8. Everybody knows that in an advertisement a product can always do every- thing. Mathematical software is no exception to that. We want to inves- tigate such promises in practice. The problem comes from the area of electric power supply: A power plant has to provide power to meet the (estimated) power demand given in the following chart. Exercises to Chapter 1 33 Chapter 1 12 pm 6 am 15 GW Expected Demand of Power 6 am 9 am 9am - 3pm 3pm-6pm 6pm - 12 pm 30 GW 25 GW 40 GW 27 GW There are three types of generators available, 10 type 1, 10 type 2 and 12 type 3 generators. Each generator type has a minimal and maximal capacity; the production has to be somewhere in between (or else the generator has to be shut off). The running of a generator with minimal capacity costs a certain amount of money (dollars per hour). With each unit above minimal capacity there arise additional costs (dollars per hour) (cf. chart). Costs also arise every time a generator is switched on. Technical information and costs for the different generator types M; ei fi Typ 1 850 MW 4000 MW 2000 4 4000 Typ 21250 MW 1750 MW 5200 2.6 2000 Typ 3 | 1500 MW 4000 MW 6000 6 1000 mi, M; : minimal and maximal capacity ei : costs per hour (minimal capacity) Ci : costs per hour and per megawatt above minimal capacity fi: costs for switching on the generator In addition to meeting the estimated) power demands given in the chart an immediate increase by 15% must always be possible. This must be achieved without switching on any additional generators or exceeding the maximal capacity. Let nij be the number of generators of type i which are in use in the j-th part of the day, i = 1, 2, 3 and j = 1, 2, ...,5, and sij the number of generators that are switched on at the beginning of the j-th part of the day. The total power supply of type i generators in the j-th part of the day is denoted by Xiji TI the founding fanation T. Typ 1 Typ 2 Typ 3 mi Mj ei 850 MW 4000 MW 2000 1250 MW 1750 MW 5200 1500 MW 4000 MW 6000 Ci 4 2.6 6 4000 2000 1000 mi, M; : minimal and maximal capacity ei : costs per hour (minimal capacity) Ci : costs per hour and per megawatt above minimal capacity fi: costs for switching on the generator In addition to meeting the estimated) power demands given in the chart an immediate increase by 15% must always be possible. This must be achieved without switching on any additional generators or exceeding the maximal capacity. = Let ij be the number of generators of type i which are in use in the j-th part of the day, i = 1, 2, 3 and j = 1, 2, ..., 5, and sij the number of generators that are switched on at the beginning of the j-th part of the day. The total power supply of type i generators in the j-th part of the day is denoted by Xij The costs can be described by the following function K: 3 5 3 5 3 5 K c;zz(tij minij) + Xeizjnij + fisij i=1 j=1 i=1 j=1 i=1 j=1 Xij are nonnegative real numbers, nij and sij are nonnegative integers. Zj denotes the number of hours in the j-th part of the day (which can be obtained from the above chart). 34 Introduction Chapter 1 a) Which simplifications are "hidden" in the cost function? b) Formulate the constraints! c) Determine nij, sij and Xij such that the total costs are as low as possible! You might not find the global minimum but only a useful suggestion. You can try a software of your choice (for example the Matlab Optimization Toolbox or Maple). Your solution has to meet all the constraints. 1 11 1) 8. Everybody knows that in an advertisement a product can always do every- thing. Mathematical software is no exception to that. We want to inves- tigate such promises in practice. The problem comes from the area of electric power supply: A power plant has to provide power to meet the (estimated) power demand given in the following chart. Exercises to Chapter 1 33 Chapter 1 12 pm 6 am 15 GW Expected Demand of Power 6 am 9 am 9am - 3pm 3pm-6pm 6pm - 12 pm 30 GW 25 GW 40 GW 27 GW There are three types of generators available, 10 type 1, 10 type 2 and 12 type 3 generators. Each generator type has a minimal and maximal capacity; the production has to be somewhere in between (or else the generator has to be shut off). The running of a generator with minimal capacity costs a certain amount of money (dollars per hour). With each unit above minimal capacity there arise additional costs (dollars per hour) (cf. chart). Costs also arise every time a generator is switched on. Technical information and costs for the different generator types M; ei fi Typ 1 850 MW 4000 MW 2000 4 4000 Typ 21250 MW 1750 MW 5200 2.6 2000 Typ 3 | 1500 MW 4000 MW 6000 6 1000 mi, M; : minimal and maximal capacity ei : costs per hour (minimal capacity) Ci : costs per hour and per megawatt above minimal capacity fi: costs for switching on the generator In addition to meeting the estimated) power demands given in the chart an immediate increase by 15% must always be possible. This must be achieved without switching on any additional generators or exceeding the maximal capacity. Let nij be the number of generators of type i which are in use in the j-th part of the day, i = 1, 2, 3 and j = 1, 2, ...,5, and sij the number of generators that are switched on at the beginning of the j-th part of the day. The total power supply of type i generators in the j-th part of the day is denoted by Xiji TI the founding fanation T. Typ 1 Typ 2 Typ 3 mi Mj ei 850 MW 4000 MW 2000 1250 MW 1750 MW 5200 1500 MW 4000 MW 6000 Ci 4 2.6 6 4000 2000 1000 mi, M; : minimal and maximal capacity ei : costs per hour (minimal capacity) Ci : costs per hour and per megawatt above minimal capacity fi: costs for switching on the generator In addition to meeting the estimated) power demands given in the chart an immediate increase by 15% must always be possible. This must be achieved without switching on any additional generators or exceeding the maximal capacity. = Let ij be the number of generators of type i which are in use in the j-th part of the day, i = 1, 2, 3 and j = 1, 2, ..., 5, and sij the number of generators that are switched on at the beginning of the j-th part of the day. The total power supply of type i generators in the j-th part of the day is denoted by Xij The costs can be described by the following function K: 3 5 3 5 3 5 K c;zz(tij minij) + Xeizjnij + fisij i=1 j=1 i=1 j=1 i=1 j=1 Xij are nonnegative real numbers, nij and sij are nonnegative integers. Zj denotes the number of hours in the j-th part of the day (which can be obtained from the above chart). 34 Introduction Chapter 1 a) Which simplifications are "hidden" in the cost function? b) Formulate the constraints! c) Determine nij, sij and Xij such that the total costs are as low as possible! You might not find the global minimum but only a useful suggestion. You can try a software of your choice (for example the Matlab Optimization Toolbox or Maple). Your solution has to meet all the constraints. 1 11 1)