Lecture 4.3 Recap

Lecture The example, and the method

“For the product of Fig. 1, made in two stages over three months, determine the production plan that minimizes total production and inventory cost, and the cost: by hand, and with Model 1 in Julia.”

“Determine the plan and its cost by hand, from the capacities and flow balance.”

“Model 2 is the flow balance of Sec. 2.2, node by node from the last, with one choice. At each node the flow out is already known: what goes on to the next stage, or to the demand at the last stage, plus the inventory carried into the next month. Production takes as much of it as the capacity allows, and the rest has to come in as inventory from the month before, which is then the inventory out of the node to its left. In month 1 the inventory coming in is the initial inventory, so production is the rest, which has to fit within the capacity. Fig. 4 draws the steps, and Table 1 lists them:”

Lecture The cost, and the one term most easily missed

“The final inventory’s cost, \(5 \times 35\), is counted because the inventory sum of Eq. 1 runs to \(j = t + 1\).”

Lecture What a setup cost is, and its share of the inventory cost

“What takes the model from a linear program to a mixed-integer linear program is a setup cost in a period, assumed to be incurred in each period in which a resource produces: literally getting the machine up and running, for example, a cost independent of how much is produced but incurred every period.”

“The setup cost is allocated to the inventory cost at a stage by assumption (e) of Model 3. Each ton is charged a share of every setup it has passed through, the setup cost divided by the capacity per setup, added to the production cost of each stage up to the stage it is at before the carrying rate is applied:”

Lecture Example 5(b): a given plan

Example 5 gives two stages that can make 40 and 80 tons a month at $200 and $800 a ton, with setups of $6,000 and $18,000 a month. Part (b) changes some of the data:

“Using some different data as noted below (otherwise, use the data given in Ex. 5) and only a three-month planning horizon, determine the total variable production, inventory, and setup cost over the horizon. The forecasted demand is 22, 3, and 33 tons per month. The product loses five percent of its value after one month. Each month, 20, 4, 40 and 25, 0, 37 tons, respectively, are produced at each stage. Currently, five tons of material is in storage at stage one and none at stage two; at the end of the planning period, seven tons of material should be in storage at stage one and four at stage two.”

“Going month by month, the inventory carried out of each node is what came in less what went on:”

Lecture The lecture's answer

“The final inventories come out at the 7 and 4 tons required, the Balance check of Lecture 1.2.”

Lecture Why a truck is a setup

“In some situations, though, the setup cost is nearly everything. A truckload is one: between two points, most of its cost is independent of the load carried, up to the truck’s maximum payload, which is why Ex. 6 treats a truck as a setup.”

“The plan has two stages. Stage 1 is the plant, which can produce any amount in a week at the product’s value, $25,000 a ton, with no setup cost. Stage 2 is the shipment by truckload, set up once for each truck sent, at one truck’s charge, with no cost per ton and a capacity per setup of the maximum payload; any number of trucks can be sent in a week. The inventory at Gainesville carries the product’s value plus the truckload charge per ton, assumption (e) of Model 3, which is Lecture 3.3’s allocated full-truckload rate, Ex. 1(h).”

“The truckload’s setup cost is one truck’s charge, $1,357.2, with no cost per ton, and its capacity per setup is the maximum payload, 6.11 tons.”

“At the higher value, inventory costs more to carry, so the shipments should be smaller and more frequent, and they are: 13 trucks of about 1.54 tons, one every 4.0 weeks, against minTLC’s 1.46 tons every 3.8 weeks.”

Example 6 ships Lecture 3.3's product, 20 tons a year, by truckload from Raleigh to Gainesville, planned week by week over a year with Model 3: (a) at $25,000 a ton, and (b) at $85,000 a ton.

Lecture One end or the other, not both

“In a rolling horizon, this month’s plan carries those four, and next month’s cuts off the first period, adds one at the end, and solves the same model again. With the cost counted every month, the inventory carried from one plan into the next would appear in both, as the final inventory of one and the initial inventory of the other, and over time the plans would show an inventory cost that is not actually incurred. Leaving out the cost of the initial inventory avoids that double counting. The final inventory could have been left out instead, and the initial included: one or the other, but not both.”

“The 10 tons on hand at the start add nothing to the cost, since their cost was incurred in the previous period and the inventory sum of Eq. 1 starts at \(j = 2\).”

The second paragraph is from Example 3(c), where stage 1 starts with 10 tons on hand.

Lecture The model decides, and only a cost makes it

“Which periods to produce in is then determined by the model, through its indicator variables, rather than by the person running it, as the zero capacities of Sec. 2 required.”

“Without the setup cost. Take \(c_i^s = 0\) in Eq. 3. The indicator still gates the capacity, row (c), but nothing in the objective makes the model turn it off, so wherever a stage may be set up its capacity is there at no cost: the model is Model 1 with \(u_{ij} = u_i\bar{z}_{ij}\), and which periods are available is determined outside it, as in Sec. 2.”

“The setup cost is the natural element: it is what makes producing in a period cost something, and so what makes in which periods to produce a decision for the model.”

Lecture Building ahead on idle capacity

“The plan builds the inventory for it on the idle capacity of the periods before it: counting back from the outage, each period’s idle capacity is added until the total covers the outage’s demand, and the horizon has to span those periods and the outage.”

“Measure demand and inventory in periods of the resource’s production, so that each period’s demand is \(u\) and its idle capacity \(1 - u\). An outage of \(\tau\) periods then has a demand of \(\tau u\), all of it to be built ahead, and \(b\) periods of build-ahead supply \(b\,(1 - u)\) of it, so the number of periods of build-ahead satisfies:”

Eq. 5, where \(H\) is the planning horizon (periods), \(\delta\) the cumulative lead time (periods), \(\tau\) the length of the outage (periods), and \(u\) the utilization of the resource in the periods before the outage.

“By Eq. 5, with materials on hand (\(\delta = 0\)), a month-long outage at 90% utilization needs a horizon of 10 months, and at 95%, 20; a lead time adds to each.”