The economic order quantity decided one item from its own data. An item can be decided alone only if nothing it consumes is consumed by something else.
A shared resource couples items through a constraint. The cost functions stay separate; one inequality ties them together. Four are common:
A shared setup couples items through a cost: several items on one truck or one machine setup pay part of the ordering cost once.
A third coupling, stock that exists to replenish other stock, closes the chapter.
Let \(n\) represent the number of items, indexed by \(j\); \(\lambda_j\) the demand rate; \(c_j\) the unit cost; \(k_j\) the ordering cost; \(h_j = ic_j\) the holding rate; and \(Q_j\) the order quantity. Let \(a_j\) represent what one unit of item \(j\) consumes of the resource, and \(A\) how much there is.
\[ \begin{aligned} \min\ C &= \sum_{j=1}^{n}\left[\frac{k_j\lambda_j}{Q_j} + \frac{h_j Q_j}{2}\right] &&\text{the sum of the relevant costs}\\ \sum_{j} a_j Q_j &\le A &&\text{a linear limit: space or budget}\\ \text{space}: a_j &= v_j &&\text{$Q_j$ units on the floor after a receipt}\\ \text{budget}: \sum_j c_j\frac{Q_j}{2} \le A &\;\Rightarrow\; a_j = \frac{c_j}{2} &&\text{average on hand is $Q_j/2$, valued at $c_j$}\\ \sum_{j} \frac{\lambda_j}{Q_j} &\le A &&\text{replenishments: not linear} \end{aligned} \]
Space and budget are the same constraint with different coefficients, so one derivation covers both. The replenishment limit is different: consumption falls as the quantities rise.
The objective is separable, a sum of terms each in one item’s quantity, and each term is convex. Without the constraint each item takes its own \(Q_j^{*} = \sqrt{2k_j\lambda_j/h_j}\). That gives three steps, and the first is free:
Step 2 is easy to skip and expensive to skip. You should perform this check first, every time: a constraint that does not bind changes nothing.
Step 3 may use equality because the objective is convex: if the unconstrained optimum violates the limit, the constrained optimum sits on the boundary, where the inequality holds with equality.
Let \(\theta\) represent the Lagrange multiplier, one scalar for the constraint, whatever the number of items:
\[ \begin{aligned} L(\vec{Q}, \theta) &= \sum_{j}\left[\frac{k_j\lambda_j}{Q_j} + \frac{h_j Q_j}{2}\right] + \theta\left[\sum_{j} a_j Q_j - A\right] &&\text{the bracket is zero on the boundary}\\ \frac{\partial L}{\partial Q_j} &= -\frac{k_j\lambda_j}{Q_j^{2}} + \frac{h_j}{2} + \theta a_j = 0 &&j = 1, \ldots, n\\ \frac{\partial L}{\partial \theta} &= \sum_{j} a_j Q_j - A = 0 &&\text{the constraint itself}\\ \frac{k_j\lambda_j}{Q_j^{2}} &= \frac{h_j + 2\theta a_j}{2} &&\text{rearrange each item's equation}\\ Q_j(\theta) &= \sqrt{\frac{2k_j\lambda_j}{h_j + 2\theta a_j}} &&\text{solve for } Q_j \end{aligned} \]
Notice that this is the economic order quantity with \(h_j\) replaced by \(h_j + 2\theta a_j\). The multiplier is a surcharge on holding, in proportion to how much of the scarce resource a unit consumes.
Substituting \(Q_j(\theta)\) into the constraint leaves one equation in one unknown. Let \(R(\theta)\) represent the resource function, what the quantities consume at multiplier \(\theta\) less what is available:
\[ R(\theta) = \sum_{j} a_j\sqrt{\frac{2k_j\lambda_j}{h_j + 2\theta a_j}} - A . \]
Every term decreases as \(\theta\) increases, so \(R\) is strictly decreasing. At \(\theta = 0\) it is positive, because that is the case where the unconstrained quantities violate the limit, and it tends to \(-A\) as \(\theta\) grows.
Thus, there is exactly one root \(\theta^{*} > 0\), and any root-finding method will find it. Bisection is enough: bracket the root, halve the interval, keep the half where \(R\) changes sign, repeat.
The whole procedure is a one-dimensional search wrapped around a formula.
| Question | Answer |
|---|---|
| What is stocked? | Three items, each from its own supplier |
| What is the demand process? | Constant and known |
| When is inventory reviewed? | Continuously, item by item |
| What triggers replenishment, and how much? | An item’s shelf reaching zero; its \(Q_j\), the three decisions |
| What happens to unmet demand? | None occurs; shortages are not allowed |
| What costs are incurred, and when? | $50 per order; 30% a year on value held; at most $10,000 of average investment |
| Item | \(\lambda_j\) | \(c_j\) | \(h_j = 0.30\,c_j\) | \(Q_j^{*}\) unconstrained |
|---|---|---|---|---|
| 1 | 12,000 | $20 | $6.00 | 447.21 |
| 2 | 25,000 | $10 | $3.00 | 912.87 |
| 3 | 8,000 | $15 | $4.50 | 421.64 |
\[ \begin{aligned} \sum_{j} c_j\frac{Q_j^{*}}{2} &= 20\,\frac{447.21}{2} + 10\,\frac{912.87}{2} + 15\,\frac{421.64}{2} &&\text{investment at the free quantities}\\ &= 4{,}472.14 + 4{,}564.35 + 3{,}162.28 = \$12{,}198.77 &&\text{item by item}\\ \$12{,}198.77 &> \$10{,}000 &&\text{the constraint is active} \end{aligned} \]
Under a budget \(a_j = c_j/2\), so the surcharge \(2\theta a_j\) becomes \(\theta c_j\):
\[ Q_j(\theta) = \sqrt{\frac{2k_j\lambda_j}{h_j + \theta c_j}}, \qquad R(\theta) = \sum_j c_j\frac{Q_j(\theta)}{2} - 10{,}000 . \]
Bracket with \(\theta = 0\) (investment $12,198.77, over) and \(\theta = 1\) (investment $5,860.10, under), then halve.
| Step | Low | High | Midpoint \(\theta\) | Investment at midpoint | Keep |
|---|---|---|---|---|---|
| 1 | 0 | 1 | 0.5 | $7,470.19 | under: lower half |
| 2 | 0 | 0.5 | 0.25 | $9,009.39 | under: lower half |
| 3 | 0 | 0.25 | 0.125 | $10,249.02 | over: upper half |
| 4 | 0.125 | 0.25 | 0.1875 | $9,569.50 | under: lower half |
| 5 | 0.125 | 0.1875 | 0.15625 | $9,891.80 | under: lower half |
| 6 | 0.125 | 0.15625 | 0.14063 | $10,065.66 | over: upper half |
| 7 | 0.14063 | 0.15625 | 0.14844 | $9,977.59 | under: lower half |
| 8 | 0.14063 | 0.14844 | 0.14453 | $10,021.33 | over: upper half |
| 9 | 0.14453 | 0.14844 | 0.14648 | $9,999.39 | under: lower half |
Each step halves the bracket, so twenty more steps pin it to six decimals: \(\theta^{*} = 0.14643\).
| Item | \(Q_j\) constrained | Ordering cost | Holding cost |
|---|---|---|---|
| 1 | 366.61 | $1,636.64 | $1,099.82 |
| 2 | 748.33 | $1,670.39 | $1,122.50 |
| 3 | 345.64 | $1,157.28 | $777.69 |
| Total | $4,464.30 | $3,000.00 |
The check. The holding cost total is $3,000, and it has to be:
\[ \begin{aligned} \sum_j h_j\frac{Q_j}{2} &= \sum_j i\,c_j\frac{Q_j}{2} = i\sum_j c_j\frac{Q_j}{2} &&h_j = ic_j\\ &= 0.30 \times \$10{,}000 = \$3{,}000 &&\text{the investment is the budget} \end{aligned} \]
The constraint fixed the investment, so it fixed the holding cost with it. You should perform that check every time a budget is solved, because it catches an error in the multiplier or the quantities immediately.
\[ \begin{aligned} C &= \$4{,}464.30 + \$3{,}000.00 = \$7{,}464.30 &&\text{relevant cost under the limit}\\ \$7{,}464.30 - \$7{,}319.26 &= \$145.04, \quad 145.04/7{,}319.26 = 1.98\% &&\text{the price of the limit}\\ \$12{,}198.77 - \$10{,}000 &= \$2{,}198.77 &&\text{the stock it removes} \end{aligned} \]
When someone proposes tightening or relaxing the working capital account, $145.04 a year is the figure that belongs in the conversation, and you should be prepared to supply it instead of merely complying with the limit.
Notice that ordering and holding are no longer equal: $4,464.30 against $3,000.00. That property holds only at an unconstrained optimum. If you check a constrained solution by looking for balanced costs, you will conclude wrongly that something is broken.
Substitute \(a_j = c_j/2\) and \(h_j = ic_j\):
\[ \begin{aligned} Q_j &= \sqrt{\frac{2k_j\lambda_j}{ic_j + \theta c_j}} = \sqrt{\frac{2k_j\lambda_j}{(i + \theta)c_j}} &&\text{a carrying charge of } i + \theta\\ \frac{Q_j(\theta)}{Q_j^{*}} &= \sqrt{\frac{2k_j\lambda_j}{(i+\theta)c_j}\cdot\frac{ic_j}{2k_j\lambda_j}} = \sqrt{\frac{i}{i + \theta}} &&\text{every item's data cancels}\\ \sqrt{\frac{0.30}{0.44643}} &= 0.81975 &&\text{the belt-tightening factor} \end{aligned} \]
The check: \(366.61/447.21\), \(748.33/912.87\) and \(345.64/421.64\) all equal 0.81975.
A budget constraint changes the scale of the whole portfolio and leaves the proportions between items untouched. Nothing is reallocated; everything shrinks.
The scaling needed \(h_j = ic_j\) and \(a_j = c_j/2\), so that the multiplier entered as a factor \((i + \theta)\) common to every item. It fails:
Thus, uniform rescaling is a property of the budget case, not of constrained lot sizing in general.
The effective carrying charge is \(0.30 + 0.14643 = 0.44643\) a year. A budget constraint and a higher cost of capital are the same thing seen from two sides: both say money tied up in stock is scarcer than the accounting rate admits.
Receiving docks, purchasing staff and inspection limit how often the operation can act. Form the Lagrangian with \(\sum_j \lambda_j/Q_j \le A\):
\[ \begin{aligned} L &= \sum_{j}\left[\frac{k_j\lambda_j}{Q_j} + \frac{h_j Q_j}{2}\right] + \theta\left[\sum_{j}\frac{\lambda_j}{Q_j} - A\right] &&\text{both terms carry } \lambda_j/Q_j\\ \frac{\partial L}{\partial Q_j} &= -\frac{k_j\lambda_j}{Q_j^{2}} - \frac{\theta\lambda_j}{Q_j^{2}} + \frac{h_j}{2} = -\frac{(k_j + \theta)\lambda_j}{Q_j^{2}} + \frac{h_j}{2} = 0 &&\text{they combine}\\ Q_j(\theta) &= \sqrt{\frac{2(k_j + \theta)\lambda_j}{h_j}} &&\text{solve} \end{aligned} \]
The multiplier is now added to the ordering cost: it is a price per order, in dollars. Quantities rise with \(\theta\), and the replenishment count falls, still strictly, so bisection applies unchanged.
The proportions do not survive. \(k_j + \theta\) is a sum, and adding a constant to unequal ordering costs changes them by unequal proportions. A frequency limit reallocates where a budget does not.
The items now differ in what they cost to order, and the dock can process 50 receipts a year.
| Item | \(\lambda_j\) | \(k_j\) | \(h_j\) | \(Q_j^{*}\) | Orders a year | \(k_j + \theta^{*}\) | \(Q_j\) held | Orders | \(Q_j/Q_j^{*}\) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 12,000 | $50 | $6.00 | 447.21 | 26.83 | $114.33 | 676.25 | 17.74 | 1.5121 |
| 2 | 25,000 | $20 | $3.00 | 577.35 | 43.30 | $84.33 | 1,185.53 | 21.09 | 2.0534 |
| 3 | 8,000 | $80 | $4.50 | 533.33 | 15.00 | $144.33 | 716.36 | 11.17 | 1.3432 |
| Total | 85.13 | 50.00 |
\(85.13 > 50\), so the limit binds; bisection gives \(\theta^{*} = \$64.33\) per order.
The check. The orders total 50.00, and each quantity is the economic order quantity with the effective ordering cost: item 1 gives \(\sqrt{2(114.33)(12{,}000)/6} = 676.25\). That confirms the multiplier and the quantities came from the same value.
\[ \begin{aligned} \$7{,}621.27 - \$6{,}815.33 &= \$805.93 &&\text{the price of the limit}\\ 805.93 / 6{,}815.33 &= 11.83\% &&\text{relative to unconstrained} \end{aligned} \]
Notice in the last column that the quantities have not grown together. Item 2, whose $20 ordering cost is most changed by adding $64.33, more than doubles; item 3, at $80, grows by only a third. The dock’s capacity has redistributed the quantities as well as lengthened them.
One more replenishment a year of dock capacity would save about $64. That is the figure to carry to whoever decides whether to add capacity at the dock.
It is a shadow price. \(\theta^{*}\) is the rate at which the minimum cost falls as the limit is relaxed: at 0.14643, one more dollar of working capital saves about 15 cents a year, a hundred more about $14.
The rate is local. As the limit relaxes the multiplier falls, reaching zero at $12,198.77, where the constraint stops binding and the last of the $145.04 is recovered. Asking for more than that buys nothing.
Where it enters depends on the constraint. Under a budget it is an increment to the carrying charge, \(i + \theta\); under a replenishment limit it is a price per order, \(k_j + \theta\).
The multiplier is not merely a device for getting an answer. It carries the economics of the constraint, and the formulation decides which reading applies.
A shared resource couples items through a constraint; the cost functions stay separate.
Check whether it binds first. If not, the item-by-item answer stands.
The multiplier is a surcharge: \(Q_j(\theta) = \sqrt{2k_j\lambda_j/(h_j + 2\theta a_j)}\), and the resource function has one root, so bisection finds it.
Under a $10,000 budget \(\theta^{*} = 0.14643\), holding is $3,000 by construction, the limit costs $145.04, and every quantity shrinks by 0.81975.
Under a dock limit the multiplier joins the ordering cost, the quantities grow, and they are reallocated: the limit costs $805.93.
Read before next session: The Exchange Curve.
Today’s material was When Items Stop Being Independent, A Constrained Lot Sizing Problem and What the Multiplier Means.
Next session:
Bring the multiplier. Next session it stands in for the carrying charge.