Mineral Economics, Mine Planning, Systems Engineering · Systems Engineering · 2 marks · Numerical answer
Assign four tasks T1–T4 to four machines, one per machine. Costs by machine are M1:(10, 40, 60, 30), M2:(90, 70, 100, 90), M3:(40, 50, 110, 70), M4:(80, 70, 80, 50). Find the optimal total cost, to one decimal.
An optimal assignment is M1→T1, M2→T3, M3→T2, M4→T4, costing 10+100+50+50=210.0. To prove optimality, take row potentials (20, 70, 50, 50) and column potentials (−10, 0, 30, 0). Every row-plus-column potential is at most its corresponding cost, and equality holds for each chosen assignment. Their total is also 210. Thus 210 is both a feasible cost and a valid lower bound, proving it is optimal.
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