109 lines
5.9 KiB
Plaintext
109 lines
5.9 KiB
Plaintext
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<20><><EFBFBD><EFBFBD><EFBFBD> <20><><EFBFBD><EFBFBD><EFBFBD> <20><><EFBFBD><EFBFBD><EFBFBD> <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
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ARRoGANT CoURiERS WiTH ESSaYS
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Grade Level: Type of Work Subject/Topic is on:
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[ ]6-8 [ ]Class Notes [Linear Programming ]
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[ ]9-10 [ ]Cliff Notes [ ]
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[x]11-12 [x]Essay/Report [ ]
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[x]College [ ]Misc [ ]
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Dizzed: 07/94 # of Words:560 School:Public State:NY
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<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>><3E><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>><3E><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>>Chop Here><3E><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>><3E><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>><3E><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>><3E><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
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Linear programming is a nonstatistical mathematical technique whereby
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the maximization or minimization of a linear expresion of variables, call
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the objective function, is determined in the presence of known or assumed
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restrictions, call constraint.
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In essence, it's a procedure for solving the problems in which there
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are more variables than simultaneous equations in which the variables are
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expressed.
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No probability or statistics are needed to study linear programming.
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The mathematics involved in linear programming is relatively easy to
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understand and to manipulate in contrast to calculus. Linear equations and
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inequations form the mathematical skeleton around which linear programming
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is built.
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A linear function called the object function is to be maximized or
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minimized in some sense, like optimzed. Most real world problems have many
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possible solutions. The purpose of optimization is to choose from among
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many possible solutions the "best" possible solutions. Some example of
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"best" are highest profit, lowest cost, largest sales, lowest production
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time, etc.
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The optimization of the objective functions take place in teh presence
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of known or assumed restriction. The technical term constraints is used to
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describe the restrictions present in linear programming problem. The
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constraints are expressed mathemically as inequalities. In a practical
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real-world situation, the constraints are generated by the presence of
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limited resources or commodities such as capital manpower and raw material.
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Mathematically, inequations can be converted to equations by the
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introduction of slack variables.
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Linear programming can be dated from the year 1947 when G.B. Dantzing
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evolved an efficent technique call the Simplex Method, for solving linear
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programming problems. The following decades, the rapid development of both
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the theory and applications of linear programming which were aided by the
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simultaneous introduction of the electronic computers. One of the first
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probelm to be solved by the simplex method was Stigler's diet problem
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(1945). Here is the diet problem
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<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
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Protein Fat Carbohydrate Cost
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<EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
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100g bread 40 5 205 2.2p
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100g cheese 60 380 60 12p
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Minimal daily requirement 300 790 1350
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The problem is determine how much bread and cheese Mrs. Jones
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should buy each day in order to minimize the cost of the diet, whilst
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fulfilling the calorie requirements. Suppose shy buys x' * 100g of
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cheese and x'' * 100g of cheese, then the mathematical problem, known
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as a linear programme is as follows.
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Minimize z=2.2x' + 12x'' (Cost Of Diet)
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Subject To 40x' + 60x'' > or = 300 (At least 300 cal of protein)
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5x' + 380x'' > or = 790 (At least 790 cal of fat)
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205x' + 60x'' > or = 1350 (At least 1350 cal of carbo.)
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x' > or = 0, x2 > = 0 (quantites must be non-negative)
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The easiest and most illustrative method of solving problems in two
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unknowns is the graphical method. The value of x' and x'' satisfying 40x'
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+ 60x'' > or = 300 lies in the upper half-plane bounded by the straight
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line 40x' + 60x'' = 300, so the x' and x'' satistying all the above
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inequalities lie in the intersection of their respective half - plane.
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Interger Solutions
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Provided the supplies and demands are positive intergers, the matrix
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minimum method always leads to and optimal solution with integer values as
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the method only involves operations on integers which results in integers.
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Obviously a non-integer optimal solution would be useless.
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Uniqueness
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It can happen that two or more differnet allocations of ships between
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ports give rise to thesame minimum cost. However, if v'=u'<c' for all x'=0
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Degeneracy
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Degeneracy occurs in a transportation problem when a partial sum of
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the supplies equals partial sum of the demans, for example when
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s'+s''''=d''+d'''. Under such circumstances, a basic feasible solution may
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be obtained in which less than m+n-1 of the value x' are positive, which
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results in too few equations to determine the u'=v'. This difficulty can
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be overcome by making the problem non-degenerate.
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