1)a) A company is manufacturing a product which requires, at the final assembly stage, three parts. These three parts can be produced by two different departments as detailed below.
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Production rate (units/hour)
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Part 1
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Part 2
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Part 3
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Department 1
Department 2
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7
6
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6
11
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9
5
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Production cost per hour is €25 in department 1 and €12.50 in department 2.
One week, 1050 finished (assembled) products are needed (but up to 1200 can be produced if necessary). If department 1 has 100 working hours available, but department 2 has 110 working hours available, formulate the problem of minimising the cost of producing the finished (assembled) products needed this week as a linear programming problem.
(b)(i) What does it mean to describe a basis as infeasible but optimal? (ii) Use the Dual Simplex method to solve the following problem:
Minimise z = 3x, +2x2
subject to 3x, + x2 ≥ 3
x1+ x2 ≤ 3
x„ x2 ≥ O.
(c) The table gives a firm's production capacity and orders for a special product. Costs are €30 per unit on regular time, €40 per unit on overtime and €3 per unit per month for inventory. Inventory costs are incurred on goods that are not sold in the month in which they are produced.
Month Production Capacities Orders
August Regular time 300, overtime 100
September Regular time 300, overtime 100 500
October Regular time 200, overtime 100 400
Set up this problem as a transportation problem, find an initial basic feasible solution, and test this solution for optimality.