Excercise: Economic planning An economy consists of three industries: Coal, steel and transportation. Each unit produced in these industries is valued at €1 and requires an input from its own and

the other two industries according to Table 1. In addition, the specified amount of labor is required. There is a time lag in production, so that a unit produced in year t + 1 requires the input in year t. Table 1: Input per year t Coal Steel Transportation Labor force per unit produced per year t + 1 Coal Steel Input per year t Coal Steel Transportation Labor force 0,1 0,1 0,2 0,6 The output of units produced in an industry can also be used to build up production capacity in this or another industry in later years. 0,5 0,1 0,1 0,3 The input required to build up an additional unit of production capacity (which then enables additional production worth €1 per year) is shown in Table 2. This build-up of production capacity also takes place with a time delay. Here, the input in year t only leads to a (permanent) increase in capacity in year t +2. Table 2: 0,0 0,1 0,2 0,4 Transportatio n 0,4 0,2 0,2 0,2 per additional unit of capacity in year t+2 Coal Steel 0,7 0,1 0,1 0,2 Transportatio n 0,9 0,2 0,2 0,1 The products can be stored from year to year. At the beginning of year 0, the stock levels of products and production capacities shown in Table 3 are available in the/navailable in various industries (in € million). There is an annual capacity of labor that is limited to € 470 million. Table 3: Industry Stock Coal Steel Transportatio n 150 80 100 Year 0 Capacity 300 350 280 The aim was to identify various growth strategies for the economy over the next five years. We are particularly interested in strategies that pursue the following objectives: (i) Maximization of the sum of the respective production capacities at the end of all five years. In addition, the following external requirements must be satisfied each year (except in year 0): € 60 million coal, € 60 million steel and € 30 million transportation. (ii) Maximization of the total production of the 4th and 5th year. The external demand from (i) is not taken into acount. (iii) Maximization of the total demand for labor over all five years, takinginto account the external demand from (i). The above-mentioned restriction on the annual capacity of labor can be ignored. Develop optimization models for these questions and solve them with suitable optimization software. In addition to the detailed explanation of the solution and the optimization model, your report should also include the source information and the files for the optimization software. Good luck!