Custom Pilot Chemical Company is a chemical manufacturer that produces batches of specialty chemicals to order. Principal equipment consists of eight interchangeable reactor vessels, five interchangeable distillation columns, four large interchangeable centrifuges, and a network of switchable piping and storage tanks. Customer demand comes in the form of orders for batches of one or more (usually not more than three) specialty chemicals, normally to be delivered simultaneously for further use by the customer. Customer Pilot Chemical fills these orders by means of a sort of pilot plant for each specialty chemical formed by inter-connecting the appropriate quantities of equipment. Sometimes a specialty chemical requires processing by more than one of these ‘‘pilot’’ plants, in which case one or more batches of intermediate products may be produced for further processing. There is no shortage of piping and holding tanks, but the expensive equipment (reactors, distillation columns, and centrifuges) is frequently inadequate to meet the demand.

The company’s schedules are worked out in calendar weeks, with actual production always taking an integer multiple of weeks. Whenever a new order comes in, the scheduler immediately makes up a precedence tree-chart for it. The precedence tree-chart breaks down the order into ‘‘jobs.’’ For example, an order for two specialty chemicals, of which one needs an intermediate stage, would be broken down into three jobs (Fig.). Each job requires certain equipment, and each is assigned a preliminary time-in-process, or ‘‘duration.’’ The durations can always be predicted with a high degree of accuracy.

Currently, Custom Pilot Chemical has three orders that need to be scheduled (see the accompanying table). Orders can be started at the beginning of their arrival week and should be completed by the end of their week due.

For example, AK14 consists of three jobs, where Job 2 cannot be started until Job 1 has been completed. Figure is an appropriate precedence tree for this order. Generally, the resources required are tied up for the entire duration of a job. Assume that Custom Pilot Chemical is just finishing week 14.

 a) Formulate an integer program that minimizes the total completion time for all orders.

b) With a more complicated list of orders, what other objective functions might be appropriate? How would these be formulated as integer programs?

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(4) Job 1 (3) Job 2 (3) Job 3 Figure F9.4 (4) Job 1 (3) Job 2 (3) Job 3 Figure F9.4