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Application of Linear Programming in Mine Systems Optimization
Article · September 2016
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Abinash Swain
National Institute of Technology Rourkela
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Application of Linear Programming
Problem for Mine System Optimization
Report for:
Mine Systems Engineering (MN 624)
Guided by- Prof. A K Gorai, NIT Rourkela
Submitted by:
Abinash Swain, Roll no- 113MN0495
Date- 28.09.2016
1|Page
CONTENTS
Item no
Name
Page no.
1.0
Introduction
2.0
Areas of Application of LPP in mining Industries 4
2.1
Blending of Ores and Production Scheduling
4
2.2
Transportation Problems
5
2.3
Assignment Problems
7
2.4
Inventory Management and Optimization
9
2.5
Queuing optimization in Opencast Mines
12
2.6
Project Management
13
2.7
Project Crashing using Optimization
techniques
16
3.0
Conclusion
17
2|Page
3
1.0 Introduction:
Linear Programming methods and Operation Research techniques help us in
perhaps all industry oriented work to optimize our system. Application of it in
mining industry is vital. Uncertainty in demand in the market, continuously varying
market conditions is the key problem for which optimization is required. Developed
during the World War II in England, operation research has emerged as a vital tool
now-a-days. It makes us take decisions scientifically giving best ever result for our
profit optimization. Different mathematical operations and iteration techniques
are applied in doing so. So many algorithms are developed in recent years for
application in many fields to optimize our requirements. There are different stages
of linear programming. First is the formulation of the liner programming problem
or model. So, converting field real life data in to mathematical formulations is the
important part. Then we can use several solving methods developed till date to get
the best ever desirable result. By doing so we can eliminate probable loss in the
business by beforehand taking measures. Hence linear programming problems
helps mining industry in optimizing various things and thereby reducing the cost of
production and increasing the profit.
In mining industry the problems that can be solved by linear programming models
are production scheduling according to market fluctuation, production from
different parts of mine with varying quality for optimum blending parameters,
transportation problem to different customers, Assigning various tasks to different
group of people having different capabilities, Optimizing Inventory by calculating
Order quantity, Reorder point etc. One of the major applications is the area of
Project management. In mining it generally takes a lot of time to start production
as it goes through various types of projects interrelated to each one. So estimating
duration of project completion, allocating more resources to complete it in less
time by crashing of project, finding critical path for optimization plays a pivotal role
in optimizing overall profit and minimizing duration of completion keeping an eye
on varying market conditions and resource availability.
Problems are solved using different mathematical models. Algorithms are used for
each type of model to solve it. May it be graphical or analytical modelling, an
optimized point is reached among input parameters. Various Linear programming
soft wares are now-a-days used for efficient planning in mining industries.
3|Page
2.0 Areas of Application of LPP in Mining Industries:







Blending of Ores and Production Scheduling
Transportation Problems
Assignment Problems
Inventory Management and Optimization
Queuing optimization in Opencast mines
Project management
Project Crashing Management and many more.
2.1 Blending of Ores and Production Scheduling:
We know the quality of ore in a single mine varies from place to place. But from a
producer point of view, we have to maintain a certain fixed quality of ore to be
supplied to the customer. We can’t give high quality ore on a day and low quality
on the next day. Hence we apply the principle of Blending. By blending of ores of
different qualities we can produce, every time, an ore of fixed parameters of
ingredients. For this purpose we need production scheduling and optimization in
blending process. What amount of different grades of ores will be mixed to produce
the required quality of ore can be decided by using optimization technique through
Linear Programming applications. As good quality ore gives us more money and low
quality earns us less, hence we have to optimize our quality and hence maximizing
the profit avoiding future losses.
For an example: A mine makes three Products of ore of different quality by
blending ores of different proportions. It has two types of ore called Ore A and Ore
B to get three blend products of Product-1, Product-2 and Product-3. For making
Product-1, it needs to blend Ores A and B in the proportion of 4:6. To make Product2, it needs to blend ores A and B in the proportion of 4:8, similarly for Product-3 it
needs the ratio of 3:8. The selling price of Product 1, 2 and 3 per ton are 30$, 50$
and 35$ respectively. So how can we decide that what product mix/blend gives us
maximum profit so that we can optimize our profit. Given that the maximum
production of Ores A and B can be 800 tons and 2000 tons respectively.
4|Page
Approach using Linear Programming:
Let’s assume that we make products 1, 2 and 3 as X1,X2,X3 tons respectively. So our
Objective function is a cost function Z. It can be written as:
Z= 30X1+50X2+35X3
We have to maximize it to get maximum profit. But we have certain Constraints to
solve the problem. These constraints can be written as:
4X1+4X2+3X3 <= 800
6X1+8X2+8X3 <=2000
This is a simple linear programming problem. We can solve it using Simplex method
to solve it easily and can get the tones of product-1, 2 and 3 giving maximum profit
and hence thereby we can plan our schedule for production of different types of
ores on day to day basis.
2.2 Transportation Problem:
A mine produces ores to be transported to consumers. For example a coal mine
sends its coal produced to different power plants or steel plant to be used as fuel
or coking coal. A single mining company usually does not have a single customer.
So it sends varying quantity of ores to different plants from its different operating
mines. As there is a large spending in transportation of ore is involved in the
process, the mining company has to optimize the quantity of ore to be sent to
different customers from different mines located at different locations and
involving varying transportation cost. We can use Linear Programming technique
to maximize our profit in transportation by optimizing the cost involved.
For an example: A mining companies has three mines named mine-1,mine-2 and
mine-2 respectively located at different places and produce coal of 150t,175t and
275t respectively. Similarly, there are three power plants to where these coals are
to get transported. Demand at those plants named plant-A, plant-B and plant-C are
200t, 100t and 300t respectively. The cost of transportation of coal to these plants
from different mines are given as:
5|Page
So we have to optimize the quantity of coal to be transported from mines to
different plants so as to minimize the cost involved in transportation. Also we have
to make sure that all the produced coal is being transported and all the demand in
the power plants are properly met.
Approach using Linear Programming:
So our objective function is a cost function Z and we have to minimize it. Let’s
assume the coal transported from mine-1 to plant-A is notated by X1A and other
notations are similar.
Objective function is given by:
Minimize Z = 6x1A + 8x1B + 10x1C + 7x2A + 11x2B + 11x2C + 4x3A + 5x3B + 12x3C
But the constraints are given as:
X1A + X1B+ X1C  150
X2A+ X2B + X2C  175
X3A+ X3B + X3C  275
X1A + X2A + X3A = 200
X1B + X2B + X3B = 100
X1C + X2C + X3C = 300, XIJ  0
This problem in linear programming can be solved by constructing Initial tableau
and Optimal Solution tableau. Initial feasible solution can be found out by methods
like:
6|Page
 North-West corner method
 Minimum Cell Cost method
 Vogel’s Approximation method
And after getting initial feasible solution we have to check the solution is optimal
or not. For that we can use Modified Distributed method (MODI). It is a modified
method of Stepping Stone method. It determines a tableau is optimal or not. Also
it tells us which non-basic variable to be chosen first to be replaced.
2.3 Assignment Problems:
In a particular mine, the man power is not uniformly capable to do different types
of works. Someone can do a certain job in less time due to his efficiency and
interest, he may not be able to do another type of work with that amount of
interest and dedication. Hence we have to optimize the distribution of work to the
man-power available so as to minimize the time required to finish the job and
hence indirectly increasing productivity and profitability of the mine. Hence use of
Linear Programming and its algorithms helps immensely to assign jobs to different
groups of man power and hence utilizing Human resources efficiently.
For an Example: A production supervisor is considering how he should assign the
four jobs that are to be performed, to four of the workers. He wants to assign the
jobs to the workers such that the aggregate time to perform the jobs is the least.
Based on previous experience, he has the information on the time taken by the four
workers in performing these jobs, as given in table:
Worker
Job
A
B
C
D
1
45
40
51
67
2
57
42
63
55
3
49
52
48
64
4
41
45
60
55
7|Page
So we can use LPP application to find out which work should be assigned to which
man to minimize the time and maximizing the productivity.
Approach using Linear Programming:
Here Hungarian Assignment model can be implemented. Different Steps are:
Step 1: Locate the smallest cost element in each row of the cost table. Now subtract
this smallest element from each element in that row.
Step 2: In this reduced cost table obtained, consider each column and locate the
smallest element in it. Subtract the smallest value from every other entry in the
column. As a result, there would be at least one zero in each of the rows and
columns of the second reduced cost table.
Step 3: Draw the minimum number of horizontal and vertical lines that are required
to cover all the ‘Zero’. If the number of lines drawn is equal to n (the number of
rows/columns) the solution is optimal, and proceed to step 6.
Step 4: Select the smallest uncovered (by the lines) cost element. Subtract this
element from all uncovered elements including itself and add this element to each
value located at the intersection of any two lines.
Step 5: Repeat steps 3 and 4 until an optimal solution is obtained.
By following this LPP algorithm, we can reach at a solution table given as:
Worker
Job
A
B
C
D
1
5
0
11
14
2
15
0
21
0
3
1
4
0
3
4
0
4
19
1
8|Page
2.4 Inventory Management and Optimization:
Inventory management is a vital part in any mining industry as far as cost and
efficiency are concerned. With fluctuations in the demand and price level in the
market for various reasons, our order, re-order, production level etc. are hampered
to a large extent. If demand is too less, we just can’t keep on producing at a regular
rate, as our inventory or storage cost will increase. Similarly, we can’t keep on
ordering our required goods at that rate if demand deceases. Similarly, if demand
increases abruptly we have to make sure that we do have required stock to meet
the growing demand to get maximum profit. So inventory management and
optimization helps us in different ways:
• demand information is distorted as it moves away from the end-use
customer
• higher safety stock inventories to are stored to compensate
• Seasonal or cyclical demand
• Inventory provides independence from vendors
• Take advantage of price discounts
• Inventory provides independence between stages and avoids work
stoppages
Various inventory costs are given below. We have to minimize the total cost of
inventory by optimizing our Economic Order Quantity (EOQ), Re-order point or
time to re-order. Costs are:
• Carrying cost- cost of holding an item in inventory
• Ordering cost- cost of replenishing inventory
• Shortage cost- temporary or permanent loss of sales when demand cannot
be met.
Benefits of Inventory management are summarized as:
• Hedge against uncertain demand
• Hedge against uncertain supply
9|Page
• Economize on ordering costs
• Smoothing
There is always a trade-off between holding cost and ordering cost. If we order in
large amount, our ordering cost gets lowered, nut of we don’t have that amount of
demand our holding cost increases. Hence, we have to optimize our cost to
minimize it and increase profitability.
But demand is rarely predictable. As our demand varies time to time. Hence we can
use some Safety Stocks in our inventory to minimize loss at peak demand hours.
Also the time lag between ordering and receipt should be optimized. By considering
all these parameters we can use linear programming and models to maximize our
profit. Models can be used are:
 Deterministic model
 Probabilistic model
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For an Example : Assuming a mining company that faces demand for 5,000 te
explosive per year, and that it costs Rs. 15,000 to have the explosive shipped to the
mines. Holding cost is estimated at Rs. 500 per te per year. Estimating the standard
deviation of daily demand to be d = 6. When should we re-order if you want to be
95% sure you don’t run out of explosive?
Approach using Operation Research Technique:
Since the expected yearly demand is 5000 te, the expected demand over the lead
time is 5000(10/365) = 137 te. The z-value corresponding to a service level of 0.95
is 1.65. So
ROP  137  1.65 10(36) 168
Order 548 te explosives when the inventory level drops to 168 te.
2.5 Queuing Optimization in Open-Cast Operations:
Queuing theory helps immensely to optimize the dumper-shovel combination
operation by increasing efficiency in production, decreasing cycle time and
increasing profit indirectly. By using previous data in this technique we can find out
average no. of dumpers being in the queue regularly, cycle time, waiting time etc.
in a probabilistic form. So that we can take remedial steps to increase productivity
by increasing paths, servers or Shovels, dumpers etc. by proper management using
mathematical tools and operation research.
For an Example: Suppose there are arrival of dumpers is 50/hour and there are
certain no. of servers or Shovels in the mine serving 40/hour. We have to decide
optimal no. of servers required to maximize profit and given that waiting time for
dumpers in the system is 50$ per hour and service cost is 30$ per hour.
Approach using Operation Research technique:
Here we can find out the total cost by doing parametric analysis for different server
numbers/ Shovel numbers. And from that the server number giving minimum cost
can be taken in to consideration.
12 | P a g e
We can find cost for waiting of dumpers by multiplying cost/waiting /hour
multiplied by the average waiting time per dumper (Wq *50) and similarly (S *30)
will give the corresponding service cost for each value of ‘S’. Hence by taking
various values of ‘S’, we can analyze the value giving minimum cost.
2.6 Project Management
Use of operation research and LPP is of paramount importance in Project
management in mining industries. As we know mining activity does not commence
at once, but it takes enough time for production after completion of certain
number of project activities. Also each activity possesses certain constraints to get
executed. Hence optimizing project completion time and scheduling is important
as our aim is to start producing as early as possible. Project managers rely on
PERT/CPM to help them answer questions such as:
 What is the total time to complete the project?
 What are the scheduled start and finish dates for each specific activity?
 Which activities are critical and must be completed exactly as scheduled to
keep the project on schedule?
 How long can noncritical activities be delayed before they cause an increase
in the project completion time?
13 | P a g e
For an Example: A mining company needs tasks named A,B,C,D and E to complete
in certain interval as a part of a project. Hence we can find the expected time of
completion of project, critical path, critical tasks etc. to optimize our project work
and minimize the duration. It is given:
Task
Immediate prerequisite
tasks
Effort
A
None
9
B
A
5
C
A
7
D
B,C
11
E
D
8
(Weeks)
Approach using Operation Research Techniques:
14 | P a g e
Blue circle gives the critical tasks that constitute the critical path. Or the path giving
the longest duration of the project to get completed. This is the strategic path as
we can minimize the project duration by changing the duration of critical tasks so
as to maximize our productivity by minimizing the duration of project. The critical
path method analyses the precedence of activities to predict the total project
duration. The method calculates which sequence activities (path) has the least
amount of flexibility.
Different Steps for PERT network analysis:
Step 1: Make a forward pass through the network as follows: For each of these
activities, i, compute:
• Earliest Start (ES) Time = the maximum of all earliest finish times for all its
immediate predecessors. (For node “START”, this is 0.)
• ESi= Maximum (EFj) for all immediate proceeding activities j.
• Earliest Finish (EF) Time = (Earliest Start Time) + (Time to complete activity i).
• EFi= ESi+ ti
Step 2: Make a backwards pass through the network as follows: Move sequentially
backwards from the last node, “FINISH” to its immediate predecessors, etc. At a
given node, j, consider all activities immediately following it and compute:
• Latest Finish (LF) Time = the minimum of the latest start times for all activities
that immediately follow j. (For node “FINISH”, this is the project completion
time.)
• LFj= Minimum (LSi) for all immediate following activities i.
• Latest Start (LS) Time = (Latest Finish Time) - (Time to complete activity j).
• LSj= LFj - tj
Step 3: Calculate the slack time for each activity by:
• Slack = (Latest Start) - (Earliest Start) or (Latest Finish) - (Earliest Finish).
• A critical path is a path of activities, from node “START” to “FINISH”, with 0
slack times.
15 | P a g e
2.7 Project Crashing using Optimization Technique:
As we saw the critical path in the PERT network analysis contains the critical tasks
that contribute to the maximum time for completion of the entire project. Hence
to reduce project duration and increase profitability we have to crash certain
critical tasks by providing more resources in to it. But obviously it will increase the
cost of production. Hence we have to optimize our duration and cost for maximum
productivity and profitability. Project crashing can be done only to certain
allowable durations and limited resources. Hence use of LPP is vital in choosing
crashing amount.
For an example: Below diagram shows the network for finishing a certain project.
It shows the expected time of completing individual tasks along with the maximum
time that a task can be crashed using limited resources and rime.
Approach using LPP technique:
So, can we crash the entire time available to each task? The answer is NO. Because
the path will not remain the critical path, hence our cost will increase for the
project. So we can crash up to that amount so that we will never have loss due to
crashing. Here we can crash all the time available because after that the path
remains as critical path as 4+3 >5. But if the situation is like this shown below:
16 | P a g e
We cannot crash all the weeks available as the path will not remain critical. In this
way we can formulate mathematical models for crashing operations. Various Soft
wares are available for executing so in mining industries. Below diagram shows the
Time-Cost Tradeoff.
3.0 Conclusion:
Hence, use of LPP in mining industries is of paramount importance. It helps
industries to optimize its systems and increase productivity. Moreover, it helps in
reducing cost and increasing profitability by optimization techniques. Use of LPP
also helps to combat unforeseen conditions due to market fluctuations in demand
and supply. Also we can utilize discount options and hence reducing total cost.
Starting from the project management to inventory management and HR
management, it has its larger role.
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