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i dont understand this

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Answered Same Day Aug 14, 2021


Sumit answered on Aug 14 2021
140 Votes
    Let’s walk through the problem.
    1.      A steel mill produces two types of steel alloy: boral and chromal.
    ·        This is the first important piece of information—the product mix (i.e., these are the decision variables, what it is we are trying to produce). So we are interested in producing steel using boral and chromal alloys. The questions is – how much of each do we use?
    2.      Production of each alloy requires three process: Box anneal, Cold Roll, and Strand anneal.
    ·        The is the next important piece of information. Each of these three processes is considered to be a constraint (restriction) because we only have a limited amount of time per month for these processes. How much time?
                                                                  i.      Box anneal: 4000 hours/month
                                                                ii.      Cold Roll: 500 hours/month
                                                              iii.      Strand anneal: 1,000 hours/month
    Note that these values are the right-hand side of the constraints.
    3.      Now look at the table of production rates for each constraint at the top of page 236. These values will make up the left-hand side of the constraint. For example, for the Box anneal constraint, it takes 4 tons/hour to process the Boral and 2 tons/hour to process the Chromal in the steel making process.
    The problem is that they give us these values as tons/hour. But the final answer is going to be how many tons of boral and how many tons of chromal we should use to meet these restrictions.
    For example, if we set up the Box anneal constraint as is, it looks like this:
    4 tons/hour*Boral + 2 tons/hour*Chromal <= 4000 hours/month
    Once we set up the constraint, Solver will replace the “Boral” placeholder with a number of tons and the “Chormal” placeholder with a number of tons and calculate the left-hand side of the constraint to see if it meets the inequality conditions of the right-hand side.
    For example, let’s substitute 2 for Boral and 3 for Chromal and calculate:
    4 tons/hour*2 tons of Boral + 2 tons/hour*3 tons of Chromal = 14 tons2/hou
    But we can’t compare this to the right hand side because the units don’t make sense (tons2/hour vs hours/month).
    So what we have to do is to invert the values in the left hand side of the constraint so that the units are hours/tons.
    For example: 4 tons/hour = 1 hou
4 tons or ¼ hou
    So the inverted constraint is:
    ¼ hou
tons*2 tons of Boral + ½ hou
tons * 3 tons of Chromal = 2/4 + 3/2 = 2 hours (the tons units cancel and leave only hours).
    Now we can compare this to the right hand side of the constraint and see that the inequality is met.
    2 hours <= 4000 hours/month
    You will need to do this for every constraint in the table. Note that the inverted values may be very, very small numbers so please ca
y them out 4 or 5 decimal places.
    4.      The last thing to set up is the objective function (goal) of the problem. This is a mathematical statement of the goal to either maximize or minimize something (e.g., maximize profit or minimize cost).
    In this case, we want to maximize the contribution/ton of the alloys. We are told that the contribution/ton for the boral is $25 and the contribution/ton for the chromal is $35.
    Right now we don’t know how many tons of the boral and chromal we will use, so let’s set up the objective function like this:
    Goal = $/ton contribution*tons of Boral + $/ton contribution*tons of Chromal
    Or: Maximize contribution =...

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