The aim of this question is to assess your ability to formulate a mixed integer problem. A small workshop has traditionally produced ornate candlesticks for use in period settings, religious buildings and for luxury use. They now wish to make a transition to metallic electric candlesticks that use low-energy LED lighting. They aim to produce a range of three models, of increasing sophistication, the Lumen, the Candela and the top-of-the-range Lux. There are three production processes which must be carried out in the following order: cutting, welding and a polishing and finishing process known in the trade as firtling. The electrical fitting is handled in a separate workshop and does not concern us here. Each process is carried out by a specially designed machine and there are two of each machine. Each machine can work on one candlestick at a time, though all six machines can operate at once. The time taken, in minutes, for each process for each type of candlestick is as follows. Process Candlestick type Lumen Candela Lux Cutting 5 8 10 Welding 7 13 Firtling 4 7 5 The manufacturer wants to produce sets of three pairs of candlesticks, one pair of each type, in the shortest possible time for each set. (a) Formulate the manufacturer's problem as an integer programming model. (b) Determine a feasible point for the model, giving values of all the variables introduced in part (a). Describe the manufacturing schedule given by your feasible point. 10 The aim of this question is to assess your ability to formulate a mixed integer problem. A small workshop has traditionally produced ornate candlesticks for use in period settings, religious buildings and for luxury use. They now wish to make a transition to metallic electric candlesticks that use low-energy LED lighting. They aim to produce a range of three models, of increasing sophistication, the Lumen, the Candela and the top-of-the-range Lux. There are three production processes which must be carried out in the following order: cutting, welding and a polishing and finishing process known in the trade as firtling. The electrical fitting is handled in a separate workshop and does not concern us here. Each process is carried out by a specially designed machine and there are two of each machine. Each machine can work on one candlestick at a time, though all six machines can operate at once. The time taken, in minutes, for each process for each type of candlestick is as follows. Process Candlestick type Lumen Candela Lux Cutting 5 8 10 Welding 7 13 Firtling 4 7 5 The manufacturer wants to produce sets of three pairs of candlesticks, one pair of each type, in the shortest possible time for each set. (a) Formulate the manufacturer's problem as an integer programming model. (b) Determine a feasible point for the model, giving values of all the variables introduced in part (a). Describe the manufacturing schedule given by your feasible point. 10
