MD. SHAHRUKH ANJUM V-SEM B –TECH (MINING ENGG.)2013JE0552

SOLVING MINING RELATED PROBLEMS USING

OPERATIONS RESEARCH (LINEAR PROGRAMMING)

Guided by Prof. M JAWED

Linear Programming

The word “linear” means the relationship which can be represented by a straight line i.e. the relation of the form ax +by=c. In other words it is used to describe the relationship between two or more variables which are proportional to each other The word “programming” is concerned with the optimal allocation of limited resources.Linear programming is a way to handle certain types of optimization problemsLinear programming is a mathematical method for determining a way to achieve the best outcome

Definition of LP

Linear Programming is a mathematical modeling technique useful for the allocation of “scarce or limited’’ resources such as labor, material, machine ,time, warehouse space ,etc…,to several competing activities such as product ,service ,job, new equipments, projects, etc...on the basis of a given criteria of optimality

Ques:- CAN EVERY PROBLEM BE DEALT WITH LINEAR

PROGRAMMING OR IT HAS ITS OWN PRE REQUISITES ?

Ans:- NO. LINEAR PROGRAMMING HAS ITS OWN PRE -REQUISITES

Requirements

There must be well defined objective function.

There must be a constraint on the resources provided

There must be alternative course of action.

The decision variables should be interrelated and non negative.

The resource must be limited in supply.

Application Of linear Programming

Business

Industrial

Military

Economic

Marketing

Distribution

Areas of application of Linear programming

o Industrial ApplicationProduct Mix ProblemBlending ProblemsProduction Scheduling ProblemAssembly Line BalancingMake-Or-Buy Problems

o Management ApplicationsMedia Selection ProblemsPortfolio Selection ProblemsProfit Planning ProblemsTransportation Problems

o Miscellaneous Applications◦ Diet Problems◦ Agriculture Problems◦ Flight Scheduling Problems

Advantages of L.P.

It helps in attaining optimum use of productive factors.

It improves the quality of the decisions.

It provides better tools for meeting the changing conditions.

It highlights the bottleneck in the production process.

Limitation of L.P.

For large problems the computational difficulties are enormous.

It may yield fractional value answers to decision variables.

It is applicable to only static situation.

LP deals with the problems with single objective.

Types of Solutions to L.P. Problem

Graphical Method Simplex Method Softwares like LINDO

LINEAR PROGRAMMING MODEL

A general representation of LP model is given as follows:Maximize or Minimize, Z = p1 x1 + p2 x2 ………………pn xn

Subject to constraints,w11 x1 + w12 x2 + ………………w1n xn ≤ or = or ≥ w1 ……………

(i)w21 x1 + w22 x2 ………………w2n xn ≤ or = or ≥ w2 …………… (ii). . . .. . . .. . . .wm1 x1 + wm2 x2 +………………wmn xn ≤ or = ≥ wm …………(m)

Non-negativity constraint,xi ≥ o (where i = 1,2,3 …..n)

A Blending ExampleEastern Steel: A Blending Example. Iron ore from 4 different mines is blended to make a metal alloy. In order to produce a blend with suitable tensile qualities, the following minimum requirements must be met on three basic elements (A, B, and C):

MIN. REQUIREMENT PER TON OF BLEND(pounds of each element) BASIC ELEMENT

A 5B 100C 30

Find the amount of ore to be borrowed from each mine to make the blend.

A Blending ExampleThe amount of these elements in the ores from the four different mines is given below:

MINE(pounds per ton of each element)

BASIC ELEMENTA 10 3 8 2B 90 150 75 175C 45 25 20 37

1 2 3 4

A Blending ExampleWe must now take into consideration the cost of ore from each mine:

DOLLAR COST PER TON OF ORE MINE1 8002 4003 6004 500

A Blending ExampleEastern Steel’s objective is to come up with a least-cost feasible blend of the ores. First, specify the decision variables:

T1 = fraction of a ton to be chosen from mine 1 T2 = fraction of a ton to be chosen from mine 2 T3 = fraction of a ton to be chosen from mine 3 T4 = fraction of a ton to be chosen from mine 4

A Blending ExampleNow, specify the amounts of the basic elements in 1 ton of blend:

A = 10T1 + 3T2 + 8T3 + 2T4

Pounds of Element in 1 Ton of Blend

B = 90T1 + 150T2 + 75T3 + 175T4

C = 45T1 + 25T2 + 20T3 + 37T4

Finally, create the constraints based on this information: 10T1 + 3T2 + 8T3 + 2T4 > 5

90T1 + 150T2 + 75T3 + 175T4 > 100

45T1 + 25T2 + 20T3 + 37T4 > 30

T1 , T2 , T3 ,T4 > 0

T1 + T2 + T3 + T4 = 1

A Blending ExampleThere are two more constraints to consider:

Nonnegativity Constraint:

Equality Constraint (also called a material balance condition) restricts the values of the decision variables in such a way that the left-hand side exactly equals the right-hand side.

Note that the constraints in a linear programming model can be equalities as well as inequalities.

A Blending Example

The objective function is to minimize cost of any blend and is given by:

Cost of 1 ton of blend = 800T1 + 400T2 + 600T3 + 500T4Now let’s put it all together and write the complete

symbolic model.

Symbolic LP ModelMin 800T1 + 400T2 + 600T3 + 500T4

Subject to (s.t.)10T1 + 3T2 + 8T3 + 2T4 > 590T1 + 150T2 + 75T3 + 175T4 > 100

45T1 + 25T2 + 20T3 + 37T4 > 30

T1 + T2 + T3 + T4 = 1

T1 , T2 , T3 , T4 > 0

LINDO RESULT

OBJECTIVE FUNCTION VALUE :- 511.1111

VARIABLE VALUE REDUCED COST

T1 0.259259 0.000000 T2 0.703704 0.000000 T3 0.037037 0.000000 T4 0.000000 91.111115

CONSTRAINT SLACK DUAL PRICES

1 0.000000 44.444443 2 31.666666 0.000000 3 0.000000 4.444445 4 0.000000 155.555557

I. LP problemVerbal description

Two Mines Company

The Two Mines Company own two different facilities that produce an ore which, after being crushed, is graded into three classes: high, medium and low-grade. The company has contracted to provide a smelting plant with 12 tons of high-grade, 8 tons of medium-grade and 24 tons of low-grade ore per week. The two mines have different operating characteristics as detailed below.

Mine Cost per day ($) Production (tons/day) High Medium LowX 180 6 3 4Y 160 1 1 6

How many days per week should each mine be operated to fulfill the smelting plant contract?

I. Solution Symbolic Formulation

Decision Variables These represent the "decisions that have to be made" or the "unknowns".

x = number of days per week mine X is operated y = number of days per week mine Y is operated

Note here that x >= 0 and y >= 0.

Constraints Ore production constraints - balance the amount produced with the quantity required under the smelting plant contract

OreHigh 6x + y >= 12Medium 3x + y >= 8Low 4x + 6y >= 24

Days per week constraint - we cannot work more than a certain maximum number of days a week e.g. for a 5 day week we have x <= 5 y <= 5

I. Solution Symbolic Formulation

Objective

Again in words our objective is to minimize cost which is given by

Z = 180x + 160y

Hence we have the complete mathematical representation of the problem as:

Minimize Z= 180x + 160ySubject to

6x + y >= 12 3x + y >= 8 4x + 6y >= 24 x <= 5 y <= 5 x,y >= 0

GRAPHICAL REPRESENTATION

Optmal solution point ( (1.7,2.85))

afsib

Feasible region

Objective function

LINDO RESULTS

LINDO RESULT

LINDO RESULTS

OBJECTIVE FUNCTION VALUE :- 765.7143

VARIABLE VALUE REDUCED COST X 1.714286 0.000000 Y 2.857143 0.000000

CONSTRAINT SLACK DUAL PRICES

1 1.142857 0.000000 2 0.000000 -31.428572 3 0.000000 -21.428572 4 3.285714 0.000000 5 2.142857 0.000000

FUTURE PROSPECT

• Going for a mine visit

• To collect data of its manpower, haulage capacities, resource deployed or contract constraints among various activities (ore production) competing for those resources

• to achieve the most desirable output using LP

• interpreting the result using linear programming identifying the bottleneck associated to optimise the profit and resource deployment.

 

Thank you For your patience