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activity algorithm artificial variables basic feasible solution basic solution basic variables basis calculation coefficients combined matrix commodity computation convex combination convex set corporate constraint corresponding cost Dalks Dantzig decentralized decision decomposition principle division divisional constraints dual linear programs dual pair dual problem dual slack dual variables duality theorem economic executive program extreme point identity matrix iteration linear equations linear inequalities linear programming model linear programming problem marginal profit mathematical matrix games Maximize Minimize negative non-basic variables non-negative non-zero objective function obtained optimal solution Philip Wolfe pivot column pivot element pivot entry pivot steps primal and dual primal problem primal solution primal variables proposed divisional solutions provisional dual prices revised simplex method routine satisfy schema schemata shadow prices Simplex Algorithm slack variables solve tableau tankers unit of output vector space x-feasible yield zero