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We study the restricted inverse optimal value problem on linear programmingunder weighted $l_1$ norm (RIOVLP $_1$). Given a linear programming problem$LP_c: \min \{cx|Ax=b,x\geq 0\}$ with a feasible solution $x^0$ and a value$K$, we aim to adjust the vector $c$ to $\bar{c}$ such that $x^0$ becomes anoptimal solution of the problem LP$_{\bar c}$ whose objective value$\bar{c}x^0$ equals $K$. The objective is to minimize the distance $\|\bar c -c\|_1=\sum_{j=1}^nd_j|\bar c_j-c_j|$ under weighted $l_1$ norm.Firstly, weformulate the problem (RIOVLP$_1$) as a linear programming problem by dualtheories. Secondly, we construct a sub-problem $(D^z)$, which has the same formas $LP_c$, of the dual (RIOVLP$_1$) problem corresponding to a given value $z$.Thirdly, when the coefficient matrix $A$ is unimodular, we design a binarysearch algorithm to calculate the critical value $z^*$ corresponding to anoptimal solution of the problem (RIOVLP$_1$). Finally, we solve the (RIOV)problems on Hitchcock and shortest path problem, respectively, in$O(T_{MCF}\log\max\{d_{max},x^0_{max},n\})$ time, where we solve a sub-problem$(D^z)$ by minimum cost flow in $T_{MCF}$ time in each iteration. The values$d_{max},x^0_{max}$ are the maximum values of $d$ and $x^0$, respectively.

Restricted inverse optimal value problem on linear programming under weighted $l_1$ norm