linear programming and game theory - 2 problems

Consider the linear programming problem: Max 6x1 + 14x2 + 13x3 subject to: ½x1 + 2x2 + x3 ≤24 x1 + 2x2 + 4x3 ≤ 60 x1, x2, x3 ≥ 0. along with the optimal dictionary x1 = 36 - 6x2 - 4s1 + s2 x3 = 6 + x2 + s1 - ½s2 _________________________ z = 294- 9x2 -11s1-½s2 (a) Without doing any computation, write down the optimal dual dictionary. (Your dictio-nary must correspond to the "maximization" version of the dual problem, which you can get by negating the dual objective function.) Please use y1, y2 as your regular variables and u1, u2, u3 to represent the dual slacks. (b) Suppose the new constraint x1 + x2 + x3 ≤ 40 is ... click for more

Job Shop Problem

Suppose you have N jobs that have to be processed on a single machine. For i = 1, 2, . . . ,N, job i requires pi units of time on the machine, and has weight wi. The objective is to schedule these jobs so as to minimize the sum of the weighted completion time of all the jobs, where the completion time of job i is the time at which job i finishes. Note that at most one job can run on the machine at any time. For example, let N = 3, p1= 1, p2 = 2, p3 = 3, and the weights are w1 = 2, w2 = 1, w3 = 0. If the jobs are scheduled in the order 231, then the completion time of job 2 is 2, completion time of job 3 is 5, and the completion time of job 1 = 6. The value of the objective function for thi... click for more

Linear Programming using Excel

Linear Programming Models in Excel (Solver) -------------------------------------------------------------------------------- TABLE: Hours for Judical Problem Jan 400 July 200 Feb 300 Aug 400 Mar 200 Sept 300 April 600 Oct 200 May 800 Nov 100 June 300 Dec 300 Suppose each judge works all 12 months and can handle up to 120 hours per month of casework. To avoid a backlog, all cases must be handled by the end of December. Determine how many judges are needed. If each judge receives 1 month of vacation each year, how does your answer change? SOLVE THIS PROBLEM USING EXCEL SOLVER

Proof in Linear programming

Please help me to find out how I can do this (See attached file for full problem description) --- Let (see attachment) It is clear that we can rewrite (attached) as (attached) , i.e. as a system of linear inequalities. (I've done this). Show that in fact we can rewrite (attached) as a system of (attached) linear inequalities. --- (See attached file for full problem description)

Proof of Dual using Farkas Lemma (PhD)

Hello, Could you please help me to prove this using Farkas Lemma? Well, I initially thought that I can use Farkas Lemma, but if it is impossible to use the lemma (though I do belive it will help), you might try other way. Thank you! --- (See attached file for full problem description)