(\"A Gentle Introduction to Optimization\", by B. Guenin, J. Konemann and L. Tun
ID: 3111793 • Letter: #
Question
("A Gentle Introduction to Optimization", by B. Guenin, J. Konemann and L. Tungel, Cambridge University Press, 2014) A chemical plant produces a noxious byproduct, called Chemical X, that is highly toxic and needs to be disposed of properly. The chemical plant is connected by a pipe system to a recycling plant that can safely process Chemical X. The amount of Chemical X produced in each hour of the day, according to a standard day's production schedule, is shown in Table 1 (Chemical X is not produced in any hour omitted in the table). The chemical plant has a storage capacity of 1000 liters for Chemical X, and, for environmental safety reasons, no Chemical X can be kept, unprocessed, overnight at the chemical plant. The cost for the recycling plant to process Chemical X varies throughout the day, as given in Table 2. Table 2: Price for processing Chemical X at the recycling plant, at each hour of the day. Formulate an LP model to assist the chemical plant manager to minimize the cost of safely disposing of Chemical X. Clearly define your variables, and briefly describe each constraint. Implement your model in the Xpress-IVe software, and solve it. What should the chemical plant manager do, and what will it cost?Explanation / Answer
The objective of the plant has to be the minimization of cost for the recycling of Chemical X as mentioned in the question- To minimize the cost of safely disposing of chemical X.
The data is given about the timings of production & cost of processing for various time intervals as follows:
It is obvious that the chemical produced after 10 am can not be processed at 10 am or the production during subsequent time-intervals can not be processed/treated during the earlier time periods of processing.
Let us define the decision variables Xij as the quantity of chemical produced during the ith time interval and processed in the jth time period and the cost of processing ($ per liter) as Cij. Therefore the objective is similar to that of a standard transportation problem given as Minimization Double summation Cij*Xij subject to the demand and supply constraints.Production data is given and for processing the storage capacity is capped at 1000 litres so the required matrix (transportation format) is as follows:
Using Excel Solver the solution is as follows:
As apparent from the data the minimum cost(30) of processing is for the first time period 10 am but it can process only the quantity produced during 9-10am, next minimum is 35 which can take care of last two intervals, similarly next minimum of 38 caters to its earlier two periods and finally production during 2-3pm has no option but to be treated at 3pm. Total minimum cost is worked as $1,02,400
Time interval 9-10am 10-11am 11am-12pm 12-1pm 1-2pm 2-3pm Chemical X (litres) 300 240 600 200 300 900Related Questions
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