J. Nig. Soc. Phys. Sci. 1 (2019) 116–121 Journal of the Nigerian Society of Physical Sciences Original Research Sensitivity of Parameters in the Approach of Linear Programming to a Transportation Problem Tolulope Latunde∗, Joseph Oluwaseun Richard, Opeyemi Odunayo Esan, Damilola Deborah Dare Department of Mathematics, Federal University, Oye-Ekiti, Nigeria Abstract Linear programming has served as a great tool in dealing with transportation problems to make a positive difference in economic and social activity. In this work, data of the National Union Of Road Transport Workers (NURTW) is analysed to minimizing the cost of maintenance and repair of buses taking the route from Sango park to different routes. Data collected from the park are represented using tables and solved using the Maple computer software application. The transportation problem is solved such that the transportation cost is minimized which leads to the profit being maximized. This is achieved by estimating the values of some identified parameters in the problem. This work will be beneficial to every other motor parks controllers to decide on some decision making that may bring to the union profit. This work will help the NURTW in Sango to spend less on the vehicles and save more as income. Keywords: Linear Programming, transportation problem, parameters, sensitivity Article History : Received: 26 September 2019 Received in revised form: 26 October 2019 Accepted for publication: 29 October 2019 Published: 12 November 2019 c©2019 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: B. J. Falaye 1. Introduction One major problem encountered by companies or organiza- tions is the transportation problem. The transportation problem was given shape as a problem in 1871 by a French economist and mathematician called Gaspard Monge, the transportation problem was first studied in 1920 as a problem by Aleskey Nikolayevich, [1]. Here, we study the Sango NURTW which runs the trans- port service both in Ogun State and outside the state. Since the transportation problem is to find a way of minimizing cost (or time spent) or to maximize profit, the goal of every transporta- tion is to meet the request of the destination. ∗Corresponding author tel. no: +2348032801624 Email address: tolulope.latunde@fuoye.edu.ng (Tolulope Latunde) In this study, we only have two destinations, one destination is seen from the other, they are respectively the place where the vehicles will stop eventually when it leaves the park and the park itself. The optimization of transportation problem can apply to so many areas like the Genetic Algorithm, Networking, Modelling and many more. The transportation method in use is a type of linear pro- gramming problem that utilises the simplex technique. It is ap- plied to the problems related to the study of the efficient trans- portation routes i.e., how efficiently the product from different sources of production is transported to the different destination such as the total transportation cost is minimum. Here, we revise the solution of the formulated transportation problem until the optimum solution is obtained, thus we varied the values of parameters in the model to study the behaviours of each parameter of the model. 116 Latunde et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 116–121 117 Since usually the objective of the transportation problem is to either minimize the distance to be covered, total transporta- tion cost or to maximize profit. The transportation problem can be defined mathematically as; suppose Xi j ≥ 0 is the number of passengers traveling from ith origin to the jth destination the model of the problem given the cost of transportation c will be, Minimize Z = m∑ i=1 n∑ j=1 ci j Xi j (1) S ub ject to n∑ j=1 Xi j ≤ ai f or i = 1,2, ...,m i.e supply m∑ i=1 Xi j ≤ b j f or j = 1,2, ...,n i.e demand. ∀Xi j ≥ 0 If ∑n i=1 ai = ∑n j=1 b j, this means that the number of passengers available is equal to the number of available spaces in the vehi- cles to transport them. If not, it will be considered unbalanced transportation in this work. 2. Literature Review In recent times, books have been written, papers published and presentation made diversely on the transportation problem, we generally that in a layman language, the transportation prob- lem is one of the optimization problems. A new way of solving the transportation problem from the park to different destinations was introduced in [2], the method used was readers friendly, easily accessible and can be quickly understood. The last table can be used to decipher the roads that serve as the best to be plied as well it explains the roads that need more vehicles to be added to increase the income rate. The cost coefficient of the objective function was dealt with thoroughly seeing into the multi-objective problems of trans- portation in [3]. A minimization work from the origin to the destination was carried out. The fuzzy programming technique was utilized to solve the problem that was converted from con- straints to a deterministic solvable problem. Asase did a great work on the transportation problem also, he used the GUINNESS GHANA LIMITED in Kumasi as a case study , he explained the transportation problem, created a table and solved using the Vogel, North-West Corner, Least Cost method as well as for test for optimality, he used the Step- ping Stone method and the Modified Distribution Method (MODI) [4]. A k-objective transportation problem was formulated in [5] by fuzzy numbers and used alpha-cut to obtain a transportation problem in the fuzzy sense expressed in linear programming form. An additive fuzzy programming model for the multi- objective transportation problem was introduced. The method aggregates the membership functions of the objectives to con- struct the relevant decision function. Weights and priorities for non-equivalent objectives are also incorporated in the method. Their model gave a non-dominated solution which is nearer to the best-compromised solution. In the paper - Optimal Solution of a Transportation Prob- lem, a method was developed in [6] to get the initial basic fea- sible solution (or near to the optimal solution) of transportation problem. Also, the algorithm provided in this paper gives the idea for the optimality in comparison with MODI method as the flow of steps by step procedure. The paper helps in explaining vividly the Algorithm used, the test of optimality and numerous numerical examples. However, some researchers have worked in the areas of sen- sitivity analysis and its application to transportation. Parameter estimation and sensitivity analysis of an optimal control model for capital asset management where the parameters of the for- mulated model of asset management are classified according to their degree of sensitivity was worked on in [7]. Also, the sen- sitivity of parameters in an optimal control model of the elec- tric power generating system to determine the behaviour of the model’s parameters in [8]. Pandian and Kavitha proposed a new bound technique for cost sensitivity ranges of solid transporta- tion problems [9]. In this work, the MAPLE software is used to derive the ex- pected income and also determine the best routes that needed to be invested in the most by analysing the sensitivity of the parameters involved in the model. 3. Statement of problem Given that some passengers are to travel to different des- tinations from an NURTW transport company, there is a need for the NURTW to optimize the cost of transportation, deter- mine the best ways of allocating vehicles by either increasing or decreasing the number of vehicles on a particular route. The total cost of transporting the passengers from the park i to the destination j is not changing from every other trans- portation problem. This is given by letting i = park and then j = destinations, prior ai and b j above xi j is the number of passengers to be transported from Sango park i to the different destinations: n∑ j=1 ci j xi j = c11 x11 + c12 x12 + ... + c1n x1n (2) The total cost Tc of transporting the passengers from Sango park to the different destinations. Tc = m∑ i=1 n∑ j=1 ci j xi j = c11 x11 + c12 x12 + ... + c1n x1n c21 x21 + c22 x22 + ... + c2n x2n . . . cm1 xm1 + cm2 xm2 + ... + cmn xmn 117 Latunde et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 116–121 118 The NURTW agrees generally in Sango to offer services of transporting her passengers from Sango both intra-state and inter- state as listed below. Intra-State Sango-Atan Sango-Ewekoro Sango-Ifo Sango-Owode Sango-Idi-Iroko Sango-Abeokuta Sango-Ijebu Ode Sango-Ijebu Remo Sango-iperu Inter-State Sango-Lagos (Ikeja) Sango-Ondo (Akure) Sango-Oyo (Ibadan) Sango-Osun (Oshogbo) Sango-Ekiti (Ado-Ekiti) 4. Data Collation and Analysis All data were gotten from Sango park in March 2019 at Sango, Ogun State. Data were gotten by conversation and in- terviewing one of the Sango NURTW official staff. The Desti- nation: This is the destination the vehicles go to from the park. It is the place the passengers paid for a service to be delivered to them. The Trip: This is the to and fro movement a vehicle com- pletes weekly. We should note that daily means all the vehicles can travel every day, turn means it depends on the total number of vehicles available and it is determined by how early the ve- hicles arrive as they will leave respectively as they have come. Fuel: This is the average amount of fuel consumption (in litres) per trip. B/Rt: (Buses per route)These are the number of buses the NURTW for Sango agreed to travel in a day. Hrs/trip: This is the total number of hours it takes a vehicle to travel to the destination and to travel back to the park if it leaves immediately. T. fare: The T.fare here means transportation fare. In essence, it is the cost a passenger is charged for transportation service, better explained as the transportation fare per person in Naira N. Table 2 is an average data from different distributors, hence it might be a little bit expensive or cheaper depending on the bargaining power. In Table 3, public drivers and Sango NURTW mode of ser- vicing their vehicles differ. The road differs, some are bad, some are fair and others are good, on that basis, some drivers will have to service their vehicles fortnightly, some every week and most service their vehicles monthly. This implies that a ve- hicle will get serviced at the rate of N8700 per month implying that averagely a vehicle will be serviced for N256.7 daily. We carefully noticed that the analysis above in Table 4 is not true for all route, as stated earlier above, some route may cost more as some roads are bad e.g the Sango to Owode-Iroko, this road attracts twice the amount most times. Fuel : This is gotten by the litres of fuel used in a trip, as at the time of this thesis, a litre is N145, therefore, the fuel consumption cost is the total number of litres multiply by 145 in (Naira). Parking Charges : This is the amount a vehicle driver will pay at the garage of the destination, this is because NURTW, Sango does not have a parking space of her own. B/Trp : This short form for Bus Per Trip. It is the number of buses per trip that can be on the road at a time. Hrs/trip : This is the short form for Hours Per Trip. The time taken for a bus to reach the destination and fro. CVS/day : This is the short form for Cost of Vehicle Ser- vice Per day. VRM/day : This is the short form for Vehicle Repair and Maintenance Per day. Income/trip : This is gotten by multiplication of the trans- port fare twice, twice because a trip is to and fro then multiplied by the number of people in a bus. The total number of passen- gers in a bus is 14 i.e (T Fare × 2 × 14), Expense/day : This is gotten from adding the Fuel Con- sumption Cost + the Parking Charges + Servicing Vehicles Cost Per Day + Vehicles Maintenance Per Day. Income/day : This is gotten from subtracting Vehicles Ex- pense Per Day from Income for a Trip per day. Parking Charges : This is the multiplication of the amount of Parking charges per Bus by the number of buses per on the road at a time. Fuel : This is the multiplication of fuel used in a trip by the number of buses on the road at a time (B/Trp). CVS : This is the cost vehicle service multiplied by B/Rt VRM : This is the cost of vehicles repair and maintenance multiplied by B/Rt, Income/Trip : This is gotten by multiplying the income per trip of the bus by the total number of buses that can be on the road at a time. Total Expense : This is the sum of by Parking Charges, Amount of Fuel used, CVS and VRM. Total Income : This is the difference between Income/Trip and the Total Expense. 5. Model Description The formulation of the problem is based on data gotten from the park and the immediate Table 5. NURTW (SANGO PARK) allows the maximum of 140 buses in total to travel the road to all road to these respective destinations every day. Meanwhile, these buses can be assigned to leave the park. Solving the above linear programming problem, we derive N1,069,345.7. However, our goal in this work is to sensitize each parameter of the model to derive optimal allocation. Thus from Table 6, we increase all routes in twos respectively from the Atan destination to the Ekiti destination, we keep repeating this process for all the routes till 10 buses are added to all the routes. 118 Latunde et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 116–121 119 Table 1: Data on Trips S/N Destination Trip(Weekly) Fuel(Ltrs) B/Rt Hrs/trip T. Fare N 1 Atan Daily 15 20 1 150 2 Ewekoro Daily 15 20 1 200 3 Ifo Daily 20 15 1.5 300 4 Owode Daily 15 20 1.5 300 5 Idiroko Daily 25 10 2 400 6 Abeokuta Turn 30 10 2.5 500 7 Ijebu-Ode Thrice 70 3 5 1500 8 Ijebu-Remo Thrice 75 3 6 1700 9 Iperu Thrice 75 3 6 1800 10 Lagos Daily 20 15 2 300 11 Ondo Five Times 35 7 3 700 12 Oyo Turn 70 2 4 1150 13 Osun Turn 85 2 5.5 2300 14 Ekiti Turn 95 2 7 2550 Table 2: Data On Vehicle Repair and Maintenance (VRM) S/N ITEM COST DURABILITY 1 Break disk 10500 2 years 2 Break lining 3000 1 months 3 Break Pad 1800 3 weeks 4 Car battery 15000 2 years 5 Front bearing 7000 5 months 6 Fuel pump 5500 2 years 7 Release bearing 3000 2 years 8 Shock absorber 11000 2 years 9 Shock filling 1800 6 months 10 Tyre 24000 4 months Total 82,600 Table 3: Cost Of Vehicle Service (CVS) S/N Service Item Amount Monthly Cost Per Day 1 Engine Oil (5ltrs) 5000 166.7 2 Oil filter 1700 56.7 3 Oil treatment 650 33.3 Total 8700 256.7 We then run every addition by the Maple software to ascer- tain the outcome and to determine the profit that will be gener- ated, it was noticed that some routes yielded more income than the others as the buses plying the routes were increased. Table 7 represents the optimal result of the formulated model. Bus: This is the representation of each route respectively from the Atan destination being X1 to the last destination i.e Ekiti represented as X14. Total No of Buses: This is the new total number of buses assumed to be on the road at once from Sango park. Former B/Rt : (Former Buses per route) This is the total number of buses plying a particular at once before the increase. New B/Rt: (New Buses per route) This is the total number of buses plying a particular at once after the increase. Income Generated: This is the result compiled by the Maple software after the buses have been respectively increased in Naira N. It is important to note that when none of the bus plying the route was increased, the income generated remains the same as the income generated in Table 6 i.e N1,069,345.7. The highest income generated is from X14 which is the Ekiti destination producing N1,617,172.7. 5.1. Discussion on Result Table 8 shows our recommendation as to know which the routes needs increment of more vehicle to ply them and the outcome of our result is represented in Figure 1 where it can be interpreted on the graph to see that the income generated by the X14 being the Ekiti destination yields the highest income. Figure 1 below shows the income generated by every bus. The buses are labelled respectively from the Atan destination as X1 to Ekiti as X14. The bus that generated the highest in- come is the X14, the bus plying Ekiti. The problem was solved in Maple software and represented in Table 8 and Figure 1 whereby the optimal value of N1,617,172.7 is obtained instead of the N1,069,345.7 after selecting the optimal allocation of ve- hicles to best routes by sensitising the parameters of the model. However, to determine the optimal vehicle allocation, we varied the values of parameters representing each vehicle in the model with respect to its location to understand how each parameter behaves. Thus we recommend that Ekiti routes should get more buses to ply it to optimize the cost transportation. This will yield extra N 547,827 more as profit. 6. Conclusion Due to some uncertainties in the cost of VRM, CVS and some other input parameters, non-linear constraint optimiza- tion problem shall be considered in future works to tackle this. 119 Latunde et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 116–121 120 Table 4: Cost Of Vehicle Repair and Maintenance Per Day S/N Item Item cost Durability Durability (Day) Cost (Day) 1 Break disk 10500 2 years 720 14.6 2 Break lining 3000 1 months 30 100 3 Break Pad 1800 3 weeks 21 85.7 4 Car battery 15000 2 years 720 20.8 5 Front bearing 7000 5 months 150 46.7 6 Fuel pump 5500 2 years 720 7.6 7 Release bearing 3000 2 years 720 4.2 8 Shock absorber 11000 2 years 720 15.3 9 Shock filling 1800 6 months 180 10 10 Tyre 24000 4 months 48 500 Total 82,600 804.9 Table 5: Total Income per day S/N Fuel(Ltrs) Fuel N Parking charges N B/Trp Hrs/trip T.Fare N CVS/day N VRM/day N Income/trip N Expense/day N Income/day N 1 15 2175 0 20 1 150 256.7 804.9 4200 3236.6 963.4 2 15 2175 0 20 1 200 256.7 804.9 5600 3236.6 2363.4 3 15 2900 0 15 1.5 300 256.7 804.9 8400 3169.6 4438.4 4 15 2175 0 20 1.5 300 256.7 804.9 1609.8 4041.5 4358.5 5 25 3625 0 10 2 400 256.7 1609.8 11200 5491.5 5708.5 6 30 4350 150 10 2.5 500 256.7 804.9 14000 5561.6 8438.4 7 70 10150 350 3 6 1500 256.7 1609.8 42000 12366.6 29633.5 8 75 10875 350 3 6 1700 256.7 1609.8 47600 13091.5 34508.5 9 75 10875 300 3 6 1800 256.7 804.9 50400 12236.6 38163.4 10 20 2900 0 15 2 300 256.7 804.9 8400 3961.6 4438.4 11 35 5075 200 7 3 700 256.7 1609.8 19600 7141.5 12458.5 12 70 10150 300 2 4 1150 256.7 1609.8 32200 12316.5 198835 13 85 12325 350 2 5.5 2300 256.7 1609.8 64400 14541.5 49855.5 14 95 13775 350 2 7 2550 256.7 2414.7 71400 16796.4 54603.6 Total 645 93525 2350 13850 3593.8 18507.7 387800 103454.54 269815.5 963.4X1 + 2362.4X2 + 4438.4X3 + 4358X4 + 5708.5X5 + 8438.4X6 + 29633.5X7 + 34508.5X8 + 38163X9 + 4438.4X10 + 12458.5X11 + 19883.5X12 + 49855X13 + 54603.6X14 X1 ≤ 20 X2 ≤ 20 X3 ≤ 15 X4 ≤ 20 X5 ≤ 10 X6 ≤ 10 X7 ≤ 3 X8 ≤ 3 X9 ≤ 3 X10 ≤ 15 X11 ≤ 7 X12 ≤ 2 X13 ≤ 2 X14 ≤ 2 X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 + X11 + X12 + X13 + X14 ≤ 132 Table 6: The total income of all buses on the road at a time S/N Dest. Parking Charges N B/Rt Fuel T.Fare N CVS N VRM N Income/Trip N Total Expense N Total Income N 1 Atan 0 20 43500 3000 5134 16098 84000 64732 19268 2 Ewekoro 0 20 43500 4000 5134 16098 112000 64732 47268 3 Ifo 0 15 43500 4500 3850.5 12073.5 126000 59424 66576 4 Owode 0 20 43500 6000 5134 3219.6 168000 80830 87170 5 Idiroko 0 10 36250 4000 2567 16098 112000 54915 57085 6 Abeokuta 1500 10 43500 5000 2567 8049 140000 55616 84348 7 Ijebu-Ode 1050 3 30450 4500 770.1 4829.4 126000 39274.5 86725.5 8 Ijebu-Remo 1050 3 32625 5100 770.1 4829.4 141000 37099.5 88900.5 9 Iperu 900 3 32625 5400 770.1 2414.7 151200 36708 114492 10 Lagos 0 15 43500 4500 3850.5 12073.5 126000 59424 66576 11 Ondo 1400 7 35525 4900 1796.9 11268.6 137200 49990.5 87209.5 12 Oyo 600 2 20300 2300 513.4 3219.6 64800 24633 39767 13 Osun 700 2 24550 4600 513.4 3219.6 128800 29083 99717 14 Ekiti 70 2 27550 5100 513.4 4828 142800 33592.8 109207.2 Total 7900 132 455875 62900 33884.4 118318.9 17579400 690054.3 1069345.7 120 Latunde et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 116–121 121 Table 7: Result of Sensitivity analysis analysis on the model S/N Bus Total No of Buses Former B/Rt New B/Rt Income Generated N 1 X1 142 20 30 1,080,770.7 2 X2 142 20 30 1,115,520.7 3 X3 142 15 25 1,115,520.7 4 X4 142 20 30 1,114,721.7 5 X5 142 10 20 1,128,221.7 6 X6 142 10 20 1,155,520.7 7 X7 142 3 13 1,367,471.7 8 X8 142 3 13 1,416,221.7 9 X9 142 3 13 1,452,766.7 10 X10 142 15 25 1,115,520.7 11 X11 142 7 17 1,195,721.7 12 X12 142 2 12 1,269,971.7 13 X13 142 2 12 1,569,691.7 14 X14 142 2 12 1,617172.7 Table 8: Optimal allocation based on sensitivity of the parameters S/N Destination Former B/Rt Recommend B/Rt 1 Atan 20 20 2 Ewekoro 20 20 3 Ifo 15 15 4 Owode 20 20 5 Idiroko 10 10 6 Abeokuta 10 10 7 Ijebu-Ode 3 3 8 Ijebu-Remo 3 3 9 Iperu 3 3 10 Lagos 15 15 11 Ondo 7 7 12 Oyo 2 2 13 Osun 2 2 14 Ekiti 2 12 Figure 1: The Graph of Income generated against number of buses by routes However, as a result of sensitivity analysis carried on the param- eters of the formulated model, we recommend that additional buses should be considered to the best route of Sango-Ekiti to optimize income and minimize cost. References [1] S. Sarbjit, “Note on Transportation Problem with New Method for Reso- lution of Degeneracy”, Universal Journal of Industrial and Business Man- agement, 3(2012) 26. [2] N. Ngutor, J. O Omolehin & K. Rauf, “Application of Linear program- ming in Modeling the Allocation of Buses to Routes in a Transport Ser- vice Authority”, Universal Journal of Applied Mathematics, 2 (2014) 125. [3] J. C. Das , Load Flow Optimization and Optimal Power Flow, Boca Ra- ton: CRC Press, 2017. [4] A. Asase, The Transportation Problem, Case study: Guiness Ghana Lim- ited”’, Master’s Thesis, Faculty of Physical Science and Technology Ku- masi, Ghana, 2011. [5] A. K. Bit, M. P. Biswal & S. S. Alam, “Fuzzy Programming Approach to Multicriteria Decision making Transportation problem”, Fuzzy sets and systems, 5 (1992) 135. [6] R. G. Patel, B. S. Patel & P. H. Bhathwala, “On Optimal Solution of a Transportation Problem”, Global Journal of Pure and Applied Mathemat- ics, 13 (2017) 6201. [7] T. Latunde & O. M. Bamigbola, “Parameter Estimation and Sensitivity Analysis of an Optimal Control Model for Capital Asset Management”, Advances in Fuzzy Systems https//doi.org/10.1155/2018/4756520 [8] T. Latunde, O. M. Bamigbola & Y. O. Aderinto, “Sensitivity of Parame- ters in an Optimal Control Model of the Electric Power Generating Sys- tem”, Ilorin Journal of Computer Science and Information Technology (ILJCSIT), 1 (2016) 54. [9] P. Pandian & K. Kavitha, “Sensitivity Analysis in Solid Transportation Problems”, Applied Mathematical Sciences,6 (2012) 6787. 121