Was effective and efficient in its entirety and have a privileged status in the market, but they gradually lose the internal capacities and its position in the market as a result of damages sustained because of war suffered by the country in General and in Basra in particular Subversion organization campaigns against facilities and production lines and partial reform cases returned good restoration company facilities and the lines already so it became necessary to renovate facilities and production lines of the company to recover and the large role by different methods To improve production lines, including **linear** **programming** method So often departments of industrial establishments difficulties in securing certain types and quantities of resources available in the productive process to good use of the resources of the (the machinery and manpower and raw materials). To achieve efficiency in the degree of achievement of the objectives established.

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The general BLP problem has been given so much attention by researchers all over the world for over half a century without a breakthrough. A difficult category of BLP models includes the traveling salesman, generalized assignment, quadratic assignment and set covering problems. The paper presented a technique to solve BLP problems by first transforming them into convex QPs and then applying interior point algorithms to solve them in polynomial time. We also showed that the proposed technique worked for both pure and mixed BLPs and also for the general **linear** integer model where variables were expanded into BLPs. We hope the proposed approach will give more clues to researchers in the hunt for efficient solutions to the general difficult integer **programming** problem.

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For **linear** bilevel **programming**, the branch and bound algorithm is the most successful algorithm to deal with the complementary constraints arising from Kuhn-Tucker conditions. However, one principle challenge is that it could not well handle a **linear** bilevel **programming** problem when the constraint functions at the upper-level are of arbitrary **linear** form. This paper proposes an extended branch and bound algorithm to solve this problem. The results have demostrated that the extended branch and bound algorithm can solve a wider class of **linear** bilevel problems can than current capabilities permit.

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proposed method is provided to solve this model and then implemented and analyzed on an actual example. Data of Milan stock market showed that the new model can be solved in shorter time. Whereas, expected return rate of 12% in based model cannot solve a problem with more than 14 types of stocks. The new model can solved a problem with 20 different stocks easily and in a short time. Furthermore, a new model greatly reduced undesirable risk very much compared to the base model. So that the process continued decreasingly with increasing the number of shares. Another study by Kuhpaei & others (2007) as a program to determine the optimal transportation of wheat was done using **linear** **programming**. Every year, millions tonnes of wheat from domestic shopping centers and import origins was transported to storage center and then distributed among regions based on consumer demand. In 2000, wheat allocated the largest volume of goods transported after cement. Using a logical model is necessary to reduce the cost of transportation. This study presents a mathematical model for determining the optimal plan of wheat transport from provincial centers and import origins to storage centers and thence to applied regions. Based on the data from 2000 and Lingo software package, studies was done separately for each month. The proposed program reduced the cost to 138 billion rials (13.5 percent) compared to the program performed in 2000. After solving the problem and determining the optimal schedule by using the concept of shadow price, time and place schedule and the method of direct or indirect transportation of wheat from shopping centers to storage centers was identified according to priorities determined by time and place. With implementation of this program, cost of transportation is reduced to 45 billion rials.

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[10] Hosseini, E and I.Nakhai Kamalabadi, Taylor Approach for Solving Non-**Linear** Bi-level **Programming** Problem ACSIJ Advances in Computer Science: an International Journal, Vol. 3, Issue 5, No.11 , September 2014. [11] Hejazi, S.R and A. Memariani, G. Jahanshahloo, (2002) **Linear** bi-level **programming** solution by genetic algorithm, Computers & Operations Research 29 1913–1925.

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This paper presents new algorithms to find the fuzzy optimal solution of the FFBLP problem with equality constraints and non-negative fuzzy variables or unconstrained fuzzy variables. In these algorithms, the fuzzy bilevel **linear** pro- gramming problem is transformed to a determin- istic bilevel **programming** problem using the rank- ing function method, and the solution of the de- terministic problem is achieved using the com- mon Kth-best method. The proposed methods are quite useful in solving the real-world prob- lems where the information is inexact. To ob- tain the fuzzy optimal solution, new efficient algo- rithms have been proposed for FFBLP problems and some numerical examples have been solved to illustrate the proposed methods.

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Recently Zeng et.al [21] considered a crop area planning problem and proposed a fuzzy multi objective linear programming problem with fuzzy numbers and transformed the fuzzy multi object[r]

vestigated the algorithm to solve **linear** **programming** problem for multi-objective functions. By new technique, we use harmonic mean (HM) of the values of objective functions. The computer application of our algorithm has also been discussed by solving some numerical examples. Finally we have showed results and comparison among the new technique and Chandra Sen’s approach [1] and Sulaiman’s approach [2].

In 2010, Moustapha Diaby provided two further proofs for P=NP. His papers **Linear** **programming** formulation of the vertex colouring problem and **Linear** **programming** formula- tion of the set partitioning problem give **linear** **programming** formulations for two well-known NP-hard problems [6], [7]. The goal of this paper is proof of proposition that kSAT is in P using multi logic formula of discrete second order logic proposed first in [8], [9] and could be solved in O (m).

Abstract - In a real world bilevel decision-making, the lower level of a bilevel decision usually involves multiple decision units. This paper proposes the Kth-best approach for **linear** bilevel multifollower **programming** problems with shared variables among followers. Finally a numeric example is given to show how the Kth-best approach works.

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Since this model has 12variables LINGO software (version 18.0) is used to get a solution and the results checked by NCSS (version 12.0.10) and TORA Optimization system windows version 2.00. Solving **linear** **programming** problem gives a Global optimal solution as

The idea of a **linear** programme’s duality and the theory of **linear** **programming** along with the duality marking manner have played a special role in economic analyses by the way in which they have emphasized the nature of prices. Ever since marginal analysis onwards, no other idea has proven to be that important to the fundamental theory of prices 1 .

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Fuzzy **linear** **programming** problem occur in many fields such as mathematical modeling, Control theory and Management sciences, etc. In this paper we focus on a kind of **Linear** **Programming** with fuzzy numbers and variables namely Fully Fuzzy **Linear** **Programming** (FFLP) problem, in which the constraints are in inequality forms. Then a new method is proposed to fine the fuzzy solution for solving (FFLP). Numerical examples are providing to illustrate the method.

Example 6 (Diet problem): A dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10 units of vitamin C. Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food ‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg to purchase Food ‘I’ and Rs 70 per kg to purchase Food ‘II’. Formulate this problem as a **linear** **programming** problem to minimise the cost of such a mixture. Solution Let the mixture contain x kg of Food ‘I’ and y kg of Food ‘II’. Clearly, x ≥ 0, y ≥ 0. We make the following table from the given data:

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In this paper, the terminal control problem is treated as a saddle-point dynamic problem with the boundary value condition. This condition is defined implicitly as a solution to the **linear** **programming** problem. The saddle-point dynamic problem generates a system of saddle-point inequalities in functional space. These inequalities are seen as strengthening the Pontryagin max- imum principle in the convex case. The saddle-point inequalities generate the differential system, which is close to a similar system in the maximum principle. Based on this system, the saddle- point process was formulated, and its convergence to the saddle point of the Lagrange function was proved. Namely, it was proved the weak convergence in controls, the strong convergence in phase and conjugated trajectories as well as the strong convergence to a solution of the boundary-value optimization problem on set of attainability.

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It is an effective method for solving **linear** **programming**, which allows us to systematically explore the set of basic solutions. The simplex method consists of a sequence of iterations, each iteration determining a basic solution with the feature that the objective function value is continuously improved up to the optimum, if any. The principle of work is changing the base, a method presented in the previous chapter (the pivot method). The question however is which vector enters the base and which one is leaving, to ensure the search on the set of basic admissible solutions.

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The existence of strongly polynomial algorithm for **linear** **programming** (LP) has been widely sought after for decades. Recently, a new approach called Gravity Sliding algorithm [1] has emerged. It is a gradient descending method whereby the descending trajectory slides along the inner surfaces of a polyhe- dron until it reaches the optimal point. In R 3 , a water droplet pulled by gravi-

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Transition from basis to basis until the optimal basis is reached is effected by changing variables one at a time, symbolically variable numbered k previously with value zero comes into [r]

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Graphs of inequalities have to show a range of possibilities. This is done by shading one side of the graph of a **linear** equation. Graphs involving the symbols and include the values on the line, while graphs involving the symbols and do not include the values on the line. The graph of a **linear** inequality is drawn using the graph of the corresponding **linear** equation. 4 The water level in the big dam on a large property is at 6.5 m. The current rate of use means

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