LINEAR PROGRAMMING

Linear Programming

Linear Programming

Task 1

(a)

Linear Programming Model adopted by Hi-tec Electronics Ltd

Every network flow model has a linear programming model, that is a model with algebraic linear expressions describing the objective function and constraints. We explain here the model for the specific case above, and will provide in the Vocabulary Section, the general model. For construction of the model, it is convenient to number the nodes and arcs for reference as in Fig.

The linear programming model is an algebraic description of the objective to be minimized and the constraints to be satisfied by the variables. The variables are the flows in each arc designated by x1 through x17. The network flow problem is to minimize total cost while satisfying conservation of flow at each node[1]. The variables must also satisfy the simple upper and lower bounds on arc flow.

(b)

The figure represents a manufacturing system producing two products labeled P and Q. The rounded rectangles at the top of the figure indicate the revenue per unit and the maximum sales per week. For instance we can sell as many as 100 units of P for £90 per unit. The circles show the raw materials used, and the rectangles indicate the operations that the products must pass through in the manufacturing process. Each rectangle designates a machine used for the operation and the time required.

For example product P consists of two subassemblies. To manufacture the first subassembly, one unit of RM1 passes through machine A for 15 minutes. The output of machine A is moved to machine C where it is processed for 10 minutes. The second subassembly starts with RM2 processed in machine B for 15 minutes. The output is taken to machine C for 5 minutes of processing. The two subassemblies are joined with a purchased part in machine D. The result is a finished unit of P. Product Q is manufactured by a similar process as indicated in the figure.

(c)

The rectangle at the upper left indicates that one machine of each type is available. Each machine operates for 2400 minutes per week. OE stands for operating expenses. For this case the operating expenses, not including the raw material cost is £6000.

There are many problems that might be posed using the figure above, but we choose the problem of allocating the times available on the machines to the manufacture of the two products. The decisions involve the amounts of the two products.

The objective is to maximize profit. From the figure we see that the profit per unit of product is its unit revenue less the raw material cost per unit. For P the unit profit is £45 and for Q it is £60. The objective is a linear expression of the amounts produced.

The constraints specify that the amounts of time required of each machine must not exceed the amount available. The amount of time required of a machine is a linear function of the production amounts.

Machine Time Constraints

Finally, we require that the amounts manufactured not exceed ...