Activity: bisecting the multiplier
Chapter 4, Multi-Item and Constrained Systems
Work in pairs. 20 minutes.
Names: ______________________________________________
The items. The distributor that stocks the mower deck belt orders three of its items from three suppliers. Each order costs \(k = \$40\), the carrying charge is \(i = 25\%\) a year, demand is constant and known, and no shortages are allowed.
| Item | \(\lambda_j\), a year | \(c_j\) | |
|---|---|---|---|
| 1 | Mower deck belt | 3,000 | $12.00 |
| 2 | Blade kit | 2,400 | $5.00 |
| 3 | Spindle kit | 600 | $40.00 |
The controller limits the average investment in these three items to $3,000. Under a budget the constrained quantities are
\[ Q_j(\theta) = \sqrt{\frac{2k\lambda_j}{(i + \theta)c_j}}, \qquad \text{investment} = \sum_j c_j\frac{Q_j(\theta)}{2} \]
Report quantities to two decimals and dollars to the cent.
Part A. Predict
The budget is about three quarters of what the unconstrained quantities need. Which item’s quantity will be cut hardest, in proportion to its unconstrained value?
Part B. Does the limit bind?
Compute each unconstrained \(Q_j^{*}\) and the investment it requires. Does the $3,000 limit bind?
Part C. Four bisection steps
Bracket the multiplier with \(\theta = 0\) and \(\theta = 1\). At each step compute the three quantities at the midpoint, then the investment, and keep the half in which the investment crosses $3,000.
Make a table with the columns Step, Low, High, Midpoint \(\theta\), \(Q_1\), \(Q_2\), \(Q_3\), Investment and Keep, one row for each of four steps.
After four steps the multiplier is bracketed. Twenty more steps give \(\theta^{*} = 0.20844\). At \(\theta^{*}\), compute the three quantities, the annual ordering and holding costs, and the cost of the limit against the unconstrained answer. Check that the holding cost is what it has to be.
Part D. What the multiplier says
- The controller asks what one more dollar of budget would be worth. Answer in dollars a year, and say why the answer holds only for a small change.
- Compute \(Q_j/Q_j^{*}\) for each item. Compare with your prediction.
Part E, if you have time. No bisection needed
Use your answer to Part D.2 to find \(\theta^{*}\) directly from the budget, without bisecting. Why would the same shortcut fail if the limit were on floor space?
Exit ticket (tear off and hand in)
Name: ____________________
- In one sentence, what does \(\theta^{*} = 0.208\) tell the controller?
- True or false, and why: at the constrained quantities the annual ordering and holding costs are equal.