Old MathJax webview

Statistics and probability

Problem 16.3 We return to the fixed-cost problem of David Masimula. He is the foreman in a factory where three large machines are used for production. The working day is divided into three shifts. Each of the three machines, A, B and C, must be used at some stage during the day. The machines are, however, not available in all shifts of the day. Machines A and B are not available in the last shift of the day since maintenance is then carried out to prepare these machines for the next day. Each of these machines may be used only in one shift of the day. In other words, machine A must be used either in shift 1 or in shift 2 but may not be used in both these shifts. Machine B is similarly restricted to either shift 1 or shift 2 and may not be used in both these shifts. After a machine has been used, it is switched off. Maintenance on machine C is carried out in the first shift of the day, which means that this machine is not available in shift 1. Machine C may be used in shift 2 only or in shift 3 only or in both shifts 2 and 3. If it is used in both these shifts, it is not switched off at the end of the second shift and there is therefore no setup cost for the third shift. The fixed setup costs, the variable production costs and the daily machine capacities are given in the following table: Machine Setup cost A B C R300 R200 R100 Production cost per unit R0,50 R0.40 R0.70 Daily capacity (in units) 2 100 1 800 3 000 Every day David receives the production requirements for the day from the production manager. Suppose the production requirements for the day are 1 800 for the first shift, 3 000 for the second shift and 1 200 for the third shift. David now has to decide which machines to use in each shift and how many units to pro- duce on these machines in each shift in order to deliver the required production at mini- mum cost. Reformulate David's model and solve it