Allocation Problem
1 (Single-Period) |
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| Minimize
the cost of operating 3 different types of machines while meeting product
demand. |
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machine has a different cost and capacity. There are a certain number of
machines |
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| available for each type. |
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| Information on machines |
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Initial
cost per day |
Additional cost per product |
Products per day (Max) |
Number of machines |
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| Alpha-1000 |
$200 |
$1.50 |
40 |
8 |
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| Alpha-2000 |
$275 |
$1.80 |
60 |
5 |
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| Alpha-3000 |
$325 |
$1.90 |
85 |
3 |
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| Number of
machines to use |
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| Alpha-1000 |
1 |
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| Alpha-2000 |
1 |
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| Alpha-3000 |
1 |
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| Number of products to make per
day |
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| Alpha-1000 |
300 |
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| Alpha-2000 |
300 |
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| Alpha-3000 |
300 |
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| Total |
900 |
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| Demand |
750 |
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| Maximum number
of products that can be made per day |
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| Alpha-1000 |
40 |
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| Alpha-2000 |
60 |
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| Alpha-3000 |
85 |
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| Cost |
$2,360.00 |
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| Problem |
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| A
company has three different types of machines that all make the same
product. Each |
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| machine
has a different capacity, start-up cost and cost per product. How should the
company |
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| produce its
product with the available machines to meet the daily demand? |
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| Solution |
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| 1)
The variables are the number of machines to use and the number of products to
make on |
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| each
machine. In worksheet Alloc1, these are given the names Products_made and |
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| Machines_used. |
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| 2) First,
there are the logical constraints. These are |
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Products_made >= 0 via the Assume Non-Negative option |
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Machines_used >= 0 via the Assume Non-Negative option |
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Machines_used
= integer. |
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| Second, there
are the demand and capacity constraints. These are: |
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Machines_used <= Machines_available |
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Products_made <= Maximum_products |
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Total_made >= Demand |
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| 3) The objective is to minimize cost. This is defined on
the worksheet as Total_cost. |
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| Remarks |
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| Notice
that we used an integer constraint to make sure no fractions of machines were
used. |
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| This
has the usual drawback; the problem is much more difficult to solve than the
'relaxed' |
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| version
without the integer constraint. When large numbers of machines are involved,
the |
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| integer
constraint can often be dropped. It is most often not critical whether 1586
or 1587 |
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| machines
are used, for example. This means that a number of 1586.4 would be
acceptable. |
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| However,
in this case there are only a few machines and it does make a big difference
whether |
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| 2 or 3 machines
are used. An answer of 2.5 would not be satisfactory. |
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