STDM - Linear Programming

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By Isuru Manawadu

B.Sc in Accounting Sp. (USJP), ACA,

AFM 31130AFM 31130

Learning Outcomes

After studying this session you will be able to:

Why linear programmingAssumptions of linear programmingGraphical method of linear programmingCost Minimization Problem Using Linear ProgrammingShadow price calculation

Why Linear Programming?Programming formulations gives solutions for the problems.

The output generated by linear programs provides useful “what-if” information.

Improves quality of decision.

Utilized to analyze numerous economic, social, military and industrial problem.

Linear-Programming Applications

Constrained Optimization problems occur frequently in economics:

maximizing output from a given budget; or minimizing cost of a set of required

outputs.

A number of business problems have inequality constraints.

Profit Maximization Problem Using Linear Programming Constraints of production capacity, time,

money, raw materials, budget, space, and other restrictions on choices. These constraints can be viewed as inequality constraints

A "linear" programming problem assumes a linear objective function, and a series of linear inequality constraints

Linear Programming Assumptions:

1. Constant prices for outputs (as in a perfectly competitive market).

2. There are no interactions between the décision variables.

3. The parameters are know with certainly.

4. Constant returns to scale for production processes.

Linear Programming Assumptions:

6. Typically, each decision variable also has a non-negativity constraint.

For example, the time spent using a machine cannot be negative. The décision variable are continuons.

.

Graphical Analysis – The Feasible Region

The non-negativity constraints

Solution Methods

Linear programming problems can be solved using graphical techniques, SIMPLEX algorithms using matrices, or using software, such as ForeProfit software.

In the graphical technique, each inequality constraint is graphed as an equality constraint. The Feasible Solution Space is the area which satisfies all of the inequality constraints.

Solution Methods Cont….

The Optimal Feasible Solution occurs along the boundary of the Feasible Solution Space, at the extreme points or corner points.

The corner point that maximize the objective function is the Optimal Feasible Solution.

Solution Methods Cont….

There may be several optimal solutions. Examination of the slope of the objective function and the slopes of the constraints is useful in determining which is the optimal corner point.

One or more of the constraints may be slack, which means it is not binding.

GRAPHICAL

X1

X2

A

B

C

CONSTRAINT # 1

CONSTRAINT # 2

Corner PointsA, B, and C

FeasibleRegion OABC

O

Extreme points and optimal solutions

If a linear programming problem has an optimal solution, an extreme point is optimal.

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Multiple optimal solutions

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• For multiple optimal solutions to exist, the objective function must be parallel to one of the constraints •Any weighted average

of optimal solutions is also an optimal solution.

Shadow Prices

Assuming there are no other changes to the input parameters, the change to the objective function value per unit increase to a right hand side of a constraint is called the “Shadow Price”

One more of the constraints may be slack, which means it is not binding.

Each constraint has an implicit price, the shadow price of the constraint. If a constraint is slack, its shadow price is zero.

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1000

500

X2

X1

500

2X1 + 1x

2 <=1000

When more plastic becomes available (the plastic constraint is relaxed), the right hand side of the plastic constraint increases.

Production timeconstraint

Maximum profit = 4360

2X1 + 1x

2 <=1001 Maximum profit = 4363.4

Shadow price = 4363.40 – 4360.00 = 3.40

Shadow Price – graphical demonstrationThe Plastic constraint

Complexity and the Method of SolutionThe solutions to primal and dual

problems may be solved graphically, so long as this involves two dimensions.

With many products, the solution involves the SIMPLEX algorithm, or software available in FOREPROFIT

Cost Minimization Problem Using Linear ProgrammingMulti-plant firms want to produce with the

lowest cost across their disparate facilities. Sometimes, the relative efficiencies of the different plants can be exploited to reduce costs.

A firm may have two mines that produces different qualities of ore. The firm has output requirements in each ore quality.

Cost Minimization Problem Using Linear ProgrammingScheduling of hours per week in each mine

has the objective of minimizing cost, but achieving the required outputs

Thank you