A food company runs a computerized processing plant and needs to formulate a series of decision rules to advise its managers on how they should react if the control panel indicates particular problems with the system. Because there is always a possibility that an indicated problem is in fact caused by a fault in the control panel itself, there is some concern that unnecessary losses will be incurred if production is halted because of a non-existent problem.

Light number 131 will illuminate on the panel if the computer detects that packs of a frozen food are being filled to below the legal weight. However, it is known that there is a 0.15 probability that this light will give a false alarm. In the event of this light illuminating, the manager would have to decide whether or not to gather further information before making a decision on whether to stop production immediately. Any stoppage would cost an estimated $150 000, but a decision to ignore the light would lead to losses of $300 000 if the bags being filled on the automatic production line really were underweight.

If the manager decides to gather further information before taking the decision on whether to stop production then this will involve taking a sample from output and weighing the selected packs. This will render the sampled packs unsaleable and cost the company $5000. The sample will indicate whether or not there is a fault in production, but there is a 0.2 probability that it will give misleading results. Despite this it has always been company policy to take a sample because of the small cost of sampling relative to the other costs.

(a) If the company's objective is to minimize expected costs, formulate a decision rule which will tell the duty manager how to react when light 131 illuminates.

(b) Explain the rationale behind your recommended decision rule in non-technical terms.

(c) Explain the role which sensitivity analysis could have in this problem.