A theoretical account is a representation of world. It can be thought of as an entity, which captures the kernel of but without the presence of world. A exposure is a theoretical account of world portrayed in the image. A mathematical equation may be used to pattern the energy contained in a given stuff. Thus, a theoretical account captures some facet of the world it attempts to stand for. A theoretical account that takes on the physical visual aspect of the object it is to stand for is called a physical theoretical account. This type of theoretical account is used to expose or prove the design of new merchandises.
Business activities are dynamic procedures that follow mathematical forms. Therefore, be represented by symbolic theoretical accounts. Symbolic theoretical accounts are algebraic, numerical and logical. Analytic theoretical accounts are mathematical theoretical accounts aimed at some simplification and abstraction of existent systems in order to supply some penetrations, and to understand some interested facets of world. Business activities are modeled to help the determination devising procedure.
Compared with mechanical theoretical accounts, mathematical theoretical accounts facilitate experimentation because all dependent variables, independent variables, invariables, and parametric quantities are explicitly related through the linguistic communication of mathematics. The determination shaper can prove the effects of different determination options, invariables, and parametric quantity values on the dependent variables much more easy than with any other type of theoretical account. Furthermore, mathematical theoretical accounts can stand for many complex jobs expeditiously and briefly and, in many instances, provide the cheapest manner to analyse these jobs.
All decision-making theoretical accounts can be classified as either deterministic theoretical accounts or probabilistic theoretical accounts. This depends mostly on how influential the unmanageable factors are in finding the result of a determination. Unlike deterministic theoretical accounts where good determination is judged by its result entirely, in probabilistic theoretical accounts the determination shaper is concerned with both the result value and the sum of hazard each determination carries.
A mathematical theoretical account is an equation, inequality, or system of equations or inequalities, which represents certain facets of the physical system modeled.
The mathematical look, which describes the behaviour of the step of effectivity, is called the nonsubjective map. Objective maps are written in mathematical look incorporating variables, the value of which is to be determined. If the nonsubjective map is to depict the behaviour of the step of effectivity, it must capture the relationship between that step and those variables that cause it to change. Therefore in the nonsubjective map the determination shaper needs to reply the inquiry, “ what values should these variables have so that the mathematical look has the greatest possible value ( maximization ) or the least possible numerical value ( minimisation ) . System variables are either determination variables or parametric quantities. Decision variables are the variables whose values are under the control of the determination shaper and influence the public presentation of the system. Parameters are variables whose values might be unsure for the determination shaper. This calls for sensitiveness analysis after happening the best scheme.
Optimization, besides called mathematical scheduling, in general, trades with the job of finding optimum allotment of limited resources to run into a given aim. The nonsubjective map must stand for the end of the determination shaper. The resources may match to, for illustration, people, stuffs, money, or land. Out of allowable allotments of the resources, it is desired to happen the one or 1s that maximize or minimise some numerical measure such as net income or cost.
Linear scheduling ( LP ) deals with a category of programming jobs which both the nonsubjective map to be optimized is additive and all dealingss among the variables correspond to resources, known as restraints, are additive.
In an LP theoretical account the nonsubjective map must be additive. That means all variables have the power of 1, and they are added or subtracted, non divided or multiplied. The nonsubjective map represents the end of the determination shaper, must be either maximization or minimisation. The restraints must besides be additive. Furthermore, the restraints must be closed, that is, they are expressed in the signifier of a system of equations or inequalities. More specifically they either have marks & lt ; = , & gt ; = or = .
Formulation of an LP theoretical account can be boring and troublesome undertaking. A incorrect theoretical account can ensue because a incorrect set of variables is included or some improper relationships among the variables are constructed. There are some guidelines in an effectual theoretical account preparation. Any LP consists of four parts: a set of determination variables, the parametric quantities, the nonsubjective map, and a set of restraints.
A minimisation job of an LP written in the matrix signifier is:
Minimize Z ( X ) = CX =
Capable to AX = B
Where A is an manganese matrix that represent rows of coefficients of the constraints1 to m each holding n coefficients. The variables are the column vector of determination variables. The C is the row vector or a ( 1n ) matrix of coefficients of the nonsubjective map and B are the parametric quantities of the restraints, which is a ( n1 ) matrix or a column vector.
A executable solution for this job is a numerical vector, X that satisfies all the restraints and mark limitations. An optimal executable solution ( or an optimal solution ) is a executable solution that minimizes the nonsubjective map, Z ( X ) among all executable solutions. Murthy ( 1983 ) has proved that if the above LP has a executable solution, it has an optimal executable solution if and merely if X ( y ) 0 for every homogenous solution Y matching to that LP. Bernard Kolman ( 1993 ) proved that a homogenous systems of thousand equations in n terra incognitas ever has a nontrivial solution if m & lt ; n, that is, if the figure of terra incognitas exceed the figure of equations.
The Proposed Models
The transit of some oil related merchandises that originate at a figure of refineries to be sent to their appropriate finishs is to be modeled. The theoretical account depicts the cargos of the merchandises from the beginnings to their finishs utilizing trucks vehicles of fixed, known capacity. Give a refinery, to where and how much a specific merchandise ( Benzene, kerosine or Diesel ) is to be sent, and the entire sum sent must be all it has produced in the given twelvemonth. The restriction for the receiving terminal is it can non take more than what it is capable of processing. The theoretical account is to happen out how much Benzene, kerosine or Diesel from specific refinery to be sent to the several finishs so that the entire transit distance is minimized.
Figure 3.1 Transportation web between Refineries and Depots
3.2.1 Commodities to be transported
A set of refineries located at different countries where petroleum oil was found, each bring forthing known sums of petroleum oil ( CO ) per twelvemonth. The CO that is produced must be transported to another topographic point called the terminals, where it is saved in large armored combat vehicles and send subsequently to the clients. These terminals have specified and fixed capacities.
3.2.2 Vehicles Used
For the CO transit, truck oilers are used. They come in assorted capacities, 36, 30 and 27 metric tons. Large bulk of oilers are the 36 metric tons, hence lone oilers of this capacity will be considered. Furthermore, heavier tonss cut down figure of trips, and therefore cut down emanation of pollutants into the environment.
3.3 Data Collection
3.3.1 CO Production
A set of two refineries in thecenter of Iraq is selected. AL-Dura Refinery in the capital metropolis for the province of Iraq sends oil production to some of terminals ( Resafa Depot, Meshahda Depot, Latefia Depot and Kut Depot ) . Beji Refinery in the South of Iraq sends oil production to ( Khanqeen Depot, Ramadi Depot and Baquba Depot ) . Information refering oil production for these refineries for the twelvemonth 2006 was gathered. Phone calls and installation visits were made to acquire moderately good estimations. This was done during the twelvemonth 2010.
3.3.2 Refinery Oil Productions Capacity
All the 7th terminals gave their existent productions capacity approved by the oil selling company ( SOMO ) . Since their combined capacity is less than the entire oil production for all the two selected refineries ( AL-Dura Refinery and Beji Refinery ) , another terminal was selected. The nearest terminal was at AL-Anbaar ( Falahat Depot ) . Now, the combined capacity of the eightth terminals exceeded oil production for the two refineries.
3.3.3 Origin-Destination Distance Appraisal
The beginnings were the Millss and the finishs were the refineries, crushers and the paper and mush mill. These distances were largely existent kilometres by tracking to all the installations in their several locations.
Table 3.1 Origin/destination distance matrix in kilometres
I / J
Perai33 14 Selama
12 B Serai 48
30 K Kangsar
80 92 50
Figure 3.2 Distance between towns where
palm oil processing installations
All oil produced at the refineries must be sent out to their several finishs.
All vehicles are stationed at the refineries, unlimited in figure and travel full-load.
Refineries and depos at the same location use the same distance to or from their finishs or beginnings.
All distances are one manner. Direct town-to-town travel is ever preferred above en path via another town.
Distance between installations within the same town location is 0.
3.5 The Models
The transit theoretical accounts that will be proposed coming sub-sections will be based on a basic theoretical account written by Winston ( 2004 ) . There are a set of thousand supply points from which a good is shipped and there are a set of n demand points to which good is shipped. Each unit produced at supply point I and shipped to demand point J incurs a variable cost of. The figure of units shipped from supply point I to demand point j peers. Therefore, giving the undermentioned transit theoretical account:
Minimize ( 3.1 )
Capable to ( Supply restraints ) ( 3.1.1 )
( Demand restraints ) ( 3.1.2 )
( 3.1.3 )
The nonsubjective map minimizes the cost of transit by summing up all merchandises of cost per unit with the figure of units transported for each origin-destination ( i-j ) brace. The supply restraints province a status that for every supply point, whatever is sent out to all finishs must non be more than the available supply sum. Similarly for the demand restraints, the entire supplies sent from all beginnings to a peculiar demand point must non be more than what is demanded by that finish. The last set of restraints is the non-negativity status. In the supply and demand restraints it is noticed as a basic regulation that supply can non be more than what is available, and satisfy demand up to what is really demanded.
Specific to the theoretical accounts that are traveling to be constructed the followerss are defined.
J finishs where oil production range
p merchandise type
The Decision Variable
is the integer figure of trips taken to transport a merchandise from beginning I to finish J.
– Distance between refinery I and finish J
– Processing capacity of merchandise P at finish J
– Sum M3 supply for merchandise P at refinery I
– M3 capacity of vehicle transporting merchandise P
In a conventional transit job, a homogenous merchandise is to be transported from several beginnings to several finishs in such a manner that the entire transit cost is minimal. Suppose there are thousand supply nodes and n demand nodes. The ith supply node can supply Sunits of a certain merchandise and the jth demand node has a demand for D unit.
S Demand nodes
Figure 3.3 Origin-destination
In the oil industry there is a set of thousand refineries each providing S M3 of oil per twenty-four hours to another set of n terminals, each with treating capacity of Dm3. More by and large there is a set of thousand refineries each providing M3 of merchandise P and send to eight terminals each with treating capacities of M3 of merchandise P.
The transit of merchandises from the ith supply node to the jth demand node carries a cost of C per unit of merchandise transported. The job is to find a executable manner of transporting all the available sums without go againsting the demand or the capacity restraints of the receiving node that minimize entire transit cost.
The job is to find a executable manner of transporting the available merchandises to their several finishs at a entire minimal draw distance.
Transportation theoretical account can be simplified and much more easy comprehended by looking at the transit job of one merchandise foremost, specifically the CO. As depicted earlier there are the refineries as the supply beginnings and the terminals as the finishs where oil will be delivered. The refineries have specific one-year oil production and the terminals have stipulated oil processing capablenesss. The job is how to administer the oil from all the refineries to their nearest terminals so that the entire transit is minimized. In kernel the theoretical account is to happen the best refinery-depot assignment so that entire distance is minimized.
Let X be the figure of vehicle trips to transport oil productions from refinery I to depot J through a distance of C. Thus theoretical accounts can be written as the followers.
3.5.1 Model 1 ( oil Transportation Model )
Minimize Z = ( 3.2 )
Capable to V= , I = 1, aˆ¦ , m ( supply restraints ) , ( 3.2.1 )
V, J = 1, aˆ¦ , N ( demand restraints ) , ( 3.2.2 )
and whole number
where X figure of vehicle trips from I to J,
distance between I and J,
treating capacity of terminal J,
supply at refinery I,
V capacity of the oiler.
The nonsubjective map minimizes the entire transit distance in presenting oil from the refineries to the terminals. The dual summing ups ( denoted by the two sigmas, one after the other ) indicate that the two variables are multiplied before their merchandises are added up. The figure of trips taken to present the oil between refinery I and terminal J ( this is the determination variable Ten, which are 20×3=60 in figure ) is multiplied by the distance between them gives the entire transit distance for the particular I and J. The supply restraints consist of m equalities, each for a peculiar refinery. For each refinery, the figure of trips that go out from that refinery to the terminals multiplied by the size of the oiler must be the entire oil production of that refinery. Equal mark for these restraints besides indicates that all the oil from the refineries must be sent out. On the other manus, the demand restraints which are wholly n in figure, the sum of oil received by the specific refineries can non be more than their processing capacities. The last restraints are the non-negative limitation on the determination variables and the figure of trips must be integer Numberss.
The Solution Approach
What the theoretical accounts are supposed to present is happening the optimum refinery – terminal assignment.
The attack in happening reply to the above is to run the plan that we programming utilizing I-Log package.
RESULTS AND DISCUSSIONS
This subdivision presents the end product when the whole number programming theoretical accounts were run on the computing machine. Data needed as the input for the scheduling tallies are the oil production for the two refineries and the capacities of the terminals. The end product for the oil transit job is presented utilizing the original locations of the refineries and terminals. The consequences show the optimum refineries -to- depots assignments ; that is which refinery will direct its oil to which terminal and by how much, so that entire transit distance is minimized.
Input signal Parameters
The trade goods to be transported is the chief concern, the chief merchandise is the oil, considered as the waste merchandise. Below, Table 4.1 depicts the annual three-dimensional metre of the trade goods for the two refineries.
Table 4.1 Oil production from refineries ( M3 /year )
Oil Output ( M3 /year )
Table 4.2 Processing capacities of the terminals
( M3 per twelvemonth )
4.3 Refineries – Depots Assignment
Given the supply of oil from Table 4.1 and the processing capacities of the terminals from Table 4.2, the aim now is to happen the optimum refineries to depots assignment so that entire transit distance is minimized.
Table 4.5 Number of trips, distance and three-dimensional metre from two refineries to terminals in
Baghdad and Beaiji
Center of Baghdad terminals
No. of Distance Tonnage
trips X ( kilometres )
No. of Distance Tonnage
trips X ( kilometres )
723 18,075 26,028
1334 46,690 48,024
750 27,750 27,000
917 60,522 33,012
N. Tebal 1
750 24,750 27,000
N. Tebal 2
778 25,674 28,002
N. Tebal 3
556 18,348 20,016
K. Kurau 1
667 40,687 24,012
K. Kurau 2
1389 84,729 50,004
K. Kurau 3
1584 96,624 57,024
B. Serai 1
1223 66,042 44,028
B. Serai 2
889 48,006 32,004
639 53,676 23,004
1250 105,000 45,000
778 65,352 28,008
639 58,788 23,004
612 48,960 22,032
1639 131,120 59,004
1389 111,120 50,004
1000 80,000 36,000
14,227 781,925 512,172
5,279 429,988 190,044
Entire capacity 580,000 500,000
Entire supply 702,216 metric tons
The entire transit distance is 1,211,913 kilometres.
The ILOG end product of the whole number programming for this theoretical account is shown in the columns labeled as ‘No. of trips X ‘ in the Table 4.5 above. These are really the values of the determination variables, X which represents the figure of trips taken by the trucks to transport all the available oil from each refinery as the beginning to the terminals as its finishs so that entire transit distance is minimized. The above consequences show that to transport 702,216 M3 of oil from the two refineries to the eight terminals the lower limit possible transit distance is 1,211,913 kilometres, which is known in additive scheduling as the Z value. The minimal one-year truck trips needed to make the transit of the monolithic trade good are 19,506. This value comes by adding 14,227, which is the entire CPO trips to the Perai refinery, with 5,279 oiler trips for the Lumut refinery.
It is clearly seen in the consequences shown by the tabular array above that the Millss located above in the north send their CPO to the Perai refineries and a few located farther south have their CPO sent to the refinery at Lumut. The 15 ‘north ‘ Millss at Padang Serai, Kulim, Serdang Selama, Nibong Tebal, Kuala Kurau, Bagan Serai and Taiping are assigned to the Perai refineries. And the staying 5 ‘south ‘ Millss at Kuala Kangsar and Terong are assigned to the Lumut refinery. Entire CPO bringings made by the ‘north ‘ Millss are 512,172 metric tons and as the receiving installations, the Perai refineries are 67,828 metric tons under capacity. The other 5 ‘south ‘ Millss in Kuala Kangsar and Terong are assigned to the ‘south ‘ refinery in Lumut with bringings numbering 190,044 metric tons.
A sum of 512,172 metric tons of CPO are received by the two refineries in Perai, ( note: these two refineries are treated as one in this first theoretical account ) although their combined capacity is 580,000 metric tons, about 70,000 metric tons under-capacity. Simple account here is Millss at Kuala Kangsar and Terong are close to Lumut than to Perai, so alternatively of directing CPO from those Millss to Perai merely to fulfill capacity demands, they are better sent to Lumut since the aim is to minimise the transit distance.
In footings of entire trips that go to Perai refineries, there are wholly 14,227 trips and to Lumut installation are 5,279 trips. At 250 working yearss a twelvemonth, a day-to-day norm of 28 trucks will line up at each of the refineries in Perai. A expression at the first factory refinery brace ( Padang Serai – Perai ) shows that it takes 723 trips to hale 26,028 metric tons of CPO in a span of a twelvemonth. 3 trucks are expected to go forth the factory per twenty-four hours, non a busy state of affairs for this little Padang Serai installation. As a comparing, the Kuala Kurau 3 factory with 1584 laden oiler trucks go forthing this premiss for its finish in a twelvemonth could be considered busier since on a day-to-day footing it is seen more than 6 trucks traveling out.
Looking at distance, to transport 512,172 metric tons of CPO to the Perai refineries distance recorded was 781,925 kilometres, whereas for 190,044 metric tons distance to Lumut was merely 429,988 kilometres. It is noticed here that although sum of CPO to Perai is 2.7 times more than that to Lumut but distance is merely 1.8 times more, the ground being bunch of Millss around Perai are closer to their refineries than those Millss around Lumut, which are further dispersed out from their assigned refineries. Detecting transit distance to Perai refineries, Taiping 2 has the highest at 105,000 kilometres, this is due to Taiping being furthest from its finish than other Millss and figure of trips is among the highest. Two Millss assigned to Lumut refinery have the highest transit distance compared to any other mill-refinery assignments because of both distance per trip consideration every bit good as CPO handiness consideration. Comparing two mill-refinery assignments it is observed that Padang Serai factory recorded a laden distance traveled of 18,075 kilometres versus that of Kuala Kurau 3 with 96,624 kilometres. This is due to Kuala Kurau being more than double farther off from Perai than Padang Serai and CPO tonnage about 3 times more.
It is noticed that entire processing capacity of the 3 refineries in the two locations exceeds entire CPO supplies from the 20 Millss by about 400,000 metric tons. Therefore, it is just to anticipate in the consequence that none of the refineries work at its full capacity, although it was informed by the directions that their installations are working at full capacity. This leads to the decision that when the regulating governments assign capacities to these finishs, distance traversed from their beginnings is ne’er of premier consideration. About 70,000 metric tons of CPO will go on to be sent to refineries in Perai, alternatively of the closer one in Lumut. Assuming the CPO in consideration comes from the factory in Terong, which is closer to Perai than the 1 in Kuala Kangsar, the entire ‘inefficient ‘ travelling is more than 180,000 kilometres. Distance saved if this trade good were to be transported to the better finish is at least 27,000 kilometres.
One manner to salvage the inefficient travelling is by increasing the CPO production in the 15 Millss that have been assigned to the Perai refineries by the proposed theoretical account, if they have non reached maximal production capacity. If they are all working at full capacity it makes sense to O.K. a hereafter factory, at the propinquity of the Perai country, with production capacity of 70,000 metric tons of CPO. At an oil extraction rate of 21 % ( minimal criterion required by MPOB is 18 % ) this factory needs more than 300,000 metric tons FFB per twelvemonth. If the hereafter factory will be in the province of Kedah which has FFB output of 18.49 metric tons per hectare, new estates that must be opened should cover countries numbering 16,000 hectares.
To see the consequence on entire draw distance, if the refineries at Perai are to work at full capacity, this can easy be done by altering the & lt ; = mark with & gt ; = mark on the Perai demand restraint. When the alteration was made the undermentioned consequences were obtained ; 246 of the CPO trips from Terong 1 now go to Perai and 366 go to Lumut ( antecedently all 612 spells to Lumut ) , and all the 1639 trips that antecedently go to Lumut, now go to Perai. The entire standard CPO in Perai is now 580,032 metric tons ; a small above its maximal capacity of 580,000 metric tons. The entire distance traveled increased to 1,238,303 kilometres, some 26,390 kilometres or 2 % excess.
If the smaller refinery at Perai were to be closed, entire transit distance now amounts to 1,432,600 kilometres, an excess 220,687 kilometres of transit distance. On the other manus if the bigger one is closed, a larger addition in entire transit distance is observed, now it becomes 1,593,300 kilometres. Closing one of the refineries at Perai does non impact the processing capablenesss of the staying two ( supply is still less than entire capacities ) , but affects the transit distance. More CPO has to be sent to the farther refinery on Lumut when the one in Perai has reached its upper limit operating capacity.
Now, say if the Lumut installation works full capacity ( Perai refineries is non closed wholly but taking the excess CPO from Lumut ) , the nonsubjective map value now is 1,692,600 ; non a really outstanding addition in entire draw distance. Had the smaller refinery in Perai been in Lumut and they both ( in Lumut ) work full capacity it can be seen a really significant addition in entire transit distance, which now becomes 2,248,200 kilometres. Therefore, it is seen if for some ground merely refineries in Lumut is forced to run at full capacity, and merely the excess CPO from there is allowed to be processed in Perai, the entire distance is about dual compared to the installations as they are now.
Another scenario worth looking at is all the refineries stay together at Perai or at Lumut. If they all operate in Perai the entire transit distance is 1,555,200 kilometres, non a really big addition compared to the reply for the original apparatus. But if they are all in Lumut so a really big aggregation of transit distance of 2,981,500 kilometres is observed. This is about a worst-case scenario where merely one port is allowed to run the refinement activities of the CPO. If merely one is allowed to run, so allow it be Perai and non Lumut because the transit distance will be more than double compared to where the installations are presently located.
At optimality it is found the best refineries -to- depots assignments. This survey found the minimal possible transit distance. Two refineries that form a bunch, from up North at Baghdad, Iraq and Beaiji, Iraq send their oil to depots. The Perai refineries with entire processing capacity of 580,000 metric tons processed 512,714 metric tons of CPO from these Millss. The other five Millss at Kuala Kangsar and Terong send 190,044 metric tons of their entire CPO end product to Lumut. The entire transit distance to implement the above assignment is 1,211,913 kilometres.