“An Arrangement of Goods According to Holding Capacity of Freight Trains”

“An Arrangement of Goods According to Holding Capacity of Freight Trains”

Joshi Dungdung, Dr. Anita Kumari

Darshana Prasad Mukherjee University Ranchi Jharkhand

DOI: https://doi.org/10.51584/IJRIAS.2025.100800085

Received: 23 August 2025; Accepted: 28 August 2025; Published: 16 September 2025

ABSTRACT

Rail transportation is more affordable than road transportation for long distances due to capacity these freights can handle. Despite being one of the most cost effective and sustainable mode of transport, freight railways have been globally losing market values. In India the availability of rakes also poses a great challenge. In this work a model is proposed for management in freight train’s holding capacity and their better utilization for different commodities. An integer programming problem model is formulated for an arrangement of goods according to holding capacity of freight trains.

INTRODUCTION

The ideal modes of transport which has a large load carrying capacity is freight railway transportation. Accepting changes and adapting to dynamic market conditions is the key to the freight segments sustainable expansion. It is less affected by external factors, however planning freight rail transportation needs most advanced structured planning. Railway freight transportation is preferred method of transport, especially when large volumes and long-distance cargoes are to be transported. Rail can also accommodate shipment of many shapes and sizes, from grains to wind turbine blades. Indian railway has modernized in technology to improve the efficiency of its operations. It has implemented several initiatives to improve the freight business. Railway freight transportation system must offer quick, reliable, and optimized flexible services.

LITRATURE REVIEW

A new problem of redirecting freight trains to revised destinations as a last-minute risk mitigation strategy is given [3]. The problem is approached from a consignee’s perspective as the demand for a change of destination is made by the consignee. An Integer Non-Linear Programming (INLP) model is formulated to minimize the cost of redirecting trains subject to various constraints. A two-stage freight demand analysis method considering customers’ choice intention and actual supply capacity was designed [1]. In this paper [2], the objective is to find the optimal sequence of jobs and the optimal resource allocation separately. This paper [4]aims to optimize the service mode and routing for hazardous freight transportation in a rail-truck network with bi-objectives optimization approach. In the paper[5], a research work on freight delays in the Indian railway setting by analyzing how section controllers make freight train stop and hold decisions while dispatching freight trains is given. The NITI Aayog report – The report focuses on the product mix of Indian Railways, which has been skewed towards the movement of bulk commodities such as coal, cement, iron ore, steel, petroleum, foodgrains , and fertilizers. In, 2018-19 , coal holds 50 percent of the total freight movement of 1,221  million tonnes , followed by Iron Ore (11%), Cement (10%), Mineral Oil (4%), Fertilizers (4%), Iron & Steel (4%), Foodgrains (3%), Limestone and Dolomite (2%), Stones (including gypsum) other than marble (2%) and other commodities (9%). The challenges faced in movement of containers via rail, and low share in domestic container movement have been discussed.

Mathematical Formulation:

An integer programming model is formulated for an arrangement of goods according to holding capacity of freight train, and hence to minimize operational cost of freight train subject to various constraints.

At first we set some notifications, script and subscripts to the problem;

A = commodity type

Fw = weight of commodity

W = wegons

R = rate charge

Parameters and Variables :

A_ij (i,j = 1,2,3,……n) = i^th freight train carry different types of commodities (material)

F_wei  =  avg least weight of any commodity (fixed)

W_i =  total no. of wagon/operational holding capacity of i^th train

Q_ij =  maximum  quantity per commodity in i^th train

R_ij  =  rate charge for any material per ton per km

w_i  =  expandation of wagons if needed

Now Multiple Objectives are:

G_i1 – avoid any underutilisation of holding capacity of i^th train

G_i2 – limit the expandation of wagons to w_i

G_i3 – achieve the total holding capacity of material Q_ij

G_i4 – minimize the overloading of wagons as much as possible

Let x_ik = no. of wagons used for holding material Aij (k depends on no. of commodity)

Since the holding capacity of i^th train is Wi wagon and expandation is allowded

Now constraints are:

 (d_i1 ^–  )  = underutilization of holding capacity  W_i

 (d_i1 ⁺  )  = overutilization of holding capacity W_i

This implies holding constraint is

Sx_ik  + d_i1 ^– –  d_i1 ⁺   = Wi

Since the maximum no. of materials A_ij are restricted to Q_ij resp. thus it is assumed that overachievement beyond maximum limits are impossible, so taking

F_wi d_i2– = underachievement of Q_i1

F_wi d_i3- = underachievement of Q_i2

And so on,

The holding constraint are:

F_wi x_i1 + d_i2–  = Q_i1

F_wi x_i1 + d_i2–  = Q_i2

And so on,

• e. x_i1  +  d_i2–  =  Q_i1

               x_i2  +  d_i3–  =  Q_i2

               ………….

                x_in  +  d_i(n+1)  =  Q_in

Overloading operational capacity constraints :

Since overloading operational capacity should be limited to w_i wagons or less, however to meet higher order of Gi2 it can be greater than w_i wagons, so

(d_i1⁺) + (d_i12^–) – (d_i13⁺) = w _i

Thus objective functions are:

Priority P_i1 – to minimize d_i1–

Priority P_i2 – to minimize d_i12+

Priority P_i3 – to minimize d_i2– , d_i3–  ,….d _i(n+1–₎

Priority P_i4 – to minimize d_i1+

*since the rate per wagon for materials are different, hence weight should be attached to priority 3

Thus the objective function of the problem is:

• Minimize Z_i =  P_i1 d_i1^–   +   P_i2 d_i12⁺   +  P_i3  S (R _in  d_i(n+1^–₎)   +   P_i4 d_i1⁺

       Subject to the constraints;

                            Sx_ik  + d_i1 ^– –  d_i1 ⁺   = W_i

                             x_i1 + d_i2^–  =  Q_i1

                             x_i2 + d_i3^–  =  Q_i2

                             ………….

                            x_in  +  d_i(n+1^– =  Q_in

                            d_i1⁺ +  d_i12^–   –  d_i13⁺  = w _i

And    x_i1 , x_i2 , ……x_in , d_i1^– , d_i2^– , …..d_in^–  ,d_i(n+1^–₎ d_i1⁺ , d_i12^– , d_i13⁺  ≥ 0

Hence the problem is formed in integer linear programming problem.

RESULT ANALYSIS

One specific result with the problem is detailed here. Data obtained for Dhanbad Division, in which there are three loading areas – Jharia area, Barkakana area, Chopan area.

Freight loading (in million tones) =  193.90

Freight loading (in tones) Avg / day = 7441

Revenue earnings (in cr) = 26,688.63

From Freight operation information system (FOIS), we have

Now, from our proposed model, we formulate the problem in ILPP problem and solving by GP programming and simplex method,

For train 1 data contains:

W – Holding capacity of freight train = 20 wagons

A₁ – Cement tiles

A₂ – Iron or Steel slab

Fwe₁ (permissible carrying capacity of commodity A₁) = 65 tones /wagon

Fwe₂ (permissible carrying capacity of commodity A₂) = 63 tones /wagon

Q₁ (holding capacity of commodity A₁) =651 tones

Q₂ (holding capacity of commodity A₂) = 634 tones

R₁ (train load freight rate profit from A₁) = 34 Rs

R₂ (train load freight rate profit from A₂) = 27 Rs

w  (expandable no. of rakes/wagons) = 5

Now by introducing slack variables and priority factors ;

i.e.,  (d₁ ^–) = underutilization of holding capacity

        (d₁ ⁺  )  = overutilization of holding capacity

        (d₂^– )  = underachievement of holding capacity of commodity A₁

        (d₃^–  ) = underachievement of holding capacity of commodity A₂

The objective function is;

Minimize Z = P₁ (d₁^– )  +   P₂ (d₁₂⁺ )   +  P₃  ( 34 d₂^– + 27 d₃^–)    +   P₄ (d₁⁺ )

Subject to constraints:

                                         X ₁ + x ₂ + (d₁ ^–) – (d₁ ⁺) = 20

                                           65 x ₁ + d₂^– = 651

                                           63 x₂ + d₃^– = 634

                                         (d ₁⁺)  + (d₁₂^–) – (d₁₃⁺) = 5

                        And    x₁ , x₂ , d₁^– , d₁ ⁺ , d₂^– , d₃^– , d₁₂^– , d₁₃⁺  ≥ 0

         Solving this step by step through simplex method

Taking   x1 = 0 =x2 = d₁ ⁺ = d₁₃⁺

We get    d₁ ^– = 20

                 d ₂^– =  651

                 d ₃^– =  634

                 d ₁₂^– = 5

And by formulation of the initial table we proceed to solve this.

At the last table we see that all goals are achieved except priority goal P₃.

 Also X₁ = 10, x₂ = 10

We get     d₁ ^–= 0, d₁₂^– = 0, d₁⁺ = 0

                 d₁₃^– = 0, d₁₂^– = 5

The Z values are;

P₁ = 0, P₂ = 0, P₄ = 0,

P₃ = 34 d₂^– + 27 d₃^– = 108

i.e., d₃^– = 4

Hence, there is an underachievement of holding capacity of material A2.

So, the freight rate for commodity A₁ (cement tiles) =  2,17,462 Rs

For commodity A₂ (iron or steel slab) = 3,26,151 Rs which are less than the indicative freight rates.

Also, the freight train i^th can carry another (634-630) = 4 tonnes extra by its holding capacity of cement tiles, which increases the revenue of freight train transportation system.

CONCLUSION

The model suggests a low-cost plan for rearrangement of goods in freight trains, which is beneficial for both the customers and the freight train management system. Future studies can focus on developing an integrated model for both bulk and non-bulk commodities. Also, the model can be tested further with diverse problem instances and bigger problem sizes. This includes coordinating with freight train management and customers to synchronize orders and bundles shipment.

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