Activity: Wagner-Whitin on six periods
Chapter 5, Dynamic Lot Sizing
Work in pairs. 25 minutes.
Names: ______________________________________________
The item. The mower deck belt’s first six months, planned as a horizon of their own, but with the ordering cost the distributor now pays for an expedited inbound truck: \(k = \$500\) an order. The belt still costs $50, and the carrying charge is 2% a month, so \(h = \$1.00\) per belt per month. The plan starts and ends empty, and carrying is charged on the ending balance.
| \(t\) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| \(d_t\) | 40 | 60 | 120 | 300 | 420 | 260 |
Notation. The window cost is \(w(t,u) = k + h\sum_{s=t}^{u}(s-t)d_s\), the cost of one order in period \(t\) covering periods \(t\) through \(u\). The recursion is \(V(u) = \min_{1 \le t \le u}\{V(t-1) + w(t,u)\}\) with \(V(0) = 0\), and \(S(u)\) is the \(t\) that attains the minimum.
Part A. Predict
How many orders will the optimal plan place in six months? Give a reason, from the economic order quantity on the average requirement or from anything else.
Part B. The window costs
Fill the upper triangle, one row at a time from its diagonal outward: each step along row \(t\) adds \(h(u + 1 - t)d_{u+1}\).
Make a 6 by 6 table with rows \(t\) and columns \(u\), and fill the cells with \(u \ge t\).
Part C. The recursion, one column at a time
Make a table with the columns \(u\), the candidates, \(V(u)\) and \(S(u)\), one row for each of \(u = 1\) to 6.
Part D. Trace back, and check
- Trace the plan back from \(S(6)\). Give each order’s period and quantity.
- Cost the plan from scratch, order by order, and confirm it equals \(V(6)\).
- \(S(5)\) names a period. Does the optimal plan order there? Why not?
Part E, if you have time. How far can \(k\) move?
- Does any month force an order at \(k = \$500\)? Would any at the chapter’s \(k = \$300\)?
- Compare the optimal plan with its two neighbors: the plan that also orders in month 6, and the plan that drops the order in month 5. Over what range of \(k\) is your plan cheaper than both?
Exit ticket (tear off and hand in)
Name: ____________________
- Why can every window cost be computed before the search begins?
- True or false, and why: the optimal plan orders in every period \(S(u)\) names.