A combined tactical and operational deterministic food grain transportation model: Particle swarm based optimization approach

A combined tactical and operational deterministic food grain transportation model: Particle swarm based optimization approach

Accepted Manuscript A combined tactical and operational deterministic food grain transportation model: Particle swarm based optimization approach Lohi...

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Accepted Manuscript A combined tactical and operational deterministic food grain transportation model: Particle swarm based optimization approach Lohithaksha M. Maiyar, Jitesh J. Thakkar PII: DOI: Reference:

S0360-8352(17)30227-9 http://dx.doi.org/10.1016/j.cie.2017.05.023 CAIE 4754

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

16 June 2016 25 February 2017 19 May 2017

Please cite this article as: Maiyar, L.M., Thakkar, J.J., A combined tactical and operational deterministic food grain transportation model: Particle swarm based optimization approach, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/j.cie.2017.05.023

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A combined tactical and operational deterministic food grain transportation model: Particle swarm based optimization approach Lohithaksha M Maiyar*, Jitesh J Thakkar Department of Industrial & Systems Engineering, Indian Institute of Technology Kharagpur, Kharagpur, WestBengal, India-721302, [email protected], +919800145081 * Corresponding author

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A combined tactical and operational deterministic food grain transportation model: Particle swarm based optimization approach This paper proposes a combined tactical and operational two stage food grain transportation model with linear formulation in the first stage and a mixed-integer nonlinear problem (MINLP) in the second stage taking the case of India. Transportation cost is minimized in both stages to fulfil a deterministic demand. First and the second stages correspond to the movement of food grains in between state and central level warehouses respectively. A novel k-parameter based method of constraint handling has been proposed. Further the two stage MINLP formulation newly incorporates vehicle capacity constraints and proposes a generic metric for measuring vehicle utilization. First stage is solved by CPLEX and for the second stage two population based random search techniques: Particle swarm optimization-composite particle (PSOCP) and PSO, have been employed. Experimentations on 10 different problem sets reveal that PSOCP performs marginally better than PSO with lesser standard deviation of global fitness and better solution quality with slightly higher CPU time. Later, sensitivity analysis is conducted on all ten problem sets and a decision support framework is proposed to assist potential stakeholders.

Keywords: deterministic demand; MINLP; food grain transportation; supply network optimization; swarm intelligence; PSOCP

1. Introduction Facts reveal that more than 50 % of the food grains are wasted (CAG report, 2013) in India annually, the reasons for which are mainly related to improper transportation planning, uneconomic utilization of resources, untimely deliveries, improper rolling stock management

and

mismatched demand-produce scenario. Effective transportation

management helps to minimize wastages and improve the overall cost across the supply

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chain especially in developing economies (Song et al. 2014). Integrating the strategic, tactical and operational levels of planning contributes to improved reliability, flexibility and sustainability (Steadieseifi et al., 2014). The combined tactical and operational transportation planning problem addressed in this paper extends the work done by Maiyar et al. (2015) while newly incorporating vehicle capacity constraints and proposes a generic metric for measuring overall vehicle utilization. The intra-state transportation problem addressed in this paper has been formulated in two stages: Intra-state state agency transportation stage and Intra-state FCI transportation stage. The first stage constitutes a linear model minimizing the transportation costs subject to demand, supply and warehouse capacity constraints. The resulting unmet demands from the Stage 1 problem are provided as input to the Stage 2. The second stage is formulated as non-linear mixed integer model that minimizes the transportation cost subject to several contextual operational constraints in addition to the demand, supply and vehicle capacity constraints. This paragraph briefs the contributions from this paper to literature. Firstly a novel mathematical model to represent the food grain transportation system has been developed and validated. Secondly an attempt to integrate the tactical and operational planning phases has been made by way of introducing vehicle capacity and mode selection constraints in the two-stage formulation along with tactical flow decisions to be made. A generic metric to measure the vehicle utilization has been introduced. Thirdly the concept of particle swarm optimization – composite particle along with traditional PSO has been translated and applied to food grain transportation domain. Finally, a brief sensitivity analysis has been conducted to understand the responsiveness of the proposed model and methodology.

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The remainder of the paper is organized as follows. The second section gives a brief overview of literature studied. Section 3 describes the problem environment. The mathematical model with objective function and constraints is described in section 4. Section 5 illustrates the importance and implementation of PSO and PSOCP for the proposed problem. The description of datasets and plan of computational experiments are discussed in section 6. Section 7 discusses results along with sensitivity analysis. A decision support framework is presented in section 8, before concluding the work in the ninth section.

2. Literature Review In the context of supply chain, transportation problems dealing with food grain commodities is limited in literature. Asgari et al., (2013) minimize transportation and storage for a case of wheat production in Iran. Hong & An (2008) modelled a Beijing based grain supply chain and reasons for inefficiency in supply chain transportation were discussed. A functional clustering based technique was used by Ng & Lam, (2014) to optimize supply networks and ensure proper allocation of industrial resources. A detailed review of planning models in agri-food supply chain context can be seen in Ahumada & Villalobos, (2009). The same authors in the subsequent years (Ahumada et al., 2012; Ahumada & Villalobos, 2011) explored modeling intricacies while dealing with harvest and distribution of perishable agricultural products. Thakur et al. (2010) focussed on bulk grain operations and suggested useful strategies to improve handling by balancing traceability and cost aspects. A recent work in bulk grain movement with major focus on silo operations was conducted by Mogale et al., (2017). Etemadnia et al. (2015) developed a mathematical model for fruit and vegetable supply chain considering travel distance, hub capacity and transportation cost. Strategic and tactical operations for crude oil supply chain were captured by Sahebi et al. (2014). A Brazil based Soyabean supply chain was studied

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by Reis & Leal (2015) and investigated the importance of temporal and spatial decisions in the presence of deterministic demand. Gunnarsson et al. (2004) developed a heuristic to solve large mixed integer linear programming model in an attempt to address transportation issues emerging in moving forest fuel. Availability of multiple transport modes gives rise to cost and priority concerns with respect to mode selection while dealing with multi-modal and intermodal transportation problems (Steadieseifi et al., 2014). A mode selection problem with vehicle capacity and lead time constraints was dealt by Eskigun et al., (2005). In network optimization problems, the focus of strategic planning models is to design a superficial network structure with decisions pertaining to continuous or discrete flows, binary node-to-node allocations, and nodal capacities. (Ahn et al., 2015; Elhedhli & Gzara, 2008). At the tactical level, ensuring optimal allocation and utilization of vehicle resources, proper facility planning and effective rolling stock management are important (Yang et al. 2012; De Meyer et al. 2015; Fodstad et al. 2015). Sahebi et al. (2014) portrayed a review on different strategic and tactical decisions relevant to crude-oil supply chains. As we approach operational planning, problems pertaining to choice of transport mode, addressing delivery delay and loading/unloading operations, scheduling, time-tabling, dispatching problems and multi-period problems are well recognized in literature ( D’Ariano et al. 2007; Upadhyay & Bolia, 2014; Goetschalckx et al. 2002). A critical review on all three levels of planning is found in Schmidt & Wilhelm, (2000). It can be seen from the literature presented that transportation related issues have been addressed less significantly in the context of food grain supply chain. Further in this area, an integrated tactical and operational models to imitate the practical scenario of operations are scarce. Hence in this paper it is attempted to address this gap by introducing certain industrially relevant novel operational constraints to achieve minimized cost of

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transportation while deciding on warehouse level movement quantities and binary rail-road allocations in food grain context. The detailed description of the problem environment, solution methodology adapted, experiments, results and conclusions drawn are presented in the subsequent sections. 3. Problem Description The tactical and operational transportation network design problem studied in this paper deals mainly with two players: Food Corporation of India (FCI) and the state. In every state, there are a finite number of regions and in each region there are finite number of warehouses. Food grains are stored in two types of warehouses: state owned warehouses and FCI owned warehouses. A state tries to satisfy the demand of its own warehouses as the first priority from the available stock at state warehouses. Fig. 1 shows a bird’s eye view of the problem environment. Stage1 deals with the movement between state warehouses and the transportation is carried out by only road. Later, the region wise unsatisfied demand, if any, is communicated to the FCI who further plan the intra FCI movement to fulfil the unmet demand of food grains in the stage 2. A vehicle is also allowed to travel from warehouse of one region to a warehouse of another region if there exists a route. It is illustrated in Fig. 1 shown by the arrow going from region 3 to region 1. All the directed arrows in the figure

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Fig. 1. Bird’s eye view of the problem environment

represent single mode of transport movement between the warehouses. In this paper, FCI is allowed to use both rail and road modes of transport to facilitate the food grain shipments, however not simultaneously. The detailed description of the mathematical model developed with objective function and constraints are presented in the next section.

4. Mathematical model The mathematical formulation of the two stage problem is described in this section. The first stage minimizes the total cost of transportation of food grains between the state warehouses and the second stage minimizes the total cost of transportation of food grains between the FCI warehouses. Considering the complexity in modelling the real life scenario following six assumptions are made (1) Number and classification of regions according to FCI is same as number and classification of regions according to the state. (2) The stock available at the FCI warehouses of the state is sufficient to satisfy the demand of the state.

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(3) At-least one FCI warehouse is present per region. (4) Seasonal fluctuations are neglected. (5) The model is solved for single time period and single food grain commodity. (6) Demands are deterministic in nature. The mathematical framework including the indices, sets and different types of variables considered are listed first and the detailed descriptions of objective function and constraints are discussed later. Various notations used to describe the indices and sets required for modelling are shown in Table 1 where as, Table 2 gives the definition of all relevant parameter notations. The first stage consists of single set of continuous decision variables Table 1. Table of sets and indices Index

Name Region Origin region Destination region FCI Origin state warehouse Destination state warehouse Origin FCI warehouse Destination of FCI Warehouse Rail Road Definition Set of state warehouses in region r

r p

q f i j m n u v Set

Wrs R Wrf

Set of regions Set of FCI warehouses in region r

jq denoted by xip which represents quantity shipped from state warehouse i of region p to

state warehouse j of region q , where, p, q  R , i, j Wrs . After the completion of first stage, the remaining quantity to be satisfied at each warehouse is calculated using Eq. (1) , where  j which represents remaining demand of food grains to be satisfied at state r

warehouse j of region r , where j Wrs , r  R . It is to be noted that Eq. (1) is rewritten as Eq. (2) by reversing notations p and q to calculate the demand deficits at destination

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warehouses, instead of the origin warehouses. It is amply clear that both these equations give identical values for  j , but two different representations are adopted because, Eq. (1) r

holds good for available stock constraints while Eq. (2) holds good for demand constraints. 

R Wqs   p p  j   D j  ( xiqjp  S jp )  , j, p q 1 i 1  

(1)

+

R W ps    qj   D qj  ( xipjq  S qj )  , j, q p 1 i 1  

(2)

Table 2. Table of parameters Parameter jq ip

C

nq Cmpu

Definition Cost of transportation by road from state warehouse i of region p to state warehouse j of region q where p, q  R , i, j Wrs Cost of transportation by rail from FCI warehouse

m of region p to FCI

n of region q where p, q  R and m, n Wrf Cost of transportation by road from FCI warehouse m of region p to FCI warehouse

C

nq mpv

warehouse n of region q where p, q  R and m, n Wrf

d ipjq

Distance by road between state warehouse i of region p and state warehouse

nq d mpu

Distance by rail between FCI warehouse

d

nq mpv

j of region q where p, q  R , i, j Wrs

m of region p and FCI warehouse

n of region q where p, q  R and m, n Wrf Distance by road between state warehouse

m of region p and state

Sir

n of region q where p, q  R and m, n Wrf Demand of food grains at state warehouse i of region r where i Wrs , r  R Current stock of food grains at state warehouse i of region r where

S mr

Current stock of food grains at FCI warehouse m of region

K ir

Capacity of state warehouse i of region

warehouse

Dir

i Wrs , r  R

r where

m Wrf , r  R

K mr

r where i Wrs , r  R Capacity of FCI warehouse m of region r where m Wrf , r  R

 'rj

Remaining demand of food grains to be satisfied at state warehouse j of



r i



region r after subtracting the current stock where j Wrs , r  R Surplus quantity available at state warehouse i of region r to be transported to other regions, where i Wrs , r  R Minimum weight of food grains required to consider loading into rake

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Cu Cv

Capacity of each rake

p lmu

Number of rakes available at FCI warehouse

p lmv

Number of trucks available at FCI warehouse m of region p where

Capacity of each truck m of region, p where

i Wps , p  R i Wps , p  R = 1 if

k

R Wrs

R Wrs

r 1 i 1

r 1 i 1

 Dir   Sir

= 0 otherwise. nq nq nq The second stage comprises of 3 sets of decision variables: ymp , amp , and bmp . The

nq continuous variable ymp represents the quantity shipped from FCI warehouse m of region nq nq p to FCI warehouse n of region q , where, p, q  R and m, n Wrf . amp and bmp are binary

decision variables that are assigned one when rail is selected and when road is selected nq respectively to transport ymp amount of food grains from FCI warehouse m of region p to

FCI warehouse n of region q . nq nq Here, we describe the importance of three other sets of variables,  p , lmp and tmp apart

from the 3 sets of decision variables defined earlier. The continuous variable  p stores the surplus quantity available at FCI in region p to be transported to other regions, where

p  R , and is calculated using Eq. (3). The use of this variable can be seen in Eq. (12).

+

W pf  Wpf  p     Sm    jp  ,p j 1  m1  p

nq nq and tmp are defined as follows: lmp

nq lmp

= 1,

nq ymp 

= 0, otherwise, p, q  R , m, n Wrf nq tmp

= 1,

nq ymp 0

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(3)

= 0, otherwise, p, q  R , m, n Wrf Finally, here we introduce the objective function and constraints specific to each stage of the tactical decision making problem. Eqs. (4)-(9) show the mathematical formulation for stage 1 while the rest of the Eqs. (10)-(23) correspond to stage 2 problem. The first stage minimizes a linear transportation cost as shown in Eq. (4). It can be noticed that writing cost and distance parameters separately would slightly increase the memory requirements but we choose to keep it separate in line with the interests of the end-user. The objective function is written as,

R Wqs

Minimize

R W ps

 C q 1 j 1 p 1 i 1

jq jq ip ip

x dipjq

(4)

subject to, R W ps

 x







 k 'qj , j ,q, where,  'qj  D qj  S qj

jq ip

 i p

p 1 i 1

R Wqs

 x q 1 j 1



jq ip

, i,p, where, i p  Sip  Dip

R W ps

 x p 1 i 1

jq ip

 S qj  K qj , j ,q

+

+

, j , q

(5)

, i, p

(6)

(7)

xipip  0 , i,p

(8)

xipjq  0, i, j,p, q

(9)

Eqs. (5), (6) and (7) are demand, available stock and warehouse capacity constraints respectively written for every warehouse i of region p. Demand constraints ensure that quantity of food grains required at state warehouse j of region q is satisfied either from the stock available at warehouse j of same region or by moving excess stock from state

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warehouse i of nearby region p. Here, the parameter k determines whether the state is facing a deficit or surplus. If the total demand of the state is less than or equal to the total stock the state is facing surplus with k=1 and the right hand side of the demand constraints remain positive. On the contrary, if demand is greater than stock with k=0, the right hand side must practically vanish as the demand of a deficit state cannot anyways be fully satisfied. In such a case the movement is driven by the excess stock available at the warehouse (Eq. (6)). Eq. (8) ensures there is no transport within a warehouse and finally Eq. (9) represents non-negativity constraints. As mentioned earlier the formulation for the second stage is shown by Eqs. (10)-(23). The minimization objective function in Eq. (10) comprises of two parts: one representing the transportation cost if rail is selected as a mode of transport and other for road. The selection of mode is driven by minimum cost considerations and few operational constraints. By simultaneously considering the flow decisions, rail-road allocations and by respecting the minimum load requirement constraints tactical planning meets operational planning in this work. R Wqf

R Wpf

R Wqf

Minimize  C q 1 n 1 p 1 m 1

nq mpu

nq mp

y d

R Wpf

nq nq nq nq a l   Cmpv ymp d mpvbmp

nq nq nq mpu mp mp

(10)

q 1 n 1 p 1 m 1

subject to, Wqf

R W pf

 y

Wqf

nq mp

n 1 p 1 m 1

W pf

S m 1

R Wqf

 y m 1 q 1 n 1

R W pf

 y p 1 m 1

nq mp

nq mp

Wqs

q m

   qj ,q,

(11)

j 1

  p ,p,

 Snq  K nq , n,q

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(12)

(13)

nq nq nq amp  bmp  tmp , m, n, p, q

(14)

nq nq nq bmp  tmp (1  lmp ), m, n, p, q

(15)

nq  max(0, ymp  )  nq lmp   nq  max(0, ymp   )  1 

t

nq mp

nq  ymp    nq   ymp  1 

R Wqf

 y q 1 n 1

nq mp

C l

p u mu

m, n, p, q

Wqf

a n 1

m, n, p, q

nq mp

C l

p v mv

Wqf

b n 1

nq mp

(16)

(17)

m, p

(18)

mp amp  0 , m,p

(19)

mp bmp  0 , m,p

(20)

mp ymp  0 , m,p

(21)

nq ymp  0, m,n,p, q nq nq nq nq amp 0,1 , bmp 0,1, lmp 0,1, tmp 0,1

(22)

m,n,p, q

(23)

Following a similar approach as in stage 1, the first three set of constraints represented by Eqs. (11), (12) and (13) are again demand, available stock and warehouse capacity constraints. It can be seen that in the first stage demand and available stock constraints are written for each warehouse and in the second stage these are written for each region. This is because the supply deficit (unmet demand) is communicated to FCI by the state authorities on a regional basis. The next four constraints represented by Eqs. (14), (15), (16) & (17) and (18) are specific to operational considerations in this model. Eqs. (14) and (15) are rail-road selection constraints. They ensure that both the modes of transport are not chosen simultaneously. Eq. (16) takes care of minimum loading tonnage requirements before selecting rail as a

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mode of transport. The possibility of having empty transport is eliminated with the help of Eq. (17). Eq. (18) ensures that the quantity transported is within vehicle capacity restrictions. Eqs. (19), (20) and (21) make default allocations to make sure variables representing transport between same warehouses take zero values. The last two Eqs. (22) and (23) represent non-negativity and binary integer constraints respectively. It is to be noted that Eq. (5) presents a novel way of representing the demand constraint for the mismatched demand-supply scenario. Also, Eqs. (14), (15), (16), (17) and (18) are newly incorporated to capture the operational aspects of rail-road allocation and vehicle capacities in this model.

5. Solution methodology The first stage network optimization problem as discussed earlier constitutes a linear formulation which can be solved using CPLEX or any other linear programming tool. In this case IBM ILOG CPLEX Concert-version 12.6 has been used to solve the first stage formulation. The outputs of the first stage are taken as input to the second stage. Since the second stage involves a NP-hard mixed integer non-linear programming problem it is computationally economical to solve the second stage problem using a meta-heuristic. The solution methodology for the second stage is discussed in further detail in following parts of this section. Researchers in evolutionary and soft computation have brought various advancements to already existing methods in order to adopt to increasing demands for solving complex real life problems in various domains. The problem formulated in the second stage is a network design problem with a third degree objective function and non-linear constraints. Zhao et al. (2016), has shown that a simple network design problem is NP hard. Further, due to presence of non-linear constraints the model classifies itself to be a constrained non-linear

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NP-hard optimization problem. Due to lack of exact approaches to deal NP hard problems, it becomes imperative to solve the problem using random search techniques. Joines & Houck, (1994) carried out a pioneer work in addressing constrained non-linear NP-hard problems using GA. Chen & Dai, (2016) implemented exact penalty method to deal with non-linear constrained optimization. Exact methods are yet to be well- explored for NPhard constrained non-linear problems. Zhu et al. (2009) employed particle swarm optimization as a solution approach for dealing with NP-hard vehicle routing problems in grain logistics. Later, Zhao et al. (2011) employed hybrid PSO to solve a food grain transportation model. However, in all the cases PSO and their variants suffer from immature convergence. An advanced version of PSO was developed by Liu et al. (2010) which was successful in overcoming immature convergence by incorporating the concept of ‘composite particle’ derived from physics and proved its superiority over other state of the art evolutionary algorithms and PSO variants. However, benefits of PSOCP over PSO in food grain context are yet to be explored. Hence, in this paper we intend to tackle the second stage problem using PSOCP and compare it with PSO. The details of both the algorithms and their adaption to the current problem are discussed in the further subsections. 5.1 Particle swarm optimization (PSO) Particle swarm optimization (Eberhart & Kennedy, 1995), is a swarm based stochastic optimization technique that explores the complex solution space in social and cognitive dimensions. For a large variety of problems PSO is known to have performed better than most traditional optimization techniques (Guo et al., 2009; Tseng & Liao, 2008). For a given swarm of dimension d, the velocity and position update rules for the (k  1)th particle is given by, vkd1  vkd  c1r1dk ( pbestkd  xkd )  c2 r2dk ( gbestkd  xkd )

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(25)

xkd1  xkd  vkd1

(26)

where,  , c1 , and c2 are inertia, cognitive and social weights respectively. The random numbers r1 and r2 are uniformly distributed in [0, 1]. PSO is known for its exploitation ability but suffers from weak exploration capability. To overcome this lacuna, PSOCP is designed with new operators and mutation schemes to improve the exploration dimension. 5.2 Particle swarm optimization - Composite particle (PSOCP) PSOCP is an extended variant of PSO that inherits benefits of incorporating the composite particle principle developed by Liu et al. (2010) for better exploration. A composite is a substance that comprises of two or more pure forms of chemical elements having better characteristics than the parent elements (Nakahata et al. 1997). Analogously in swarm optimization, each particle is subjected to certain series of transformations in each iteration to form a composite particle aimed to contain better solution. It is characterized by the following key features: (1) Internal dissimilarity and compatibility, internal action and concerted action are the attributes that govern the selection of a composite particle. The initial composite particle is generated by using a worst particle and two equidistant random particles from the existing population. A population size of N generates (N-1)/3 number of composite particles and (N-(N-1)/3) independent particles. The best elementary particle is selected as the pioneer particle in each iteration. (2) Each composite particle is subjected to the reflection and scattering operator to generate a new composite particle. The reflection operator reflects the current position of the worst particle of the composite to a new position using Eq. (27), provided the worst particle of the new composite is fitter than the old particle. The

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Velocity-Anisotropic Reflection (VAR) scheme helps to define the interaction and sustain the dimensional diversity between the particles.

PR  PM  Rstep   PM

(27)

where, Rstep is the reflection step size for the worst particle to move from point M towards  WM . In a D-dimensional space  is the VAR vector such that,

m n  

(28)

where, m, n 1, 2,3,..., D , and i, j /  m   n . In Eq. (28),  is the maximum difference in reflection velocities to be maintained. (3) The scattering operator diversifies the current solution and multiplies the possibility of arriving at newer optimums. The fittest particle is scattered to two different points as per Eq. (29)

FSi    Ni F , i  1, 2

(29)

where, S1 and S 2 are scattered points in the direction of randomly selected nonpioneer particles N1 and N 2 , and  is the random vector that assumes values between [Smin, Smax], where S min and Smax are the minimum and maximum scattering step size. (4) An integral movement scheme is employed. It transfers the velocity of the pioneer particles to the elementary particles for proper guidance. It ensures that all the particles are moving in the right direction by preserving the valuable information. Finally, the position and velocity are updated using Eqs. (25) and (26). For detailed explanation and relevant proofs with respect to PSOCP, readers are advised to refer Liu et al. (2010). The algorithm has been accordingly translated to match the current scenario. 17

Fig. 2 shows the various interactions between the different kinds of particles present in PSOCP. 5.3 Adaption of PSO and PSOCP to food grain transportation problem 5.3.1 Particle representation The solution string in swarm intelligence techniques is represented by a particle. The structure of the basic particle resembles the diagram shown in Fig. 3 where V1, V2, and so on represent the decision variables. The dimension of each particle is equal to the total number of decision variables present in the problem instance. For example, in the problem instance 1 (illustrated in section 6) each particle stores 36 x 3 number of decision variables (second stage problem) in 108 cells with first 36 representing quantity variable and the next

Fig. 2. Inter-particle interaction and evolution in PSOCP 72 representing the rail- road binary decision variables. The locus of the particle is guided by the position and velocity vectors which are iteratively updated. 5.3.2 Initial feasible solution generation Generation of an initial feasible solution greatly effects the problem solving time (Huang and Yao 2008). In this paper, all the instances are run for 10 times and the initial feasible

18

solutions are respectively generated. Each population is given 10 different starting points to maintain solution diversity right from the beginning. Undesirable values are eliminated using the ‘xmod’ operator. Boundary condition handling, default allocations, and binary rounding are the main features handled by the operator. 5.3.3 Boundary condition handling All the decision variables are operated within upper and lower bounds. The boundary limits are chosen suitably within a specified range for all instances. The continuous variables falling

Fig. 3. Structure of particle

outside the respective bounds are directed to the feasible region using the absorption principle as described Eq (24) for both continuous and discrete variables. For binary d variables, the values for xlbd and xub are taken as 0 and 1 respectively. The same expression

is responsible for maintaining non-negativity of all variables. d d d   xlb , if xold  xlb d xnew  d d d   xub , if xold  xub

(24)

5.3.4 Default allocations This feature allows the default allocations to be activated before each particle is passed for fitness evaluation. The handling of constraints represented by Eqs. (19), (20), (21) is a part of default allocation procedure. Variable starting point allocations are also made while dealing the binary and continuous default allocations. 5.3.5 Binary rounding

19

Particle swarm optimization and their variants originally deal with continuous variables. The random search nature of the algorithms may cause certain binary variables to lie in the range (0, 1) after updating the particle. Binary rounding ensures that such variables are rounded to binary integers using Eq. (29) d new

x

d  0, if 0  xold  0.5  d  1, if 0.5  xold  1

(29)

5.3.6 Parameter tuning As discussed earlier, the parameters for PSO and PSOCP are tuned by executing the program for several runs before the start of computational analysis. Initially, the starting particle velocities are randomly generated and are modified according to size of the instance. The velocity is set at lower range for low size instances to make sure the solution does not become infeasible, and at higher range for larger instances to kick start the exploration process.

20

Fig. 4. Step-wise flow diagram for implementation of PSOCP Table 3. Algorithmic settings Algorithm

Npop

w

c1

c2

Smin

Smax

Rstep



PSOCP PSO

200 200

0.9 0.9

0.1 0.3

0.98 0.95

2 -

3 -

6 -

0.5 -

21

Inertia weight (  ) and accelerating coefficients (social weight ( c1 ) and the cognitive weight ( c2 )),common to PSO and PSOCP are selected based on the minimum cost obtained over 10 trial runs. Similar procedure is adopted to determine inter-particle Euclidian distance limit for PSOCP. Population size for both algorithms is carefully chosen to adjust the trade-off between execution time and convergence rate.

The stretching and

diversification parameters of PSOCP are taken from Liu et al. (2010). The best values obtained after parameter tuning of the two algorithms, for the given problem are shown in Table 3. Fig. 4 shows the flow of steps involved in implementation of the algorithm to the current problem. The next section describes the variety of problem sets adopted to carry out the computational study.

6. Computational Experiments The computational experiments for this work are conducted on 10 problem sets covering 5 different problem sizes with two instances in each category. The number of variables and regions occurring in a particular instance are taken as basis for determining the problem size. The smallest and largest category consist of 3 and 15 regions with 189 and 7371 number of variables respectively including both the stages. The region-wise warehouse distribution for each problem set and corresponding number of variables are provided in Table 4. All instances are run on Windows 8, 64-bit Operating System consisting of 8 GB RAM and Intel Core i7 1.8 GHz processor. IBM ILOG CPLEX Concert-version 12.6 is used to solve the Stage 1 linear programming problem whereas Stage 2 problem is solved in MATLAB environment.

22

The value of parameter k is explicitly provided to the program prior to the start of execution. The total number of decision variables in each stage is a function of total number of

Table 4. Region wise warehouse distribution for Stage 1 and Stage 2

Stage 1

Stage 2

No. of Regions 3 5 10 13 15 3 5 10 13 15

No. of variables 81 289 1225 1156 1296 108 675 3888 4800 6075

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

2 3 1 1 1 1 2 1 1 1

3 2 3 3 3 3 3 3 3 3

4 4 4 4 3 2 1 2 2 2

3 3 3 3 5 5 3 3

5 4 4 4 4 4 4 4

3 3 3 3 3 3

6 3 2 5 5 5

4 4 4 6 4 4

2 2 2 4 4 4

5 2 2 3 3 3

2 2 2 2

1 1 2 2

2 2 4 4

2 3

2 2

warehouses present and is given by, zi (tnwi )2 , where zi is the number of type of variables present in stage i , and tnwi is the total number of warehouses present in stage i . Reiterating that this is a single commodity food grain transportation model, rice has been taken as a sample commodity. The same set of experiments can be carried out for the case of wheat and other food grain commodities. This formulation is restricted to single commodity type of transport with more focus on transportation network design decisions excluding inventory considerations. The first stage solutions are solved to optimality and the second stage solutions are found to converge at near optimal solutions.

7. Results and Discussions In this section, we summarize the outcome of experiments relevant to Stage 1 and Stage 2. Warehouse-wise demands, available stock at the state warehouses, distance matrix, and unit cost estimates are preliminary inputs to the Stage 1 model.

The first stage solution

includes, region wise-unmet demands, optimal global fitness values and their

23

corresponding mean execution times obtained for each problem set. Table 5 presents global fitness values, shipment quantities and the execution times for the stage 1 model. Table 6 shows the unmet demand at each region p which is evaluated by summation of unmet demands at warehouse Table 5. Optimal global fitness for Stage 1 problem Problem set

Regions

1 2 3 4 5 6 7 8 9 10

3 3 5 5 10 10 13 13 15 15

Optimal Global Fitness (Rs) 10000 100016.570 169039.200 195074.400 124624.300 163957.844 38958.6420 102400.917 26478.121 62081.437

CPU Time (s) 14.26 14.88 17.32 16.29 62.15 60.47 178.22 179.56 284.33 296.98

level (  jp ) over the set of warehouses present in region p ( W pf ).

In Stage 2, the region-wise unmet demand, available stock, available capacity of vehicle resources, and inter-warehouse distances are given as inputs. The near-optimal quantity of food grains to be shipped from FCI warehouse m of region p to FCI warehouse n of region q and corresponding rail allocations are reported in Table 7 for each of the problem sets. Table 8 specifies the best, average and worst global fitness values (Total transportation cost + Table 6. Region-wise unmet demand of food grains after executing Stage 1 model. Region 1 2 3 4 5 6

Region wise unmet demand of rice (MT) for each problem set 1 100 400 150

2 140.03 344.97 184.99

3 80.96 142.42 0.00 61.79 96.36

4 151.61 124.94 0.00 29.20 102.61

5 100.00 23.00 229.40 10.00 30.00 0.00

24

6 129.52 0.00 217.28 14.72 39.94 70.00

7 129.87 307.29 305.30 93.55 126.95 95.66

8 136.78 303.39 309.61 40.39 80.51 48.57

9 159.87 409.76 439.89 119.53 189.96 159.97

10 116.55 240.76 251.77 28.99 54.55 25.93

7 8 9 10 11 12 13 14 15 *MT- Metric Ton

542.60 169.39 65.00 257.00

548.86 166.01 64.94 285.52

90.80 126.78 75.71 58.46 72.66 31.88 109.68

11.78 40.04 33.89 51.87 21.10 30.00 96.30

99.68 199.58 99.98 85.64 45.89 22.79 45.79 86.95 108.96

5.93 13.47 5.91 29.80 37.86 14.73 15.92 32.98 13.05

nq nq Table 7. Inter-FCI warehouse shipment quantities ( ymp ) and rail allocation ( amp )

mpnq

nq ymp

Problem Set 1 1112 50 1132 50 1312 50 Problem Set 2 1112 5.9011 1132 10.0969 1322 31.9984 2322 6.9743 Problem Set 3 1221 50 2221 5.9557 Problem Set 4 1145 0.0049 2144 0.0001 1324 0.0055 1415 0.0298 2411 42.8056 2413 0.0445 3412 40.4189 3425 0.0394 4445 0.0105 4413 0.0246 1511 50.0000 1524 0.0032 1521 0.2276 2515 0.1927 2535 0.0003 3513 0.0283 3515 0.1121

nq amp

1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0

mpnq

nq ymp

Problem Set 4 4524 0.0003 Problem Set 5 3211 50.0000 3223 36.0989 543(10) 50.0000 4538 0.0003 4513 50.0000 4547 5.1158 6837 2.4851 6813 3.3057 681(10) 47.0000 Problem Set 6 2213 50.0000 241(10) 33.8746 4426 0.0035 4511 29.5488 1623 9.2648 181(10) 41.6526 183(10) 50.0000 3813 18.0872 3817 1.6329 6811 0.0407 2911 50.0000 4917 12.5010 Problem Set 7 1113 4.0946 1411 0.0021 1422 4.4594 2512 0.1054

nq amp

0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 0

mpnq

nq ymp

Problem Set 7 3637 0.0093 4613 1.6252 4627 0.0006 1947 0.0173 3932 8.8509 Problem Set 8 1411 30.4172 1423 50.0000 2411 0.3810 3422 50.0000 3424 0.0206 2557 0.1806 1612 50.0000 1723 19.6118 3723 50.0000 373(13) 0.0001 283(10) 0.0199 4822 23.4131 4847 0.0278 492(11) 0.0191 Problem Set 9 3211 0.0004 2414 50.0000 3411 8.0983 343(14) 0.0222 1513 50.0000 1523 50.0000 1557 0.0004 151(13) 2.2583

25

nq amp

0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1

mpnq

nq ymp

Problem Set 9 3513 27.5889 4532 50.0000 2623 2.3503 4728 0.0162 5724 0.0140 2832 1.9795 482(10) 0.0003 2932 42.0564 3932 14.4186 2(10)11 50.0000 2(10)32 3.4951 2(10)47 0.0136 2(10)3(10) 0.0156 2(11)2(13) 0.0103 1(13)37 4.3116 3(13)45 0.0218 1(14)22 0.0009 2(14)11 21.8183 2(14)45 0.0800 3(14)23 50.0000 Problem Set 10 2222 1.8749 1426 1.6848 3445 0.0017 173(14) 0.0167 3723 17.8295 4729 0.0471 4827 7.9040 3914 0.0001

nq amp

1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 0

4521 4532

0.0014 0.0014

0 0

1613 1623

42.2765 1 1.6095 1

2532 7.8834 254(13) 0.0101

1 0

2(11)15 2(11)28

0.0028 0.0503

Penalty cost) for the Stage 2 problem averaged over 10 trails. Table 8 also shows the estimated standard deviation and the observed computational time required for both the algorithms to converge. It is observed that with slightly higher CPU time, PSOCP gives better result (lower standard deviation) than PSO for all the instances. The resulting convergence Fig. (a) Convergence Graph for Problem Set 1

Fig. (b) Convergence Graph for Problem Set 2

Fig. (c) Convergence Graph for Problem Set 3

Fig. (d) Convergence Graph for Problem Set 4

Fig. 5: Convergence Graph of Stage 2 for problem sets with 3 and 5 regions Table 8. Summary of results for problem sets 1 to 10. Problem set 1 2 3 4

Algorithm

Best Global Fitness

PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP

428211.6 427536.8 112725 95893.5 82243.8 82114.6 166418.4 159401.2

Average Global Fitness 429950.5 428702.0 117962.7 97053.1 83539.2 83133.5 167671.7 160386.6

26

Worst Global Fitness 431835.6 429354.8 119374.0 97916.5 85777.8 84610.6 170114.4 162224.2

Standard Deviation

CPU Time (s)

1194.0107 546.5847 1907.0810 618.4617 1002.7499 727.2237 1163.5970 784.1179

12.2834 20.4365 12.021 19.8815 104.619 130.241 105.8867 129.5612

0 0

5 6 7 8 9 10

PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP

358288.2 358254.2 586040.4 383218.0 106744.3 58668.1 479509.9 204500.0 604147.6 571650.1 110696.2 23600.0

359291.4 359273.2 587259.4 384114.3 109003.6 59501.2 481690.8 205431.7 606013.7 572483.6 112830.6 24738.0

361554.2 360637.2 588753.4 385673.0 110237.3 61357.1 483654.9 206318.0 607389.6 573790.1 114737.2 26299.0

931.6465 710.2647 1008.2852 740.2928 1122.1999 753.9856 1248.2308 628.8980 1033.6621 686.4003 1304.4551 700.2041

344.2737 412.924 338.4416 410.1861 512.3625 581.7724 509.095 570.113 677.8122 754.6232 694.0051 783.1182

Fig. (a) Convergence Graph for Problem Set 5

Fig. (b) Convergence Graph for Problem Set 6

Fig. (c) Convergence Graph for Problem Set 7

Fig. (d) Convergence Graph for Problem Set 8

Fig. (e) Convergence graph for Problem Set 9

Fig (f) Convergence Graph for Problem set 10

Fig. 6: Convergence Graph of Stage 2 for problem sets with 10, 13 and 15 regions

27

graphs are shown in Figs. 5 and 6. During the first few iterations Fig. 5 (a), PSO converges faster than PSOCP. Here, the scattering operator used by PSOCP allows to jump across large domain of solution space witnessed by the sudden drop of global fitness in between iterations 6 and 11. It can be seen that PSO converges to a local optimum in between iterations 11 and 16, whereas PSOCP continues to search for global optimum guided by its scattering operator and integral movement principle. Similar behaviour is observed in other cases. The managerial implication of penalty cost could be intuitively interpreted as cost incurred for pilferage, theft and improper planning. It can be seen from tables 7 and 8 that cost increases drastically with increasing size of problem due to explosion in number of nodes and arcs in the transportation network. So a small percentage of reduction in transportation cost amounts to significant savings. However, we stick the focus of this paper to showcase the development, solution methodology and behaviour of the model and algorithm under different problem environments. 7.1 Sensitivity Analysis In this section we carry out sensitivity analysis by altering the problem environment under two scenarios to visualize the effects of these changes on the performance of the model and the algorithms. 7.1.1 Effect of maximum number of iterations on quality of convergence Though it is a known fact that executing the algorithm for higher number of iterations would improve the quality of solution, it is important to observe the problem specific performance of the algorithm. Keeping this in mind, both the algorithms are executed for three levels of maximum iteration limit, 75,150 and 300 respectively on all the problem sets. The average of difference in global fitness for last 10 iterations is considered to quantitatively determine the quality of convergence at each level (Table 9). The average

28

difference in consecutive solutions was found to be very high at 75 iterations for majority of the problem sets. While, reaching 150 iterations, the difference was observed to be zero for the first, third and fourth problem sets and significantly high for other instances. Finally, at 300 iterations the difference in global fitness was found to be low for all the problem sets. The highest level below which the difference in average global fitness is nonzero considering both algorithms is noted as the optimal

Table 9. Maximum iteration size limit recommendation for problem sets 1 to 10 Problem set 1 2 3 4 5 6 7 8 9 10

Algorithm PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP PSO PSOCP

Average Global Fitness difference for last 10 iterations Level 1 Level 2 Level 3 515.8235 0 0 1028.091 0 0 0 0 0 8.62 26.484 0 252.3458 0 0 8963.493 0 0 387.663 0 0 0 0 0 18347.5 126.7969 0 9502.455 8.92 0 5211.07 1757.38 0 8200.14 121.162 0 11280.7 1110.47 0 6059.87 42.502 0 13355.4 1089.37 0 9880 710 0 13871.1 3206.93 0 34217.1 1711.03 0 12886.5 93.5021 0 11830 1600 0

Recommendation (Level) 2 2 3 3 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3

maximum iteration size level. Table 9 shows the optimal recommendations. Thus adhering to these would ensure equal justice to storage memory, computational time and solution quality. 7.2.2 Effect of minimum loading requirement (theta) on percentage utilization of rakes

29

The level of minimum loading requirement plays a key role in selection of mode of transport. In this subsection we look forward to understand how percentage utilization of rakes varies with changing levels of minimum loading requirement. For this purpose a dimensionless metric, known as overall vehicle utilization factor ( ) has been proposed and is given by Eq. (24). According to this formula,  depends on three factors: number of empty rakes available per region, capacity of the vehicles and quantity to be transported. The correlation of  with these factors is evident from the mathematical relation. However, it is not evident how  and  are functionally related. Hence they are plotted on cartesian coordinates as shown in Fig. 7 for the ten problem sets respectively. R Wqf

Number of trucks/rakes used   Number of trucks/rakes available

R W pf

  (

nq nq amp ymp

Cu

q 1 n 1 p 1 m 1

R W pf

  (l p 1 m 1

p mu



nq nq bmp ymp

Cv

) (24)

l ) p mv

As shown in Fig. 7, the vehicle utilization factor has been plotted for three levels of minimum loading requirement: low theta (25%), medium theta (50%) and high theta (75%). A simple yet useful observation finds that  and  are positively correlated. As the problem size increases the factor assumes to take a nearly equal value. The peculiar behaviour of problem set 7 may be attributed to the low levels of unmet demands from Stage 1 and consequent less amount of total food grain movement in stage 2. The analysis presented here is for PSOCP and the nature of the result is similar for PSO. However, several other factors that can influence this relation such as varying level of demand, changing vehicle capacity and temporal considerations, remain ignored and can be considered as possible future extensions of this work.

30

Fig. 7. Graph showing positive correlation between minimum loading requirement and overall vehicle utilization factor 8. Decision Support framework In this section a systemic view of the proposed model has been presented with the help of a decision support architecture. The various entities which make up the architecture, inputs and deliverables are summarized in Fig. 8. In the front end the user visualizes the system in three interfaces: the input interface, run-time interface and the out-put interface. The user provides required inputs to the system through input interface and these values get stored in the database. The real-time parameters (or inputs) stored in a data-base management system can be accessed how and when required by the intelligence module. The model and solution methodologies would be embedded in the intelligence module. Run-time interface facilitates the user to visualize the output information in the form of convergence graphs, global fitness values and constraint penalties. The output interface prints the final values of the near optimal solutions including fitness values and decision variables on the screen. All entities are encapsulated into two classes: the front end and the back end, as shown in Fig. 8. The directed arrows represent interdependencies between the entities. The DSS helps in taking real-time quick efficient decisions for FCI and contributes to

31

Fig. 8. The decision support framework

significant savings in time and money. It also contributes to the improved resilience of the food grain supply chain. Real time decisions provided by the DSS are origin-destination routes, inter-warehouse shipment quantity, rail- road allocations between each origindestination pair. Successful implementation of this model would also contribute to reduction in duplication of information, improved traceability and less documentation. The modules can be coded in MATLAB, Java or Python. Some of the challenges encountered while developing the framework are compiler compatibility issues, model complexity, nature and scale of data, etc. However, the choice of technology should balance developer skills and client requirements to minimize overall implementation cost.

9. Conclusion In this study, a two-stage optimization problem with linear formulation in the first stage and mixed integer nonlinear formulation in the second stage has been proposed to address food grain transportation problem in Indian context. An attempt to bridge the gap between tactical and operational levels of transportation planning by considering vehicle

32

capacities, movement quantity decisions, rail-road binary allocations simultaneously was made. A novel k-parameter based constraint handling method has been proposed to deal with the demand-supply mismatch. The first stage model has been solved to optimality by CPLEX and the second stage model has been tackled by the use of two population based random search techniques: PSOCP and PSO. The best, average and worst global fitness values are evaluated for ten different instances and PSOCP is found to have better standard deviation than PSO in terms of solution quality with slightly higher CPU time for all cases. Further, an overall vehicle utilization metric has been proposed for the model. Sensitivity analysis reveals that that the overall vehicle utilization factor and minimum rake loading requirements are always positively correlated. Optimum iteration sizes to be maintained while solving different size of instances were found out. A decision support architecture has been put-forth to assist stakeholders and potential end-users. Further extensions of the formulation may aim to model the problem in dynamic, stochastic and sustainable environments by incorporating stochastic demand, multiple commodities, and multiple periods and individual vehicle utilization metrics.

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Acknowledgements This research is conducted in the context of project titled, ‘Transportation management to implement NFSA (TMI)’ sponsored by Ministry of Human Resource and Development (MHRD), Department of Higher Education, Government of India [sanction letter No. F. NO. 4-25/2013-TS-I].

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The following points summarize highlights of this work: 

A combined tactical and operational mixed-integer non-linear deterministic formulation.



K- parameter based novel constraint handling methodology to deal unique nature of food grain transportation scenario.



A Decision support framework with application of PSOCP and PSO to solve and validate the MINLP is proposed.



A generic metric for measuring overall vehicle utilization has been proposed for this model.



Sensitivity Analysis is carried out to evaluate performance of model and algorithm under diverse conditions.

40