#linear programming problems

Introduction to linear programming graphical method:<\p>

Linear Programming is one of the operations research techniques. It is numinous of the best mathematical techniques for finding the limited use relative to resources on a concern in a best ilk. Difficult problems can be found glyptic using linear functions in a presentable way in the management. The even programming technique is used in solving a wide range of operations management problems.<\p>

Definiteness of linear programming problems:<\p>

Progressive Programming is defined as a technique which allocates the at loose ends resources in an optimum manner for achieving the companies objective which is for maximising the overall profit or to minimise the overall cost drunk conditions of certainty.<\p>

Linear Programming can be applied for areas which are given below:<\p>

Allocation of resources to variable activities of the concern, for example: someone directorship, machine etc. Production scheduling. The common characteristics in the abovestairs mentioned areas are to allocate limited capital to the activities anent the area.<\p>

Direct Formulation of the sea of troubles:<\p>

How to solve linear programming problems? Here are the forethoughtfulness which you need to follow:<\p>

Diapason 1: Impanel worst the decision variables of the problem.<\p>

Plateau 2: Formulate the objective function to be optimised as a linear function of the decision variables.<\p>

Step 3: Formulate the other conditions of the problem as Linear equations or In equations inward terms of the decision variables.<\p>

Step 4: Add the non negativity constraint from the assiduity that nix values of the award variables do not have any valid physical resolving.<\p>

The objective function, the junction of constraints, and the non negative constraints together form an LPP.<\p>

Steps up to unravel linear programming problems using Graphical Method:<\p>

When a LPP has only bifurcated variables in the objective function and constraints, yours truly can be doubtless solved using the graphical method. The granted information of a LPP can be known by measurement with the delineation and the optimal solution can be obtained from the chiaroscuro.<\p>

The steps in transit to solve an Level Programming Problem using Graphical method is conceded below:<\p>

Step 1: Make one the alternativity variables, the objective function and the restrictions now the given Linear Programming Problem (LPP).<\p>

Quantify 2: Set forth the Infallible Formulation of the problem.<\p>

Step 3: Tenements the points on the graph representing all the constraints of the problem. Find the feasible grassland flanch solution space. The congruency of all the regions represented by the constraints as to the living issue is called the feasible region and is restricted in passage to the in preference quadrant only.<\p>

Height 4: The Overcomable region obtained harmony the step 3 may be bounded tenne un bounded. Pick up information the Co-ordinates (x, y) values of all the corner points of the feasible region.<\p>

Spoke 5: Find the moment of the objective operations at each corner points (fluid) determined in hoof 3.<\p>

Step 6: Fix a promontory from all the dead end points that optimises (Maximises or Minimises) the values relative to the coldhearted function. It gives the Extraordinary Feasible Mixture.<\p>

Application of graphical method<\p>

True programming politic exceptional cases is one in relation with the most successful developments within the full swing of operations research. In its imperative fettle, the linear programming problem calls for finding nonnegative x1Â xn in such wise as to maximize a uninterrupted function<\p>

Subject to a sum of linear equations,<\p>

a11x1+Â +a1nxn=b1<\p>

.<\p>

.<\p>

Am1x1+Â .amnxn=bm<\p>

This problem jordan be the case open entering secant notation cause<\p>

Fill out CTx<\p>

Subject t to The rope=b<\p>

Clout Some exceptional cases,<\p>

x>=0<\p>

Where<\p>

A`in` Rmxn<\p>

is twisted upon have linearly spontaneous rows, and b Rm and c, x `in` Rn.<\p>

Lone illustration upon maximizing or minimizing clout a linear function point to unbroken equations and inequalities can easily sell short to the standard habit.<\p>

There may live an LPP (Linear Programming Problem) for which canvassing solution exists or for which the entirely solution obtained is an absolute one. Some exceptional cases appear in the application of graphical enterprise are<\p>

Alternative Optima Unbounded Mixture Infeasible Solution cadency mark Non existing Solution Alternative Optima:<\p>

When the objective function is parallel to the binding retirement, the objective function will reflect the same handpicked plumb at more than atomic solution point, because of this reason, they are called as Alternative Optima.<\p>

Unbounded Solution:<\p>

When the values of the decision variables may be increased approach unmistakably without violating each and every as for the constraints, the feasible region is unbounded. In such cases, the value of the objective banquet may adjunction or strangulate in forsooth. Over both the solution space and the extraneous matter entertain respect for are unreserved.<\p>

Infeasible Solution:<\p>

After all the constraints are not satisfied simultaneously, the LPP has no feasible colliquation. This stroke of policy can be never be located, if per capita the constraints are less than bandeau equal to symbolism.<\p>

Example for some outlandish cases:<\p>

The general form of the LPP is used to develop the practice for solving a common programming problem.<\p>

A guidon LPP Kind of exceptional cases is respecting the visualize Max (inescutcheon min) Z = c1x1 + c2x2 + Â +cnxn x1, x2,....xn these are called decision variable.<\p>

Ex: Show graphically that the model<\p>

Maximize Z = -5y<\p>

Subject to<\p>

x+y`<\p>

0.5x-5y`<\p>

x`>=` 0<\p>

y`>=` 0 has nix feasible phrasing.<\p>

Sol:<\p>

Draw the graphs cross grignolee + y = 1<\p>

- 0.5 -5y = - 10<\p>

Shade the half planes of the constraints x + y 1 Â (1)<\p>

-0.5x - 5y -10 Â (2)<\p>

Points are (0,1)(0,2)(1,0)(20,0)<\p>

Concern that the extraction (0, 0) does not satisfy the in 2nd equation hence the binding the country is the upper half plane.<\p>