22-Mec-B4 Integrated Manufacturing Systems · December 2016

Question 4 of 6: Economic Order Interval and Total Annual Inventory Cost

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 07-Mec-B4 — Integrated Manufacturing Systems, National Exams December 2016. Three hours, open book, any non-communicating calculator permitted. Six questions are printed; any five constitute a complete paper and all questions are of equal value, so each is worth 20 marks on a five-question basis. Only the first five questions appearing in the answer book are marked. All six are solved here.

Reference texts. The paper draws on the operations and facilities side of manufacturing engineering rather than on process metal cutting, so the useful shelf is:

E. S. Buffa and R. K. Sarin, Modern Production / Operations Management, 8th ed. — facility layout and operation sequence analysis, materials handling, inventory systems, production planning and control, dispatching.
M. P. Groover, Automation, Production Systems, and Computer-Integrated Manufacturing, 5th ed. — group technology, cellular manufacturing and rank order clustering (Ch. 15).
R. B. Chase and F. R. Jacobs, Operations and Supply Chain Management, 16th ed. — forecasting model selection, inventory control, statistical quality control.
S. Nahmias and T. L. Olsen, Production and Operations Analysis, 7th ed. — economic order quantity, order intervals, safety stock.
D. C. Montgomery, Introduction to Statistical Quality Control, 8th ed. — inspection strategy, quality information systems, process capability.

Canadian practice is assumed throughout: handling and lifting design is governed by the applicable provincial occupational health and safety regulation and by CSA standards (for example CSA B335 for lift trucks), and quality records are kept to satisfy ISO 9001 as adopted by CSA.

Question 4: Economic Order Interval and Total Annual Inventory Cost (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.

Given. Two independent inventory situations, the first stated in dollars of purchase value and the second in physical units.

Given data
Symbol	Quantity	Value
Part (a)
D	Annual purchase value from the supplier	$260,000 per year
a	Order cost, as a fraction of value	0.01
i	Carrying cost, fraction of average inventory value per year	0.18
Part (b)
D	Annual demand, steady rate	40,000 parts per year
S	Procurement cost per order	$60.00
H	Carrying cost on average inventory	$0.20 per unit per year
B	Minimum planned inventory level (policy floor)	500 parts
—	Replenishment	Instantaneous; lead time reliable

Find. (a) the number of weeks of supply that should be bought on one order, and (b) the total annual inventory cost of the assembly part under its minimum-stock policy.

Check — reading of "1% of the value of each order". Taken literally as one per cent of whatever is ordered, the ordering charge would be $0.01Q$ per order and the annual ordering cost would be $(D/Q)(0.01Q)=0.01D$, a constant independent of the order size; the economic order quantity would then collapse to zero and the question would have no answer. The reading that makes the question well posed, and the one intended by the textbook problem, is that the cost of placing an order is one per cent of the annual value bought from that supplier, so $S=0.01D$, i.e. $2,600 per order. This is stated as an assumption in accordance with Note 1 on the cover page. The method below is unchanged under any fixed value of $S$; only the numerical interval moves.

Approach. Both parts are economic order quantity problems: part (a) is worked in dollars and converted to a time supply, and part (b) is worked in units with the policy minimum added as a constant block of stock that is carried but never cycled.

Part (a) — fix the ordering cost per order. On the reading declared above, each order costs one per cent of the annual purchase value, $$S=aD=0.01\times 260{,}000$$ which is $2,600 per order. The carrying charge is 18 per cent per year applied to the average dollar value held, so both cost elements are already expressed in dollars and no unit price is needed.
Write the economic order quantity in dollar terms. With $Q$ measured as the dollar value bought on one order, the annual ordering cost is $(D/Q)S$ and the annual carrying cost is $i(Q/2)$. Differentiating their sum and setting it to zero gives the familiar square-root form $$Q^{*}=\sqrt{\frac{2DS}{i}}=\sqrt{\frac{2(260{,}000)(2{,}600)}{0.18}}$$ which evaluates to $86,666.67 of material on each order.
Convert the order size into a time supply. Dividing by the annual usage rate turns dollars into years of cover, $$T^{*}=\frac{Q^{*}}{D}=\frac{86{,}666.67}{260{,}000}=0.3333\ \text{yr},$$ so on a 52-week year the answer is $\boxed{T^{*}=0.333\ \text{yr}=17.3\ \text{weeks of supply}}$, that is, three orders a year at intervals of about four months.
Note the shortcut the question is really testing. Substituting $S=aD$ into the previous two steps, the annual dollar volume cancels completely: $$T^{*}=\frac{1}{D}\sqrt{\frac{2D(aD)}{i}}=\sqrt{\frac{2a}{i}}=\sqrt{\frac{2(0.01)}{0.18}}=\sqrt{0.1111}=0.3333\ \text{yr}.$$ The economic time supply depends only on the two percentages, not on how much is bought, which is why the question can be answered in weeks without knowing a unit price or a quantity. It also means the same interval applies to every supplier billed on the same percentage terms.
Confirm the answer by costing it. At three orders a year the ordering cost is $3\times 2{,}600=7{,}800$ dollars and the carrying cost is $0.18\times(86{,}666.67/2)=7{,}800$ dollars, giving a total relevant cost of $15,600 per year. The two components are equal, which is the signature of a correctly located economic order quantity and is worth writing down as a check on any answer of this type.
Part (b) — compute the economic order quantity in units. Here the ordering cost is a fixed $60.00 per order and the carrying cost is $0.20 per unit per year on the average level, so $$Q^{*}=\sqrt{\frac{2DS}{H}}=\sqrt{\frac{2(40{,}000)(60)}{0.20}}=\sqrt{24{,}000{,}000}$$ which is 4,898.98, say $\boxed{Q^{*}\approx 4{,}899\ \text{parts per order}}$. The corresponding order frequency is $40{,}000/4{,}899=8.16$ orders per year, about one every six and a half weeks.
Build the average inventory from its two parts. Because replenishment is instantaneous, the cycle stock ramps linearly from $Q^{*}$ down to the policy floor and averages $Q^{*}/2$; the floor of 500 parts is never consumed, so it sits underneath the sawtooth as a constant. Hence $$\bar{I}=\frac{Q^{*}}{2}+B=\frac{4{,}898.98}{2}+500=2{,}949.49\ \text{parts}.$$ It is the whole of this average, floor included, that attracts the carrying charge.
Cost the three components. Ordering costs $(D/Q^{*})S=8.165\times 60=489.90$ dollars a year; carrying the cycle stock costs $H(Q^{*}/2)=0.20\times 2{,}449.49=489.90$ dollars a year, again equal to the ordering cost as it must be at the economic quantity; and carrying the policy floor costs $HB=0.20\times 500=100.00$ dollars a year, which is fixed and plays no part in choosing $Q^{*}$.
Add them for the total annual inventory cost. $$TC=\sqrt{2DSH}+HB=979.80+100.00$$ so $\boxed{TC=1{,}079.80\ \text{dollars per year}}$. The purchase price of the parts is deliberately not included, because it is not given and, being independent of the order size at a constant unit price, would not affect the decision in any case.
Read what the safety policy is buying. The $100 spent carrying the 500-part floor is 9.3 per cent of the total inventory cost, and it buys about four and a half days of cover at the usage rate of 40,000 a year. Given that the supplier is described as exceptionally reliable, that is a cheap and defensible insurance premium; the important point for the answer is that the floor changes the cost but not the order quantity, since it enters the total as an additive constant.

Final results — Question 4
Quantity	Value
(a) Ordering cost per order, on the declared reading	$2,600
(a) Economic order quantity, in purchase value	$86,666.67
(a) Economic order interval	0.333 yr = 17.3 weeks of supply (3 orders per year)
(a) Total relevant cost at that interval	$15,600 per year
(b) Economic order quantity	4,899 parts (8.16 orders per year)
(b) Average inventory carried	2,949 parts
(b) Annual ordering cost	$489.90
(b) Annual carrying cost, cycle stock	$489.90
(b) Annual carrying cost, 500-part floor	$100.00
(b) Total annual inventory cost	$1,079.80 per year

Topic. Economic order quantity and economic order interval, including the percentage-cost form and the effect of a minimum-stock policy.

Key relations. $Q^{*}=\sqrt{2DS/H}$; $T^{*}=Q^{*}/D$; with an ordering charge set as a fraction $a$ of the annual value and a carrying rate $i$, $T^{*}=\sqrt{2a/i}$ independently of the dollar volume; total cost $TC=\sqrt{2DSH}+HB$ where $B$ is a constant safety or policy floor.

Why this works. The economic order quantity is where the falling ordering cost curve and the rising carrying cost curve cross, which is why the two annual components are equal at the optimum — a one-line check that catches most arithmetic errors. A constant floor of stock adds a fixed carrying charge to the total but contributes no derivative, so it shifts the cost curve upward without moving its minimum; the order quantity is therefore unaffected by safety stock, and only the cost is. The cost curve is also famously flat near the optimum, which is what makes rounding 17.33 weeks to a convenient four-month cycle harmless.

Common pitfall. Two errors are common here. The first is taking the average inventory in part (b) as $Q^{*}/2$ alone and losing the $100 that the policy floor costs. The second is trying to apply the percentage ordering charge to the order quantity itself in part (a), which makes the annual ordering cost constant and destroys the optimum altogether; recognising that and declaring the working assumption is part of the answer.

Source. Nahmias and Olsen, Production and Operations Analysis, 7th ed., Ch. 4 (deterministic inventory models, EOQ sensitivity, order intervals); Buffa and Sarin, Modern Production / Operations Management, 8th ed., inventory chapters; Chase and Jacobs, Operations and Supply Chain Management, 16th ed., Ch. 20.

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