At the current basic feasible solution (bfs) ν0 (ν0 ∈ \(\R^5\)), the simplex method yields the following form of a linear programming problem in standard form.
minimize z = -x1 - 2x2
s.t. x3 = 2 + 2x1 - x2
x4 = 7+x1-2x2
x5 = 3-x1
x1, x2, x3, x4, x5 \(\geq\) 0
Here the objective function is written as a function of the non-basic variables. If the simplex method moves to the adjacent bfs v1 (v1 ∈ \(\R^5\)) that best improves the objective function, which of the following represents the objective function at v1, assuming that the objective function is written in the same manner as above?
minimize z = -x1 - 2x2
s.t. x3 = 2 + 2x1 - x2
x4 = 7+x1-2x2
x5 = 3-x1
x1, x2, x3, x4, x5 \(\geq\) 0
Here the objective function is written as a function of the non-basic variables. If the simplex method moves to the adjacent bfs v1 (v1 ∈ \(\R^5\)) that best improves the objective function, which of the following represents the objective function at v1, assuming that the objective function is written in the same manner as above?
- z=-4-5x1 + 2x3
- z = -3+x5 - 2x2
- z=-4-5x1 + 2x4
- z = -6-5x1 + 2x3
The Correct Option is A
Solution and Explanation
Top Questions on Linear programming
A factory produces \( m \) (i = 1, 2, ..., m) products, each of which requires processing on \( n \) (j = 1, 2, ..., n) workstations. Let \( a_{ij} \) be the amount of processing time that one unit of the \( i^{th} \) product requires on the \( j^{th} \) workstation. Let the revenue from selling one unit of the \( i^{th} \) product be \( r_i \) and \( h_i \) be the holding cost per unit per time period for the \( i^{th} \) product. The planning horizon consists of \( T \) (t = 1, 2, ..., T) time periods. The minimum demand that must be satisfied in time period \( t \) is \( d_{it} \), and the capacity of the \( j^{th} \) workstation in time period \( t \) is \( c_{jt} \). Consider the aggregate planning formulation below, with decision variables \( S_{it} \) (amount of product \( i \) sold in time period \( t \)), \( X_{it} \) (amount of product \( i \) manufactured in time period \( t \)) and \( I_{it} \) (amount of product \( i \) held in inventory at the end of time period \( t \)). \[ \text{max} \sum_{t=1}^{T} \sum_{i=1}^{m} (r_i S_{it} - h_i I_{it}) \] subject to \[ S_{it} \ge d_{it} \forall i, t \] <capacity constraint>
<inventory balance constraint>
\[X_{it}, S_{it}, I_{it} \ge 0; \, I_{i0} = 0 \] The capacity constraints and inventory balance constraints for this formulation are
A cement company has three factories which transport cement to four distribution centres. The daily production of each factory, the demand at each distribution centre, and the associated transportation cost per tonne from factory to distribution centre are given in the Table.
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