Resource Allocation Problem

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Resource Allocation Problem

Statement
The type of problem most often identified with the application of linear program is the problem of distributing scarce resources among alternative activities. The Product Mix problem is a special case. In this example, we consider a manufacturing facility that produces five different products using four machines. The scarce resources are the times available on the machines and the alternative activities are the individual production volumes. The machine requirements in hours per unit are shown for each product in the table. With the exception of product 4 that does not require machine 1, each product must pass through all four machines. The unit profits are also shown in the table. The facility has four machines of type 1, five of type 2, three of type 3 and seven of type 4. Each machine operates 40 hours per week. The problem is to determine the optimum weekly production quantities for the products. The goal is to maximize total profit. In constructing a model, the first step is to define the decision variables; the next step is to write the constraints and objective function in terms of these variables and the problem data. In the problem statement, phrases like "at least," "no greater than," "equal to," and "less than or equal to" imply one or more constraints. Machine data and processing requirements (hrs./unit)

MachineQuantityProduct 1Product 2Product 3Product 4Product 5 M141.21.30.70.00.5
M250.72.21.60.51.0
M330.90.71.31.00.8
M471.42.80.51.20.6
Unit profit, $——1825101215

Model
Variable Definitions
Pj : quantity of product j produced, j = 1,...,5
Machine Availability Constraints
The number of hours available on each machine type is 40 times the number of machines. All the constraints are dimensioned in hours. For machine 1, for example, we have 40 hrs./machine  4 machines = 160 hrs. M1 :1.2P1 + 1.3P2 + 0.7P3 + 0.0P4 + 0.5P5 < 160

M2 :0.7P1 + 2.2P2 + 1.6P3...
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