Application of Linear Programming in Modeling the Allocation of Buses to Routes in a Transport Service Authority

The objective of the study was to allocate the available buses of the transport authority to the authority’s service intra and inter states routes in a manner that will yield optimum profit, taking all the constraints into consideration. The problem was modeled using Linear Programming (LP) and the TORA computer software result yielded a maximum objective value of N897, 214 per day after 20 iterations, which is a better result compared to the current intuitive schedule by the authority that yields N766, 046 per day.


Introduction
Niger State Transport Authority (NSTA) is a state-own transport service operator in Minner, Niger state, Nigeria with the following seven commuting services i. Inter State services ii.
Intra State services iii.
City service within Minna municipal, Bida and Kontagora towns iv.
Towing, Haulage and Hire services to all parts of Nigeria v.
Ferry services from Rofia across to Zamare in Kebbi State and from Shiroro across to Lakpama, in Niger State vi.
Passenger Boat service from kofa to kabo in Suleja area council vii.
Civil servants and students Bus services within Minna municipal (NSTA, n.d) However, this study was limited to the first two servicesinter and intra state services due to interest and inadequate finance. These routes are as follows:

Objective
The objective of the study was to apply linear programming model to optimally allocate the available buses of the Transport Authority to the service routes.

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Application of Linear Programming in Modeling the Allocation of Buses to Routes in a Transport Service Authority Niger State transport Authority (NSTA) operates ten intra and ten inter States routes transport service (see section 1.0). The authority has 80, fourteen sitter buses (Fifty Hiace and thirty King Long Buses); from which six of the Hiace buses are usually on stand-by for emergency hire purposes. Twenty-five of the buses are used for city service, Civil servants and students Bus services within Minna municipal, Bida and Kontogora towns of the state. Forty-Nine are used for commuting the intra and inter states routes. According to findings, daily, the buses incur costs in four ways: fuel, percentage parking levy, routine service and maintenance (repair). Buses going to New-Busa, Mokwa, Ilorin and Lagos incur thrice the cost of repair (maintenance) due to bad road. The authority might want to turn from intuition and consider mathematically, an optimized way of allocating buses to routes in order to obtain daily optimal gain though there are seasonal variations in terms of passenger patronage during Christmas and Salla periods.

Brief History of NSTA
According to NSTA (n.d.), Niger State Transport Authority was established by the then Military Governor Col. Lawal Gwadabe on the 11 th August, 1988 under edict No. 11 of 1988 enacted by the State Government. A task force was immediately constituted, headed by a military administrator to oversee the affairs of the organization. Since inception, as argued by NSTA (n.d), the authority has become a household name and has lived up to expectation been the only viable transport venture in the state and one of the best in the country vis-à-vis its contemporaries.
NSTA has the following organizational chart: Source: NSTA (n.d), Niger State Transport Authority at a Glance, Yan Ju Prints, Minna, Nigeria According to NSTA (n.d), Passengers are to note the following adherences: i. The authority does not accept liability for loss of goods. Passengers are therefore advised to take good care of their goods/properties while waiting to board our vehicles and while on transit ii.
Preaching is strictly prohibited in our vehicles iii.
The habit of smoking in our vehicles is strictly prohibited iv.
You can only enter our vehicle when you have paid correct money and obtained a ticket for the journey v.
Tickets should be in passengers' possession until the end of the journey vi.
Ticket can only be issued to prospective commuters when they maintain a single queue vii.
Heavy loads, bags and boxes are paid for and tickets obtained before they are loaded in our vehicles viii.
Commuters are expected to present their tickets on demand to our traffic inspectors during the course of the journey when asked to do so ix.
Intentional damage to our vehicle seats, glasses etc. will not be accepted x.
Female passengers are seated at the back of the vehicle while male are seated at the front xi.
Hawking of goods inside our vehicles is strictly prohibited xii.
Passengers are to help us so that we can serve them better

Brief Literature Review on Linear Programming
Linear Programming (LP) is an optimal decision making tool in which the objective is a linear function and the constraints on the decision problem are linear equalities and/or inequalities. It is the most commonly applied form of constrained optimization. The four main elements of any constrained optimization are decision variables, objective function, constraints and variable bounds. In LP, all the mathematical expressions for the objective function and constraints are linear. One might imagine that the restriction to linear models severely limits the ability to model real-world problems; but this is not so. An amazing range of problems can be modeled using LP from airline scheduling to least cost petroleum processing and distribution (Chinneck, 2001;Ramsey, 2012) The popular Simplex method of solving LP problems obtains the optimum solution by moving along edges of the solution space from one extreme point to another.
Linear programming problems have the property that the constraints and the objective function are all linear functions of the input variables. The existence of a polynomial time algorithm for solving linear programs and the multitude of optimization problems that they can encode makes them particularly useful in practice.
Generally, linear programming problem can be stated as follows: Maximize the objective function Z = c 1 x 1 + c 2 x 2 + c 3 x 3 +…………..+ c n x n Subject to the constraints a 11 x 1 +a 12 x 2 + ……………….+ a 1n x n1 a 21 x 1 +a 22 x 2 +………………..+ a 2n x n2 . . a m1 x 1 + a m2 x 2 +………………..+ a mn x nm We can write the problem in abbreviated form called the standard form or canonical form as follows: Minimize Here, x is a vector of real-valued variables (sometimes assumed to be non-negative), C and b are vectors of real constants, and A is a matrix of real constants. 14. An Augmented solution: Is a solution for a problem that was originally in inequality form that has been augmented by the corresponding values of the slack or surplus variables to change the problem into equality form (Hillier and Lieberman, 1986).

Advantages of LP
1. Linear programming helps in dealing with the problem of allocation of limited resources among different competitive activities in the most optimal manner. 2. It is concerned with determining the optimal allocation of scarce resources to meet certain objectives. 3. It provides practical and better quality of decision that reflects very precisely the limitations of the system. I.e. various restrictions under which the system must operate for the solution to be optimal. 4. Linear programming is an adaptive and flexible mathematical technique and hence can be used in analyzing a variety of multi-dimensional problem quite successfully. 5. The minimization of a function is equal to the maximization of the negative of that same function.
i.e. P min = 0 = -P max 6. The techniques help to make the best possible use of available productive resources. 7. Linear programming is applicable to transportation problem, diet problem, product mix problems, investment planning problem, marketing and distribution management etc. 8. According to Adler et al (1995), linear programs are expressed in an inequality form, which allows for the inexact computation of the algorithms direction of improvement, resulting in a significant computational advantage.

Raw Data Collected
The data below were collected in December, 2012 from NSTA headquarters which is located at Shango by Paiko Road, Opposite Trade Fair Complex, Minna. Data on bus routes were obtained from the Interview with the desk officer, Mal. Babangida; while that of bus service and maintenance were obtained from the interview with the Assistant Workshop Manager, Mal. Ali Baba Suleiman.  Total 3700 Note: Buses in NSTA are serviced twice a month, which is after 15 days. Note: Buses going to New-Busa, Mokwa, Ilorin and Lagos incur thrice the cost of repair/maintenance due to bad road. Note: Buses in NSTA are serviced twice a month, which is after 15 days. This means that a bus in NSTA consumes N7400 in a month (Amount per 15 Days × 2). Dividing service amount per month by 30 days gives us N247 as the cost of servicing a bus in a day. Note: The cost of maintaining a bus in a day is calculated to be N513 (Cost ÷ Duration in Days). Since buses going to New-Busa, Mokwa, Ilorin and Lagos incur thrice the cost due to bad road, their repair/maintenance costs will be 513×3 = N1539 per bus per day.

NSTA Problem Formulation
The Problem Formulation is based on the information under the Problem Situation and Table 6. The problem is formulated under the assumption that all the 49 buses that commute intra and inter states routes are working daily.
Based on the interview with NSTA, availability of passengers and the number of other transport services plying the same routes, among other factors, determine the number of buses that NSTA can assign to these routes. The following shows the possible number of buses that can be scheduled to routes:

Result of the Problem Formulation
The formulated problem was solved using TORA -computer software used in solving Linear Programming which has been developed by Hamdy Taha (2002 edition). The 20 th iteration reached the optimal solution and the objective value of 897, 214 is obtained as can be seen on TORA window under figure 2 below 134 Application of Linear Programming in Modeling the Allocation of Buses to Routes in a Transport Service Authority  Table 8 below shows the summary of our recommendation which is based on the result of our study: