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việt 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikiquote wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from mixed integer programming method to solve optimization problems for the retronym referring to television broadcasting see broadcast programming a pictorial representation of a simple linear program with two variables and six inequalities the set of feasible solutions is depicted in yellow and forms a polygon a 2 dimensional polytope the optimum of the linear cost function is where the red line intersects the polygon the red line is a level set of the cost function and the arrow indicates the direction in which we are optimizing a closed feasible region of a problem with three variables is a convex polyhedron the surfaces giving a fixed value of the objective function are planes not shown the linear programming problem is to find a point on the polyhedron that is on the plane with the highest possible value linear programming lp also called linear optimization is a method to achieve the best outcome such as maximum profit or lowest cost in a mathematical model whose requirements and objective are represented by linear relationships linear programming is a special case of mathematical programming also known as mathematical optimization more formally linear programming is a technique for the optimization of a linear objective function subject to linear equality and linear inequality constraints its feasible region is a convex polytope which is a set defined as the intersection of finitely many half spaces each of which is defined by a linear inequality its objective function is a real valued affine linear function defined on this polytope a linear programming algorithm finds a point in the polytope where this function has the largest or smallest value if such a point exists linear programs are problems that can be expressed in standard form as find a vector x that maximizes c t x subject to a x b and x 0 displaystyle begin aligned text find a vector mathbf x text that maximizes mathbf c mathsf t mathbf x text subject to a mathbf x leq mathbf b text and mathbf x geq mathbf 0 end aligned here the components of x displaystyle mathbf x are the variables to be determined c displaystyle mathbf c and b displaystyle mathbf b are given vectors and a displaystyle a is a given matrix the function whose value is to be maximized x c t x displaystyle mathbf x mapsto mathbf c mathsf t mathbf x in this case is called the objective function the constraints a x b displaystyle a mathbf x leq mathbf b and x 0 displaystyle mathbf x geq mathbf 0 specify a convex polytope over which the objective function is to be optimized linear programming can be applied to various fields of study it is widely used in mathematics and to a lesser extent in business economics and some engineering problems there is a close connection between linear programs eigenequations john von neumann s general equilibrium model and structural equilibrium models see dual linear program for details 1 2 3 industries that use linear programming models include transportation energy telecommunications and manufacturing it has proven useful in modeling diverse types of problems in planning routing scheduling assignment and design history edit leonid kantorovich john von neumann the problem of solving a system of linear inequalities dates back at least as far as fourier who in 1827 published a method for solving them 4 and after whom the method of fourier motzkin elimination is named in the late 1930s soviet mathematician leonid kantorovich and american economist wassily leontief independently delved into the practical applications of linear programming kantorovich focused on manufacturing schedules while leontief explored economic applications their groundbreaking work was largely overlooked for decades the turning point came during world war ii when linear programming emerged as a vital tool it found extensive use in addressing complex wartime challenges including transportation logistics scheduling and resource allocation linear programming proved invaluable in optimizing these processes while considering critical constraints such as costs and resource availability despite its initial obscurity the wartime successes propelled linear programming into the spotlight post wwii the method gained widespread recognition and became a cornerstone in various fields from operations research to economics the overlooked contributions of kantorovich and leontief in the late 1930s eventually became foundational to the broader acceptance and utilization of linear programming in optimizing decision making processes 5 kantorovich s work was initially neglected in the ussr 6 about the same time as kantorovich the dutch american economist t c koopmans formulated classical economic problems as linear programs kantorovich and koopmans later shared the 1975 nobel memorial prize in economic sciences 4 in 1941 frank lauren hitchcock also formulated transportation problems as linear programs and gave a solution very similar to the later simplex method 7 hitchcock had died in 1957 and the nobel memorial prize is not awarded posthumously from 1946 to 1947 george b dantzig independently developed general linear programming formulation to use for planning problems in the us air force 8 in 1947 dantzig also invented the simplex method that for the first time efficiently tackled the linear programming problem in most cases 8 when dantzig arranged a meeting with john von neumann to discuss his simplex method von neumann immediately conjectured the theory of duality by realizing that the problem he had been working in game theory was equivalent 8 dantzig provided formal proof in an unpublished report a theorem on linear inequalities on january 5 1948 6 dantzig s work was made available to public in 1951 in the post war years many industries applied it in their daily planning dantzig s original example was to find the best assignment of 70 people to 70 jobs the computing power required to test all the permutations to select the best assignment is vast the number of possible configurations exceeds the number of particles in the observable universe however it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the simplex algorithm the theory behind linear programming drastically reduces the number of possible solutions that must be checked the linear programming problem was first shown to be solvable in polynomial time by leonid khachiyan in 1979 9 but a larger theoretical and practical breakthrough in the field came in 1984 when narendra karmarkar introduced a new interior point method for solving linear programming problems 10 uses edit linear programming is a widely used field of optimization for several reasons many practical problems in operations research can be expressed as linear programming problems 6 certain special cases of linear programming such as network flow problems and multicommodity flow problems are considered important enough to have much research on specialized algorithms a number of algorithms for other types of optimization problems work by solving linear programming problems as sub problems historically ideas from linear programming have inspired many of the central concepts of optimization theory such as duality decomposition and the importance of convexity and its generalizations likewise linear programming was heavily used in the early formation of microeconomics and it is currently utilized in company management such as planning production transportation and technology although the modern management issues are ever changing most companies would like to maximize profits and minimize costs with limited resources google also uses linear programming to stabilize youtube videos 11 representation of the problem of finding the minimum of a finite number of linear programming maximization problems as a single linear programming problem is well known representation of the problem of finding the maximum of a finite number of linear programming maximization problems as a single linear programming problem is available in topic 2 of section 1 of a monograph entitled some topics in linear programming and bi matrix game theory 12 standard form edit standard form is the usual and most intuitive form of describing a linear programming problem it consists of the following three parts a linear or affine function to be maximized e g f x 1 x 2 c 1 x 1 c 2 x 2 displaystyle f x_ 1 x_ 2 c_ 1 x_ 1 c_ 2 x_ 2 problem constraints of the following form e g a 11 x 1 a 12 x 2 b 1 a 21 x 1 a 22 x 2 b 2 a 31 x 1 a 32 x 2 b 3 displaystyle begin matrix a_ 11 x_ 1 a_ 12 x_ 2 leq b_ 1 a_ 21 x_ 1 a_ 22 x_ 2 leq b_ 2 a_ 31 x_ 1 a_ 32 x_ 2 leq b_ 3 end matrix non negative variables e g x 1 0 x 2 0 displaystyle begin matrix x_ 1 geq 0 x_ 2 geq 0 end matrix the problem is usually expressed in matrix form and then becomes max c t x x r n a x b x 0 displaystyle max mathbf c mathsf t mathbf x mid mathbf x in mathbb r n land a mathbf x leq mathbf b land mathbf x geq 0 other forms such as minimization problems problems with constraints on alternative forms and problems involving negative variables can always be rewritten into an equivalent problem in standard form example edit graphical solution to the farmer example after shading regions violating the conditions the vertex of the unshaded region with the dashed line farthest from the origin gives the optimal combination its lying on the land and pesticide lines implies that revenue is limited by land and pesticide not fertilizer suppose that a farmer has a piece of farm land say l hectares to be planted with either wheat or barley or some combination of the two the farmer has f kilograms of fertilizer and p kilograms of pesticide every hectare of wheat requires f 1 kilograms of fertilizer and p 1 kilograms of pesticide while every hectare of barley requires f 2 kilograms of fertilizer and p 2 kilograms of pesticide let s 1 be the selling price of wheat and s 2 be the selling price of barley per hectare if we denote the area of land planted with wheat and barley by x 1 and x 2 respectively then profit can be maximized by choosing optimal values for x 1 and x 2 this problem can be expressed with the following linear programming problem in the standard form maximize s 1 x 1 s 2 x 2 displaystyle s_ 1 cdot x_ 1 s_ 2 cdot x_ 2 maximize the revenue the total wheat sales plus the total barley sales revenue is the objective function subject to x 1 x 2 l displaystyle x_ 1 x_ 2 leq l limit on total area f 1 x 1 f 2 x 2 f displaystyle f_ 1 cdot x_ 1 f_ 2 cdot x_ 2 leq f limit on fertilizer p 1 x 1 p 2 x 2 p displaystyle p_ 1 cdot x_ 1 p_ 2 cdot x_ 2 leq p limit on pesticide x 1 0 x 2 0 displaystyle x_ 1 geq 0 x_ 2 geq 0 cannot plant a negative area in matrix form this becomes maximize s 1 s 2 x 1 x 2 displaystyle begin bmatrix s_ 1 s_ 2 end bmatrix begin bmatrix x_ 1 x_ 2 end bmatrix subject to 1 1 f 1 f 2 p 1 p 2 x 1 x 2 l f p x 1 x 2 0 0 displaystyle begin bmatrix 1 1 f_ 1 f_ 2 p_ 1 p_ 2 end bmatrix begin bmatrix x_ 1 x_ 2 end bmatrix leq begin bmatrix l f p end bmatrix begin bmatrix x_ 1 x_ 2 end bmatrix geq begin bmatrix 0 0 end bmatrix augmented form slack form edit linear programming problems can be converted into an augmented form in order to apply the common form of the simplex algorithm this form introduces non negative slack variables to replace inequalities with equalities in the constraints the problems can then be written in the following block matrix form maximize z displaystyle z 1 c t 0 t 0 a i z x s 0 b displaystyle begin bmatrix 1 mathbf c mathsf t mathbf 0 mathsf t mathbf 0 mathbf a mathbf i end bmatrix begin bmatrix z mathbf x mathbf s end bmatrix begin bmatrix 0 mathbf b end bmatrix x 0 s 0 displaystyle mathbf x geq mathbf 0 mathbf s geq mathbf 0 where s displaystyle mathbf s are the newly introduced slack variables x displaystyle mathbf x are the decision variables and z displaystyle z is the variable to be maximized example edit the example above is converted into the following augmented form maximize s 1 x 1 s 2 x 2 displaystyle s_ 1 cdot x_ 1 s_ 2 cdot x_ 2 objective function subject to x 1 x 2 x 3 l displaystyle x_ 1 x_ 2 x_ 3 l augmented constraint f 1 x 1 f 2 x 2 x 4 f displaystyle f_ 1 cdot x_ 1 f_ 2 cdot x_ 2 x_ 4 f augmented constraint p 1 x 1 p 2 x 2 x 5 p displaystyle p_ 1 cdot x_ 1 p_ 2 cdot x_ 2 x_ 5 p augmented constraint x 1 x 2 x 3 x 4 x 5 0 displaystyle x_ 1 x_ 2 x_ 3 x_ 4 x_ 5 geq 0 where x 3 x 4 x 5 displaystyle x_ 3 x_ 4 x_ 5 are non negative slack variables representing in this example the unused area the amount of unused fertilizer and the amount of unused pesticide in matrix form this becomes maximize z displaystyle z 1 s 1 s 2 0 0 0 0 1 1 1 0 0 0 f 1 f 2 0 1 0 0 p 1 p 2 0 0 1 z x 1 x 2 x 3 x 4 x 5 0 l f p x 1 x 2 x 3 x 4 x 5 0 displaystyle begin bmatrix 1 s_ 1 s_ 2 0 0 0 0 1 1 1 0 0 0 f_ 1 f_ 2 0 1 0 0 p_ 1 p_ 2 0 0 1 end bmatrix begin bmatrix z x_ 1 x_ 2 x_ 3 x_ 4 x_ 5 end bmatrix begin bmatrix 0 l f p end bmatrix begin bmatrix x_ 1 x_ 2 x_ 3 x_ 4 x_ 5 end bmatrix geq 0 duality edit main article dual linear program every linear programming problem referred to as a primal problem can be converted into a dual problem which provides an upper bound to the optimal value of the primal problem in matrix form we can express the primal problem as maximize c t x subject to a x b x 0 with the corresponding symmetric dual problem minimize b t y subject to a t y c y 0 an alternative primal formulation is maximize c t x subject to a x b with the corresponding asymmetric dual problem minimize b t y subject to a t y c y 0 a fundamental idea of duality theory is the fact that for the symmetric dual the dual of a dual linear program is the original primal linear program the dual variables y displaystyle mathbf y may be understood as coefficients in a linear combination of inequalities from the primal problem formed in the hope of obtaining a bound on the objective function such coefficients need to be nonnegative since otherwise inequalities would become reversed whence we require y 0 displaystyle mathbf y geq mathbf 0 if in addition c y a displaystyle mathbf c top leq mathbf y top a then it follows that c x y a x y b b y displaystyle mathbf c top mathbf x leq mathbf y top a mathbf x leq mathbf y top mathbf b mathbf b top mathbf y text i e the value of the dual objective function b y displaystyle mathbf b top mathbf y at any feasible dual point y displaystyle mathbf y is an upper bound on the value of the primal objective function c x displaystyle ma...