A Markov Chain Process in Industrial Replacement

IN the manufacture of rubber tyres a machine is used which contains two "bladders", and which produces one tyre on each bladder simultaneously. Bladders can fail in service, a faulty bladder being discovered when a tyre fails to pass inspection. Thus there is a cost of a scrap or "second" tyre associated with a bladder which fails in service. Further, when a bladder fails the machine must be stripped down before replacement can be made, a process requiring fitters, labour, and production time with an associated cost. However, once a machine has been stripped, both bladders can be replaced at the same time. The cost of replacing the second bladder is virtually the cost of purchasing the bladder alone. It is found that the probability of failure is a function of the number of tyres made with a particular bladder. It thus seems likely that the cost of replacing bladders could be kept at a minimum with the following policies: (A) Replace bladders which have made a predetermined number of tyres without failure. (B) When a machine is stripped to replace one bladder, replace the other bladder if it has produced more than a given number of tyres. For brevity the number of tyres produced on a given bladder will be called the "age" of the bladder. Normal production records show the number of tyres produced on each bladder, and a curve of the probability of failure with age is readily obtained. It is the purpose of this paper to show how the ages required in A and B can be determined so as to minimize the average costs of replacing bladders per useful tyre produced; that is, to minimize the sum of: (i) The costs of stripping a machine for bladder replacement. (ii) The costs of lost production time due to a bladder failing and being replaced on a production shift. (iii) The costs of "second" or scrap tyres produced. (iv) The costs of purchasing bladders.