In this chapter, we consider stocking problems in which stock is ordered in anticipation of depletion by demand processes. A different kind of problem arises in transportation and distribution systems such as freight railroads, express and delivery services, freight forwarders, etc., in which the inventory accumulates according to supply processes and is depleted by dispatches. Consider such a dispatching problem, in which an inventory of dissimilar items accumulates according to a time-varying process. A periodic review is assumed, whereby at the end of each time period, there is an opportunity to dispatch, that is, to reduce the inventory to zero. A cost trade-off similar to the stocking problem is at issue, that is, there are inventory holding costs, and a significant, fixed dispatching cost.

We suppose the system starts (at the beginning of period 1) with an initially empty inventory (i.e., a dispatch cleared the system at the end of period 0). In the tth period, t =

1, 2, ..., we let Wt denote the set of items supplied to the inventory whose first opportunity for dispatch is at the end of period t, and HC(Wt) shall denote the cost of holding items Wt in the inventory for one period. (E.g., if the set of items Wt are kept in the inventory for 3 periods before dispatching, the associated holding costs are 3 HC(Wt)). We let A denote the fixed cost per dispatching event. A dispatching policy is desired, that is, a determination in each time period of whether to dispatch or not dispatch in that period?

a. Develop a decision procedure, such as the Silver–Meal heuristic, based on minimizing the relevant costs per unit time between dispatch moments.

b. Illustrate your procedure (for just the first two dispatches) with the following example involving the dispatching of freight trains from a switchyard terminal:
At a terminal, freight cars with a particular destination W are switched to departure tracks where they accumulate until a train to destination W is dispatched. The holding costs per hour vary from car to car according to ownership, age, car type, etc., but we suppose that an information system at the terminal allows the calculation of holding costs for a given group of cars. Let Wt denote the set of cars destined to W switched in hour t, and let HC(Wt) denote the hourly holding costs for Wt. We consider the case where a train was dispatched to W just before hour 1, and the switching results of the next several hours are as follows:
Hour, t 1 2 3 4 5 6 7 Holding cost, HC(Wt) 220 85 145 210 310 175 300 The fixed cost of dispatching a train to W is A = 800.