBABILITY THAT REPACED BULBS FAIL YEAR 1 10% fail 10 £60 £1,000 £1,060 YEAR TWO TOTAL £96,460 TWO YEARS LATER Year 3 20% fail 200 £1,200 £20,000 £21,200 Year 4 30% fail 300 £1,800 £30,000 £31,800 Year 5 20% fail 200 £1,200 £20,000 £21,200 ADD: PROBABILITY THAT REPACED BULBS FAIL YEAR 1 20% fail 20 £120 £2,000 £2,120 YEAR 2 10% fail 20 £120 £2,000 £2,120 YEAR THREE TOTAL £78,440 THREE YEARS LATER Year 4 30% fail 300 £1,800 £30,000 £31,800 Year 5 20% fail 200 £1,200 £20,000 £21,200 ADD: PROBABILITY THAT REPACED BULBS FAIL YEAR 1 20% fail 20 £120 £2,000 £2,120 YEAR 2 20% fail 40 £240 £4,000 £4,240 YEAR 3 10% fail 20 £120 £2,000 £2,120 YEAR THREE TOTAL £59,360 FOUR YEARS LATER Year 5 20% fail 200 £1,200 £20,000 £21,200 ADD: PROBABILITY THAT REPACED BULBS FAIL YEAR 1 30% fail 30 £180 £3,000 £3,180 YEAR 2 20% fail 40 £240 £4,000 £4,240 YEAR 3 20% fail 20 £120 £2,000 £2,120 YEAR 4 10% fail 30 £180 £3,000 £3,180 YEAR 2 replacements 20% fail 2 £12 £200 £212 YEAR 3 replacements 10% fail 8 £48 £800 £848 YEAR FOUR TOTAL £28,620 FIVE YEARS LATER ADD: PROBABILITY THAT REPACED BULBS FAIL YEAR 1 20% fail 20 £120 £2,000 £2,120 YEAR 2 30% fail 12 £72 £1,200 £1,272 YEAR 3 20% fail 16 £96 £1,600 £1,696 YEAR 4 30% fail 30 £180 £3,000 £3,180 YEAR 2 replacements 30% fail 6 £36 £600 £636 YEAR 3 replacements 20% fail 16 £96 £1,600 £1,696 YEAR 4 replacements 10% fail 1 £6 £100 £106 YEAR FIVE TOTAL £3,392

## Bibliography

Broyles, D., Beims, J., Franko, J. & Bergman, M. (2005) Just-in-time inventory management strategy & lean manufacturing. Web.

Ficalora, J. (2010) Using fault tree analysis instead of failure mode and effects analysis. [Internet], Improvement and Innovation.com. Web.

Lean Deployment (n.d.) Just in time. [Internet] Web.

Kim, C. E., Ju, Y. J. & Gens, M. (1996) Multilevel fault tree analysis using fuzzy numbers. Computers & Operations Research, 23 (7), pp. 695-703

Reliability Analysis Center (n.d.) Fault tree analysis. [Internet], IIT Research Institute. Web.

### Book

Mohr, L.B. (1996) Impact analysis for program evaluation. 2nd ed. London, Sage.

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Bennett, H., Gunter, H. & Reid, S. (1996) Through a glass darkly: images of appraisal. Journal of Teacher Development, , pp.39-46.

Clarke, T.J. (1995) Freud’s Cézanne. Representations, No. 52 Fall, pp.94-122.

Kim, C. E., Ju, Y. J. & Gens, M. (1996) Multilevel fault tree analysis using fuzzy numbers. Computers & Operations Research, 23 (7), pp. 695-703

### WEB

Rutter, L. & Holland, M. (2002) Citing references: the Harvard system [Internet], Poole, Bournemouth University Academic Services.

Lean Deployment (n.d.) Just in time. [Internet] 2010.

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