Original Research Article
A Modified Ant Colony Optimization Algorithm for
Solving a Transportation Problem
ABSTRACT:
Transportation of products from sources to destinations with minimal total cost plays a key role in logistics and supply chain management. The transportation problem (TP) is a special type of Linear Programming problem where the objective is to minimize the total cost of distributing a product from several sources to several destinations. Initial feasible solution (IFS) acts as a foundation of an optimal cost solution technique to any TP. Better is the IFS lesser is the number of iterations to reach the final optimal solution. This paper presents a meta-heuristic algorithm, modified ant colony optimization algorithm (MACOA) to attain an IFS to a Transportation Problem. The proposed algorithm is very simple, easy to understand, easy to implement, and gives us proximity optimal solutions an infinite number of iterations. The efficiency of this algorithm is also been justified by solving validity and applicability examples. An extensive numerical study is carried out to see the potential significance of our modified ant colony optimization algorithm (MACOA). The comparative assessment shows that both the MACOA and the existing JHM are efficient as compared to the studied approaches of this paper in terms of the quality of the solution. However, in practice, when researchers and practitioners deal with large-sized transportation problems, we urge them to use our proposed MACOA due to the time-consuming computation of JHM. Therefore this finding is important in saving time and resources for minimization of transportation costs and optimizing transportation processes which could help significantly to improve the organization’s position in the market.
Keywords: Modified ant colony algorithm; Transportation problem; Initial feasible solution; Vogel’s
approximation method; Juman and Hoque’s method
1. INTRODUCTION
Dorigo and Maniezzo [14] developed the first foraging ant’s algorithm which is called Ant Colony Algorithm (ACA). This is based on the probabilistic technique for solving various types of optimization problems. Such as vehicle routing problem, traveling salesman problem, traffic assignment problem, shortest path problem, minimum spanning tree problem, etc. The algorithm is examining for an optimal path in the graph based on the behavior of ants.
On the other hand, the development of this algorithm was ants can find the shortest roots between food sources and their colony. Ants are social insects and walk randomly. They are walking between their colony and the food source, ants deposit pheromones (communication medium) along the paths they move. The pheromone level on the paths increases the probability of path (quantity and the quality of the
life dynam The trans various w from each The TP w Transport Dantzig (1 Besides, method [1 Goyal’s v Initial Fea balanced transporta This pape solving bo and it alw modificati less iterat The rema transporta example p instances conclusio 2. MATH The TP is destinatio transporta transporta Hence th problem) M S mic traveling s portation prob warehouses. T h source to ea was first devel
tation System 11) developed several heur 18], VAM -Vo ersion of VAM asible Solutio TP. Sharma ation problem er presents a oth types of b ways tries to ons to this AC tion. ainder of this ation problem problem. The are carried ns along with HRMATICAL s focused on ons having de ation cost. A ation cost per he above pro can be given Min Subject to salesman pro blem (TP) de The objective ach destinatio loped by Hitc m. After that, d the transpo ristic solution gel’s approxi M [17], EHA-on (IFS) to th a[37]presente , Jumanand N an overview o balance and reach a min CA are made s paper is as m. In Section en, its compar d out to sho
limitations an L FORMULA
the distributio emand 𝑏 , 𝑏 Arc (i, j) join r unit,𝑐 , and oblem (some in the followi
where
blem. als with the d e of this prob on with a mini chcock [19] an Charnes and rtation Simple s approach mation metho -An Efficient e TP. Moreo d a new sol Nawarathne [ of the conce unbalance TP nimum cost s e and ensured s follows: Se n 3 the modi rative studies ow the pote nd future rese ATION on of the prod , … , 𝑏 ) to m ing source i the amount s etimes called ng form: distribution of lem is to det mum total tra nd then Koop d Cooper [8] ex Method to such as Nor od [31], JHM Heuristic App over, Sabbagh lution proced [24]presented pt of ACA an P. The propo solution to th d a solution th ection 2 deals fied ACA is
with the exis ential signific earch directio
duct from m meet the dem to destinatio shipped, 𝑋 , w as the gen a product ma termine the fe ansportation c pmans [25] de developed th this problem. rthwest Corne –Juman and proach [22], e h [33]develop dure to solve d an alternativ nd provides osed one is v e concern pr hat is very clo
s with the m proposed an sting ones on
ance of the ons are presen
sources havi and of each on j carries tw where i = 1, 2 neral, classic anufactured a easible amou cost. eveloped Opt he stepping s . er Method [1 Hoque’s met etc were prop ped a new hy the dual of ve approach t a review of i ery simple, e roblem. In t oser to the op mathematical nd illustrated the results of e proposed a nted in Sectio ng capacities destination w wo pieces o 2, ..., m and j cal or Hitchc 2 | at factories to unts to be sh imum Utilizat stone method 18], minimum thod [23], GV posed to obta ybrid algorith the incapac to solving a T its application easy to under his paper, se ptimal solution formulation o with a num f some bench approach. F on 5. s 𝑎 , 𝑎 , … , 𝑎 with the least
f information = 1, 2, ..., n . ock transpor Page some hipped tion of d and m cost VAM – ain an hm for citated P. ns for rstand everal n with of the merical hmark inally, ) to 𝑛 t total n: the rtation
A transpo demands problem. Result a 3. SOLU H (ACOA) to has been agreeable as follows Step 1:ma Step 2: A o s p u 3.1. Now ortation proble at all destin and Discuss UTION PRO Here, we prop o solve the tr successfully e performance s:
ants are rand is initialized t A transition rul option at the solutions are procedure is used in the or describe the em is said to ations. That sion OCEDURE pose a meta-h ransportation y applied to s es [16]. Follo domly placed to some prop e is used at e current decis incrementally repeated unt riginal ant sys
e ant appeara o be balanced is, heuristic solut problem. Thi solve several owing Afshar on the n deci per value at th each decision sion point is s y created by til all decision stem is define ances d if the total s otherwise tion method b s method ins l combinatori [3] the basic ision points a he start of the n point i to dec elected, the a ants as they n points of th ed as follows supply from a e, it is called based on Ant spired by the al optimizatio steps on this nd the amoun computation cide which op ants move to y move from o he problem a (Dorigo et al. all sources is d an unbalan t Colony Opti behavior of r on problems s algorithm [1 nt of pheromo . ption is to be s the next dec one point to are covered. T [14]): s equal to the ced transpor mization Algo real ant colon
and has ach 14] can be de
one trail on al
selected. Onc cision point an the next one The transition e total rtation orithm nies. It hieved efined ll arcs ce the nd the . This n rule
4 | Page probability the path with the higher the amount of pheromone concentration. But consider the above case, the better path is 0 𝑝ℎ𝑒𝑟𝑜𝑚𝑜𝑛𝑒 𝑙𝑒𝑣𝑒𝑙 1.
An ant iteratively moves from sources to various destinations is based on the pheromone trail. Thus an ith ant decides to move next unvisited jth destination is given by the expression
𝑃 𝑐 ∑
∈ (Transition Rule)
This equation calculates the probability of selecting a single component of the solution. Here, denotes the amount of pheromone on a component between states i and j, and denotes its heuristic value or visibility of arc (i,j), is
: 𝛼 0 , 𝛽 0 .Where is linear and 𝜃 min.value𝑐 ; cost between node i and node j, 𝜃 0.
For our scenario, we assumed 𝛼 𝛽 1 in the transition rule,
𝑃 𝑡 ∑∈
; ith ant visits the jthcity
0 ; Otherwise
.
Pheromone update rules are used to calculate the amount of pheromone level on each edge between node i and node j. The solution of each ant is updated using following update function:
𝑡 1 1 𝜌 𝜑 𝑡 ∑ ∆𝜑 𝑡 (Pheromone Update Rule) Where,
∆𝜑 ℵ; if component 𝑖, 𝑗 was used by ant (best route) = 0 ; Otherwise
Here, 𝐿 is the distance of the best route. Is simply a parameter to adjust the amount of pheromone deposited, typically it would be set to 1. We sum 𝐿𝑘for every solution which used component (i, j), then that value becomes the amount of pheromone to be deposited on component (i, j). In our case ,
𝑡 1 1 𝜌 𝜑 𝑡
Where 𝑡 = . , . ,ρ represents the pheromone evaporation rate and 0 𝜌 1.
3.2 A new Algorithm: Modified Ant Colony Optimization Algorithm (MACOA)
The steps of the proposed method can be expressed as follows:
Step 1: Formulate the Transportation Cost Matrix from the given transportation problem. If the matrix is balanced then go to step 3. Otherwise, go to step 2.
Step 2: If the matrix is unbalanced, then make it a balanced by added dummy row or a dummy column accordingly.
Step 3: C Step 4: A the proba Step 5: A Step 6: A supply an Step 7: If move to th Step 8: If Step 9: S The MAC proposed MACOA is Flow cha l Compute the p Ants in the col bility matrix to Allocate 𝑋 Allocate this m d demand va f the demand he next maxim the terminati Stop and Calc
COA is valida method (MA s also provide
art represent
path according onies are pla o make the fir 𝑚𝑖𝑛 𝑎 , 𝑏 in minimum value alues with res
in the colum mum probabi
on condition ulate the initia
ated using a ACOA) is pro ed below:
tation of the
g to the proba aced at the sta rst allocation. n the i-jth with e to the selec pect to the se mn (or supply lity cell. is satisfied (𝑎 al feasible so numerical ex ovided in Ap new algorith ability row ma arting nodes respect to the cted cell in Ste
elected cell. in the row) i 𝑎 𝑏 0 ), lution. xample. Deta pendix F. T hm, MACOA atrix or colum with the Max e maximum p ep 5 and redu s satisfied, d then go to St
ailed computa The flow char
n matrix (usin imum value o probability cel uce the minim elete that col ep 9. Otherw ation of this e rt representa ng eq. (2)). of the Probab l.
mum value fro lumn (or row) ise, go to Ste example usin ation of the a ility in m the ) then ep 4. ng the above
6 | Page Flow chart representation of the new algorithm, MACOA
Note that, in order to check the optimality of the initial solution obtained by the proposed algorithm, one can use the method of Stepping Stone / MODI. If it is not optimal, then proceed with the Stepping Stone / MODI.
4. COMPARATIVE ASSESSMENT
This section provides performance comparisons across the various well-known methods – LCM, NWCM, ZSM, JHM, VAM, and some other popular methods by the solutions obtained from disparate problems. Comparative assessments are performed and illustrated in the immediately following sections. The detailed representation of the numerical data of Table 1 is provided in Appendix A.
Table 1.Comparative results of LCM, VAM, and New method for 12 benchmark instances Problem
chosen from TCIFS(𝐼 Optimal cost(𝑂 Percentage minimal Deviation total From cost
LCM VAM NEW LCM VAM NEW
Deshmukh (2012) 555 555 555 555 0.00 0.00 0.00 Deshmukh (2012) 114 112 112 112 1.78 0.00 0.00 Ahmed (2016) 2,900 2,850 2,850 2,850 1.75 0.00 0.00 Ahmed (2016) 3,500 3,320 3,320 3,320 5.42 0.00 0.00 korukoglu (2011) 72,174 59,356 59,356 59,356 21.59 0.00 0.00 Imam at el.(2009) 475 475 475 475 0.00 0.00 0.00 Taylor(Module B) 4,550 4,525 4,525 4,525 0.55 0.00 0.00 Deshmukh (2012) 305 290 260 260 17,3 11.53 0.00 Sen et al.(2010) 2,404,500 2,164,000 2,146,750 2,146,750 11.20 0.80 0.00 Ahmed (2016) 9,800 9,200 9,200 9,200 6.52 0.00 0.00 Ahmed (2016) 14,625 13,225 10,375 10,375 40.96 27.46 0.00 Ahmed (2016) 6,450 6,000 5,600 5,600 15.17 7.14 0.00 The comparative results obtained in Table 1 are also depicted using bar graphs and the results are given in Fig.1.
Fi Line grap cost solut Fig.2.Perc It can be than LCM method of Perfor Table 2. T ig.1.Comparat hs for the pe ion) obtained centage of Dev seen from T M and VAM in f this paper p mance meas The detailed r
ive Study of the rcentage dev in Table 1 ar viation of the Re Table 1, and F n every case erforms bette ure of NEM o representation e result obtaine viation (of the re presented esults obtained Fig. 1 and Fi e where an im er compared t over ZSM, VA n of the nume ed by LCM, VA LCM, VAM w in Fig.2. d by LCM, VAM g. 2 that the mprovement to LCM and V AM and JHM rical data of T
AM with the pro with the propo
M and the propo proposed me in efficiency VAM. M for 9 bench Table 2 is pro oposed method osed method osed method ethod (MACA was possible mark instanc ovided in App d from minima A) is more eff e. Thus, prop es is shown pendix B. al total ficient posed in the
Table 2.A Problem from Juman and (2015) Srinivasan on (1977) Sen et al.(2 Deshmukh Ramadan Ramadan ( Kulkarni&D (2010) Schrenk (2011) Samuel (20 Imam et al Adlakha Kowalski (2 The comp in Fig.3: Fig.3.Com Line grap total cost A comparativ chosen d Hoque TC 𝐼 ZS &Thoms 910 2010) 215 h (2012) 798 and (2012) 5,6 Datar 1,2 et al. 59 012) 28 . (2009) 460 and 2009) 400 parative resul parative Study hs for the per solution obta ve study of Z IFS( M VAM 0 955 58500 2164 0 8 779 600 5600 00 880 60 28 0 475 0 390 ts obtained in y of the Result o rcentage devi ined in Table SM, VAM, JH M JHM 880 400 214675 0 743 0 5600 840 59 28 460 390 n Table 2 are obtained by ZS iation (of the Z
2 are presen HM and NEM O l co 𝑂 NEW 880 88 214675 0 20 743 74 5600 56 840 84 59 59 28 28 435 43 390 39 also depicted SM, VAM, JHM ZSM, VAM, J nted in Fig.4. M for 9 bench Optima ost( Perce e min ZSM 80 3.40 14675 0.55 43 7.40 600 0.00 40 42.85 9 0.00 8 0.00 35 0.06 90 2.56 d using bar g
with the propo JHM and the p
mark instanc entag
nimal Devian tota
VAM 8.52 0.80 4.84 0.00 5 4.76 1.69 0.00 3.26 0.00 graphs and th osed method proposed met 8 | ces atio al Fm co JH M NW 0.0 0 0 0.0 0 0 0.0 0 0 0.0 0 0 0.0 0 0 0.0 0 0 0.0 0 0 0.0 0 0 0.0 0 0 e results are thod) from m Page ro m ost NE W .00 .00 .00 .00 .00 .00 .00 .00 .00 given inimal
Fig.4. Perc According to all prob Comparat instances provided i Table 3. benchma Problem chosen from Juman a Hoque (201 Problem 1. Problem 2 Problem 3. Problem 4. Problem 5. Problem 6. Problem 7. The comp in Fig.5. centage of Dev g to the simul blems in Table tive results o are shown i in Appendix A compara ark instances m nd 5) TCIFS(𝐼 ZSM . 4525 2. 3460 . 920 . 864 . 475 . 3598 . 136 parative resul viation of the Re ation results e 2 compared btained by Z n the followin C: tive results s VAM 5125 3520 960 859 475 3778 112 ts obtained in esults obtained (Table 2, Fig d with ZSM an SM, VAM, JH ng Table 3. obtained by JHM NE 4525 452 3460 346 920 920 809 809 417 417 3458 345 109 109 n Table 3 are d by ZSM, VAM g. 3 and Fig. nd VAM. It pro HM ,and the Detailed data y ZSM, VAM Optim cost(𝑂 EW 25 4525 60 3460 0 920 9 809 7 417 58 3458 9 109 also depicted M, JHM and the 4),the propos ovides the sa proposed me a representat M, JHM and al 𝑂 Percentaminimal ZSM 3.40 0.55 7.40 0.00 42.85 4.04 0.00 d using bar g e proposed me sed method y ame results as
ethod for the ion of these New metho age Deviatio total VAM 8.52 0.80 4.84 0.00 4.76 9.25 0.00 graphs and th thod yields better re s JHM. seven bench seven proble od for the s on Fr co JHM NE 0.00 0.0 0.00 0.0 0.00 0.0 0.00 0.0 0.00 0.0 0.00 0.0 0.00 0.0 e results are esults hmark ems is seven om ost EW 00 00 00 00 00 00 00 given
Fig.5. Com Line grap cost solut Fig.6. Perc Based on provides modified from the o In additio problems mparative Study phs for the pe ion obtained centage of Dev the above re the same re ant colony a optimal result n to the abo chosen from y of the Result ercentage dev in Table 3 are viation of the Re esults (Table esults as JHM lgorithm. Not . This calcula ove benchma m Ahmed [4] obtained by ZS viation (of the e depicted in esults obtained 3, Fig. 5 Fig M. However, te that, here, ation is carried ark instances in order to d SM, VAM, JHM e ZSM, VAM Fig.6: d by ZSM, VAM . 6), the prop this cannot the formula d out to evalu , we have a determine the M with NEW Me , JHM and N M, JHM and the posed method deny the va is used to o ate how muc lso studied s e performanc ethod New method) e proposed me d outperforms alue of our p obtain the per
h nearer the i some other n ce of our new 10 | from minima thod s ZSM and VA proposed me rcentage dev is to. numerical exa w method ove Page al total AM. It ethod, viation ample er the
available 14 approaches. The obtained results are presented in Table4. Detailed data representation of these four example problems is provided in Appendix D.
Table 4.Comparative results of new method along with the 14 available approaches for 4 benchmark instances
Methods. Total
initial cost feasible for solution the
Ex.1 Ex.2 Ex.3 Ex.4
North West Corner Method(NWCM) 4400 4,160 540 1,50
0
Row Minimum Method (RMM) 2,850 4,120 470 1,45
0
Column Minimum Method (CMM) 3,600 3,320 435 1,50
0
Least Cost Method (LCM) 2,900 3,500 435 1,45
0
Vogel’s Approximation Method (VAM) 2,850 3,320 470 1,50
0
Extremum Difference Method (EDM) 2,900 3,620 415 1,39
0
Highest Cost Difference Method(HCDM) 2,900 3,620 435 1,45
0
Average Cost Method (ACM) 2,900 3,320 455 1,44
0 TOCM-MMM Approach 2,900 3,620 435 1,45 0 TOCM-VAM Approach 2,850 3,620 430 1,45 0 TOCM-EDM Approach 2,850 3,620 435 1,45 0 TOCM-HCDM Approach 2,900 3,620 435 1,45 0 TOCM-SUM Approach 2,850 3,320 455 1,44 0
ATM Approach ATM 2,850 3,320 415 1,39
0
Proposed New Method 2,850 3,320 410 1,39
0
Fig.7 Comp Note that instanc benchm LCM, V Table 5.P Solutio NWCM LCM VAM EDM PM Optima The comp in Fig.8be Fig.8Comp parative Study t, although o ces, the ava marks in det VAM, and ED Performance on methods al Solution parative resul elow: parative Study of the result ob our proposed ailable fourte ermining the DM. Detailed d measure of Total cost for Ex.1 107 83 80 83 76 76 ts obtained in of the result ob btained by the method yie een approach performance data represen new method r the initial fea Ex.2 19,70 13,10 12,25 12,25 11,50 11,50 n Table 5 are btained by PMa proposed meth lds the optim hes do not. e measure o ntation of the d (NM) over N asible solution 00 00 50 50 00 00 also depicted
against the exis
hod against the mal solution We have a f the propos problems is g NWCM, LCM, n d using bar g sting NWCM, L e existing 14 ap to the above also studied ed method ( given in Appe , VAM and ED graphs and th LCM, VAM, ED 12 | pproaches e four bench a further s PM) over NW endix E [5]. DM e results are DM Page hmark set of WCM, given
Note that, although PM yields the optimal solution to both benchmark instances above, the available four approaches (NWCM, LCM, VAM, and EDM) do not. Hence, the comparative assessments of the above different cases show that both the modified ant colony algorithm and JHM are efficient as compared to the studied approaches of this paper in terms of the accuracy of the solution.
5. CONCLUTION
Transportation of product distribution from several original points to several destinations with minimizes the cost of transportation. Transportation problem (TP) is one of the special class of Linear Programming problems in which the objective is transportation problem solution methods to reach the optimal solution. Many researchers have paid attention to solve this problem using different approaches. Out of all the existing methods in the literature, Northwest, Least Cost, Vogel’s Approximation, and Juman and Hoque’s methods are the most prominent and renowned methods in finding an initial feasible solution to a TP. Modified Distribution (MODI) Method and Stepping Stone Method are the most acceptable methods in finding the minimal total cost solution to the transportation problem. These well-known minimal total cost solution techniques start with an Initial Feasible Solution (IFS). Thus an IFS acts as a foundation of an optimal cost solution technique to any TP. Better is the IFS lesser is the number of iterations to reach the final optimal solution. However, in this research paper, we discuss a new alternative method, a modified ant colony optimization algorithm which gives often an optimal solution to the transportation problem. In this research paper, we first examine different initial solutions providing methods for attaining initial feasible solutions to balanced and unbalanced transportation problems. This research paper presents an overview of the concept of an Ant colony algorithm and provides a review of its applications to solve transportation problems. The PMis very simple, easy to understand, and easy to implement. Several modifications to the ant colony algorithm are made and ensured a solution which is very closer to the optimal solution. This method requires a minimum number of steps to reach the optimality as compared to the existing methods. An extensive numerical study is carried out to see the potential significance of our modified ant colony algorithm (MACOA). The comparative assessment shows that both the MACOA and JHM are efficient as compared to the studied approaches of this paper in terms of the quality of the solution. However, in practice, when researchers and practitioners deal with large-sized transportation problems, we urge them to use our proposed MACOA due to the time-consuming computation of JHM. Although our proposed Modified Ant Colony Optimization algorithm provides better IFS that is often optimal, it does not always guarantee the exact optimal cost solution. Since the existing well-known exact optimal cost solution technique (SSM) deals fully with the path tracing technique, it becomes very difficult in solving large-scaled transportation problems. Thus, we intend to devote ourselves in near future in proposing an alternate exact optimal approach that gets rid of this difficulty.
REFERENCE
1. Acierno LD, Gallo M, Montella B.(2010) Ant Colony Optimisation approaches for the transportation assignment problem. WIT Transactions on the Built Environment2010; Vol 111. 2. Adlakha V, Kowalski K.Alternate Solutions Analysis for Transportation Problems. J.Bus.
Econ.Res.2009;7 (11):41-49.
3. Afshar MH. A new transition rule for ant colony optimization algorithms: application to pipe network optimization problems. Engineering Optimization.2005; 37(5): 525-540.
4. Ahmed MM, Khan AR, Uddin MS, Ahmed F.A New Approach to Solve Transportation Problems. Open Journal of Optimization.2016; 5: 22-30.
5. Ahmed MM, Khan AR, Uddin MS, Ahmed F. Incessant Allocation Method for Solving Transportation Problems. American Journal of Operations Research.2016; 6: 236-244.
6. Ahmed, M.M., Tanvir, A.S.M., Sultana, S., Mahmud, S.,& Uddin, M.S. An Effective Modification to Solve Transportation Problems: A Cost Minimization Approach. Annals of Pure and Applied Mathematics.2014; 6, 199-206.
7. Blum C. Ant colony optimization: Introduction and recent trends. Physics of LifeReviews.2005; 2 : 353–373.
14 | Page modified Ant Colony Optimization algorithm to solve a dynamic traveling salesman problem: A case study with drones for wildlife surveillance. Journal of Computational Design and Engineering 10. Dantzig GB. Linear programming and extensions. Princeton, NJ: Princeton University press, !963. 11. Dantzig GB . Application of the Simplex Method to a Transportation Problem, Activity Analysis of Production and Allocation. Koopmans, T.C., Ed., John Wiley and Sons, New York.1951; 359-373. 12. Deshmukh NM. An Innovative Method For Solving Transportation Problem International Journal
of Physics and Mathematical Sciences .2012; Vol. 2 (3) , pp.86-91.
13. Dorigo M, Stutzle T. An Experimental Study of the Simple Ant Colony Optimization Algorithm. WSES International Conference on Evolutionary Computation (EC’01):2001; 253–258.
14. Dorigo ACM, Maniezzo V. Distributed optimization by ant colonies. Proceedings of the 1st European Conference on Artificial Life:1991; pp.134-142.
15. Dorigo ACM, Maniezzo V, Coloni A. The Ant system; Optimization by a colony of cooperating agents. IEEE Trance action of system ,Man, and cyberneties part B,1996; 26(1):29-41.
16. Dorigo M, Gambardella LM. Ant colony system: a cooperative learning approach to the traveling. IEEE Transactions on Evolutionary Computation.1997; 1: 53-66.
17. Goyal SK. Improving VAM for unbalanced transportation problems. Journal of Operational Research Society.1984; 35 : 1113-1114.
18. Hamdy AT . Operations Research: An Introduction. 8th Edition, Pearson Prentice Hall, Upper Saddle River.2007.
19. Hitchcock FL.The distribution of a product from several resources to numerous localities, J. Math. Phy. 1941; 20,224-230.
20. Imam T, Elsharawy G, Gomah M, Samy I.Solving Transportation Problem Using Object-Oriented Model .Int.J.comput.Sci.Netw.Secur.2009;9(2):353-361.
21. Jaiswal U, Aggarwal S.Ant Colony Optimization. International Journal of Scientific & Engineering Research. 2011;2(7): 1-7.
22. Juman ZAMS.,Hoque MA.An efficient heuristic approach for solving the transportation problem. Proceedings of the 2014 International Conference on Industrial Engineering and Operations Management Bali, Indonesia.2014.
23. Juman ZAMS, Hoque MA. An efficient heuristic to obtain a better initial feasible solution to the transportation problem, Applied Soft Computing.2015; 34 : 813-826.
24. Juman ZAMS, Nawarathne NGSA .An efficient alternative approach to solve a transportation problem. Ceylon Journal of Science.2019; 48(1):19-29.
25. Koopmans TC . Optimum Utiliztion of Transportation System, Econometrica, Supplement .1949;vol 17.
26. Korukoglu S, Bali S.A improve Vogel Approximation Method for the transformation Problem.Mathematical and computational Applications.2011; 16(2), 370-381.
27. Kulkarni SS, Datar HG. On Solution To Modified Unbalanced Transportation .Problem.Bulletion of the Marathwada MathematicalSociety.2010;11(2):20-26.
28. Matteucci M, Mussone L.Ant colony optimization technique for equilibrium assignment in congested transportation networks. Conference: Genetic and Evolutionary Computation Conference, GECCO .2006.
29. Quddoos A, Javaid S, Khali MM. A New Method for Finding an Optimal Solution for Transportation Problems. International Journal on Computer Science and Engineering (IJCSE) 2012.
30. Ramadan SZ, Ramadan IZ. Hybrid two stage algorithm for solving transportation problem.Mod.Appl.Sci.2012;6(4): 12-22.
31. Reinfeld NV, Vogel WR. Mathematical Programming, Englewood Cliffs, New Jersey: Prentice-Hall.1958;59-70.
32. Royo B, Sicilia J, Oliveros MJ, Larrod E. Solving a Long-Distance Routing Problem using Ant Colony Optimization. Applied Mathematics & Information Sciences.2015; 9(2):415-421.
33. Sabbagh MS, Ghafari H, Mousavi SR.A New hybrid algorithm for the balanced transportation problem. Computers and Industrial Engineering2015;82 (2015) :115-126.
34. Samuel AE. Improved zero point method (IZPM) for the transportation problems. Appl.Math.Sci.2012;6(109):12-22.
35. Schrenk S, Finke G, Cung VD.Two classical transportation problem revisited: pure constant fixed charge s and the paradox.Math.Comput.Model.54 (2011): 2306-2315.
36. Sen N,Som T, Sinha BA. Study of Transportation Problem for an essential item of southern part of north eastern region of India as an OR modal and use of object oriented programming,Int.J.Comput.Sci.Netw.Secur.2010;10(4):78-86.
37. Sharma RRK, Sharma KD. A new dual based procedure for the transportation problem. European Journal of Operational Research.2000;122 :611-624.
38. Shtovba S. Ant Algorithms: Theory and Applications.Programming and Computer Software.2005;31(4):167-178.
39. Sudhakar VJ, Kumar VNA.New Approach for Finding an Optimal Solution for Integer Interval Transportation Problems. Int. J. Open Problems Compt. Math.2010; Vol. 3: No. 5.
40. Taha HA. Operation Research: An introduction (8thed.). Prentice-Hall, India.2006.
41. Taylor IMS. Transportation and Assignment Solution Methods (Modulo B). http://web.tecnico.ulisboa.pt/mcasquilho/compute/_linpro/TaylorB_module_b.pdf
42. Vannan SE, Rekha S.A New Method for Obtaining an Optimal Solution for Transportation Problems.International Journal of Enginerring and Advanced Technology(IJET) .2013;2(5):369-371.
Appendix A Problem
chosen from.
Size Data of the problem
1.BTP-1 (12) 3 3 𝑐 6,4,1; 3,8,7; 4,42 , 𝑠 50,40,60 , 𝑑 20,95,35 2.BTP-2(12) 4 6 𝑐 9,12,9,6,9,10; 7,3,7,7,5,5; 6,5,9,11,3,11; 6,8,11,2,2,10 , 𝑠 5,6,2,9 , 𝑑 4,4,6,2,4,2 3.BTP-3(4) 3 4 𝑐 3,1,7,4; 2,6,5,9; 8,3,3,2 , 𝑠 300,400,500 , 𝑑 250,350,400,200 4.BTP-4(4) 3 4 𝑐 50,60,100,50; 80,40,70,50; 90,70,30,50 , 𝑠 20,38,16 , 𝑑 10,18,22,24 5.BTP-5(24) 5 5 𝑐 46,74,9,28,99; 12,75,6,36,48; 35; 199,4,5,71; 61,81,44,88,9; 85,60,14,25,79 , 𝑠 461,277,356,488,393 , 𝑑 278,60,461,116,1060 6.BTP-6(18) 3 4 𝑐 10,2,20,11; 12,7,9,20; 4,14,16,18 , 𝑠 15,25,10 , 𝑑 5,15,15,15 7.BTP-7(21) 3 3 𝑐 6,8,10; 7,11,11; 4,5,12 , 𝑠 150,175,275 , 𝑑 200,100,300 8.UTP-1 (12) 3 5 𝑐 4,1,2,4,4; 2,3,2,2,3; 3,5,2,4,4 , 𝑠 60,35,40 , 𝑑 22,45,20,18,30 9.UTP-2(35) 5 4 𝑐 60,120,75,180; 58,100,60,165; 62,110,65,170; 65,115,80,175; 70,135,85,195 , 𝑠 8000,9200,6250,4900,6100 , 𝑑 5000,2000,10000,6000 10.UTP-3 (5) 3 5 𝑐 5,8,6,6,3; 4,7,7,6,5; 8,4,6,6,4 , 𝑠 800,500,900 , 𝑑 400,400,500,400,800 11. UTP-4(5) 𝑐 12,10,6,13; 19,8,16,25; 17,15,15,20; 23,22,26,12 , 𝑠 150,500,600,225 , 𝑑 300,500,75,100 12.UTP-5(5) 4 4 𝑐 5,4,8,6,5; 4,5,4,3,2; 3,6,5,8,4 , 𝑠 600,400,1000 , 𝑑 450,400,200,250,300
16 | Page Appendix B
Problem chosen from
Juman&Hoque (2015) Data of the problem
Problem 1 𝑐 3,6,3,4; 6,5,11,15; 1,3,10,5 , 𝑠 80,90,55 , 𝑑 70,6035,60 Problem 3 𝑐 19,30,50,10; 70,30,40,60; 40,8,70,20 , 𝑠 7,9,18 , 𝑑 5,8,7,14 Problem 4 𝑐 32,40,120; 60,68,104; 200,80,60 , 𝑠 20,20,45 , 𝑑 30,35,30 Problem 5 𝑐 3,4,6; 7,3,8; 6,4,5; 7,5, , 𝑠 100,80,90,120 , 𝑑 110,110,60 Problem 6 𝑐 3,6,1,5; 7,9,27; 2,4,2,1, 𝑠 6,6,6 , 𝑑 4,5,4,5 Problem 7 𝑐 1,2,3,4; 4,3,2,0; 0,2,2,1 , 𝑠 6,8,10 , 𝑑 4,6,8,6 Problem 8 𝑐 10,2,20,11; 12,7,9,20; 4,14,16,18 , 𝑠 15,25,10 , 𝑑 5,15,15,15 Problem 9 𝑐 2,1,3,2,2; 3,2,1,1,1; 5,4,2,1,3; 7,5,5,3,1 , 𝑠 20,70,30,60 , 𝑑 50,30,30,50,20 Appendix C
Problem chosen from
Problem 1 𝑐 6,8,10; 7,11,11; 4,5,12 , 𝑠 150,175,275 , 𝑑 200,100,300 Problem 2 𝑐 20,22,17,4; 24,37,9,7; 32,37,20,15 , 𝑠 120,70,50 , 𝑑 60,40,30,110 Problem 3 𝑐 4,6,8,8; 6.8.6.7; 5,7,6,8 , 𝑠 40,60,50 , 𝑑 20,30,50,50 Problem 4 𝑐 19,30,50,12; 70,30,40,60; 40,10,60,20 , 𝑠 7,10,18 , 𝑑 5,7,8,15 Problem 5 𝑐 13,18,30,8; 55,20,25,40; 30,6,50,10 , 𝑠 8,10,11 , 𝑑 4,6,7,12 Problem 6 𝑐 25,14,34,46,45; 10,47,14,20,41; 22,42,38,21,46; 36,20,41,38,44 , 𝑠 27,35,37,45 , 𝑑 22,27,28,33,34 Problem 7 𝑐 9,12,9,6,9,10; 7,3,7,7,5,5; 6,5,9,11,3,11; 6,8,11,22,10 , 𝑠 150,175,275 , 𝑑 200,100,300 Appendix D Problem chosen from Ahmed et al. (2016)
Data of the problem
Problem 1 𝑐 3,1,7,4; 2,6,5,9; 8,3,3,2 , 𝑠 300,400,500 , 𝑑 250,350,400,200 Problem 2 𝑐 50,60,100,50; 80,40,70,50; 90,70,30,50 , 𝑠 20,38,16 , 𝑑 10,18,22,24 Problem 3 𝑐 7,5,9,11; 4,3,8,6; 3,8,10,5; 2,6,7,3 , 𝑠 30,25,20,15 , 𝑑 30,30,20,10 Problem 4 𝑐 4,3,5; 6,5,4; 8,10,7 , 𝑠 90,80,100 , 𝑑 70,120,80 Appendix E Example 1(Balanced TP).
A company manufactures motor tyres and it has four factories 𝐹 , 𝐹 ,𝐹 and 𝐹 whose weekly production capacities are 5,8,7 and 14 thousand pieces of tyres respectively. The company supplies tyres toits three showrooms located at 𝐷 , 𝐷 and 𝐷 whose weekly demand are 7, 9 and 18 thousand pieces respectively. The transportation cost per thousand pieces of tyre is given below in the TT;
Showrooms F 𝐷 𝐷 𝐷 Supply a 𝐹 2 7 4 5 c 𝐹 3 3 1 8 t 𝐹 5 4 7 7 o 𝐹 1 6 2 14 Demand 7 9 18
We want to schedule the shifting of tyers from factories to showroom with a minimum cost.
Example 2(Unbalanced TP).
A company has four plants located at A,B,C and D, which supply to warehouses located at E,F,G,H and I. Monthly plant capacities are 300,500,825 and 375 units respectivelyand monthly warehouses
18 | Page A 10 2 16 14 10 300 B 6 18 12 13 16 500 C 8 4 14 12 10 825 D 14 22 20 8 18 375 Requirements 350 400 250 150 400
Determine a distribution plan for the company in order to minimize the total transportation cost. Appendix F (Validation of the new Algorithm via numerical example)
Consider the following Transportation problem:
𝐷 𝐷 𝐷 𝐷 Supply 𝑆 60 120 75 180 8,000 𝑆 58 100 65 170 9,200 𝑆 62 110 65 170 6,250 𝑆 65 115 80 175 4.900 𝑆 70 135 85 195 6,100 Demand 5,000 2,000 10,000 6,000
Step 1: Formulate the Transportation Cost Matrix. The problem is unbalanced, make it a balanced problem by introducing a dummy destination.
𝐷 𝐷 𝐷 𝐷 𝐷 Supply 𝑆 60 120 75 180 0 8,000 𝑆 58 100 65 170 0 9,200 𝑆 62 110 65 170 0 6,250 𝑆 65 115 80 175 0 4.900 𝑆 70 135 85 195 0 6,100 Demand 5,000 2,000 10,000 6,000 11,450
Step 2: Compute the path according to the probability matrix (using eq.(2)).
𝐷 𝐷 𝐷 𝐷 𝐷 Supply 𝑆 .204 .194 .195 .197 0 8,000 𝑆 .208 .219 .220 .210 0 9,200 𝑆 .201 .205 .212 .204 0 6,250 𝑆 .196 .200 .189 .201 0 4.900 𝑆 .188 .179 .182 .185 0 6,100 Demand 5,000 2,000 10,000 6,000 11,450
Step 3, Step 4, Step 5, Step 6 and Step7 Iteration 1 .204 .194 .195 .197 0 8,000 8,000 8,000 8,000 8,000 8,000 .208 .219*2000 .220*7,200 .210 0 9,200 7,200 0 0 0 0 .201 .205 .212*2800 .204*3,450 0 6,250 6,250 6,250 3,450 0 0 .196 .200 .189 .201*2,550 0 4.900 4,900 4,900 4,900 4,900 2,350 .188 .179 .182 .185 0 6,100 6,100 6,100 6,100 6,100 6,100 5,000 2,000 10,000 6,000 11,450 5,000 0 10,000 6,000 11,450 5,000 0 2,800 6,000 11,450 5,000 0 0 6,000 11,450 5,000 0 0 2,550 11,450 5,000 0 0 0 11,450 Iteration 2 .204*5000 .194 .195 .197 0*3,000 8,000 3,000 0 0 .208 .219*2000 .220*7,200 .210 0 0 0 0 0 .201 .205 .212*2800 .204*3,450 0 0 0 0 0 .196 .200 .189 .201*2,550 0*2,350 2,350 2,350 2,350 0 .188 .179 .182 .185 0*6,100 6,100 6,100 6,100 0
20 | Page
0 0 0 0 11,450
0 0 0 0 8,450
0 0 0 0 0
Step 8: Stop and Calculate the initial feasible solution.
𝐷 𝐷 𝐷 𝐷 Supply 𝑆 60*5000 120 75 180 8,000 𝑆 58 100*2000 65*7200 170 9,200 𝑆 62 110 65*2800 170*3450 6,250 𝑆 65 115 80 175*2550 4.900 𝑆 70 135 85 195 6,100 Deman d 5,000 2,000 10,000 6,000 Total cost = 60x5000 + 100x2000 + 65x7200 + 65x2800 + 170x3450 + 175x2550 = 2,146,750
