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Z=50x+275y+175z where x,y,z are inventory of the product required by ware house 1,2,3, now coming to equation 5x+8y+4z greater than equal to 100,equation 2 7x+9y+3z greater then equal to 250,equation 3-6x+10y+11z greater than 150 then you can make graph and find the value

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Hint Since $A$ is a $m \times n$ matrix, with $m<n$, a Singular Values Decomposition of it will represent a first useful step. In particular you can determine the null-space of it, i.e. the set of vectors $v_1, v_2, \cdots, v_{n-m}$ (or more) for which $${\bf A}\,{\bf v}_{\,k} = {\bf 0}\quad \Rightarrow \quad {\bf A}\,\sum\limits_k {\lambda _{\,k} } ... 1 This blog post demonstrates one approach that uses mixed integer linear programming: https://blogs.sas.com/content/operations/2014/11/10/do-you-have-an-uncle-louie-optimal-wedding-seat-assignments/ 1 The disaggregated cuts t_i \le \mu_i will yield a tighter feasible region (and hence a better lower bound) than the aggregated cut \sum_i t_i \le \sum_i \mu_i, but it is worth trying both ways. 1 The second row is multiplied by 3, then the third row is subtracted from it. This is equally valid. 1 The second part (not returning to base) can be finessed by adding a zero-length arc from every node other than base back to the base node. So wherever you end up last, you return to base at no cost. As for revisiting nodes, if you know in advance how many times each node must be visited, and if there are no time windows or other constraints on when you ... 1 As you have noted, this (implicit) feasibility region on y can be formulated as the projection (via y=cx) of an explicit feasibility region in x. I had to deal with this problem myself recently. One method, as people have commented, is to obtain the vertices from your half-space representation and then project these down; the convex hull of these ... 1 Firstly we define the variables: x_1: Multiple of pancake recipe, x_2: Multiple of waffle recipe. We want to maximize the total amount of serves. Therefore the objective function is$$\text{max} \ \ 6x_1+5x_2$$Next we have several constraints for the ingredients. For instance, that you have 24 cups of Bisquick. The corresponding constraint is$$3x_1+...

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The following is a proof for the only if side of your problem, which requires a good understanding of the duality theorems in linear programming, espeically the complementary slackness thoerem. Proof$\;$ Suppose $F$ is a face of $P$ defined by $c^Tx\le \delta$, i.e., $F=\{x\in P|c^Tx =\delta\}$. Note that $F$ is the set of optimal solutions for linear ...

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