Developing a mathematical model to optimize foodgrains storage and transport for public distribution system in India

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1 Developng a mathematcal model to optmze foodgrans storage and transport for publc dstrbuton system n Inda Ajnkya Tanksale*, J.K. Jha Department of Industral and Systems Engneerng, Indan Insttute of Technology Kharagpur West Bengal, *e-mal: ajnkya.tank@gmal.com Abstract Food Corporaton of Inda (FCI) s a nodal agency responsble for procurement, storage and movement of the foodgrans across the states n Inda. Movement of the foodgrans between the states s the key functonal area due to varaton n procurement and demand levels. However, n many nstances, mproper plannng and schedulng of the movement of the foodgrans results n unlfted foodgrans, excess transportaton cost and underutlzaton of avalable storage capacty, whch would n turn ncreases operatonal cost of FCI. In order to cater the need of streamlnng the process of foodgran dstrbuton, ths paper tres to answer the questons- When and how much foodgrans are to be transported from one state to other to meet the demand of foodgrans of each state. In ths paper, an LP based mathematcal model s formulated to mnmze the storage and transportaton cost of foodgrans. The model s solved wth Cplex optmzaton studo and the results are presented whch would be helpful to FCI n producng the monthly movement plans. Key words- Publc Dstrbuton System, foodgrans, storage, transportaton, FCI, supply chan Introducton Inda has acheved self-suffcency n the agrcultural sector, wth enormous efforts to mprovse, wtnessng Green revoluton after ndependence. Today, Inda s a leadng country n the world n producton of major agrcultural crops, area under rrgaton, yeld of the crops etc. Inda holds one sxth stock of the wheat and rce of the total stock of the world ( Lke many other countres n the world, to ensure the avalablty of foodgrans to the needy elements of socety and to acheve food securty, Inda has adopted a model of publc dstrbuton system (PDS). Ths system started durng Brtsh regme n Inda and got modfed from tme to tme after ndependence. Today, Inda has the largest PDS of ts knd n the world wth a chan of over a half mllon Far Prce Shops servng to more than sxty fve mllon poor households (Saxena N. C., 2010). In spte of these facts, performance measures lke Global Hunger Index (GHI), starvaton, malnutrton, under nutrton among chldren are not satsfactory n Inda. Accordng to the GHI report 2013, Inda s at 63 rd poston out of 78 under developed and developng countres wth a GHI score of 21.3 and there s 17.5% undernourshed populaton n the country. On the contrary, 1

2 as much as 1.94 lakh MT of foodgran was wasted due to nadequate storage facltes, n transt losses and mproper management by FCI n Inda durng whch worth bllons of rupees ( Many studes have been conducted wth the context of Indan PDS and food securty ssue n Inda (Jha et al. 2013; Kumar et al. 2007; Qu et al. 1999; Ramaswam and Balakrshnan 2002; Saxena N. C. 2010; Zhou and Wan 2006). These studes have ponted out that mproper polces and operatonal neffcences are the root causes behnd the poor performance and nonachevement of targets of Indan PDS. Ths s one of the motvatons of the present study to nvestgate the storage and transportaton ssues of foodgrans n Indan PDS. Movement of the foodgrans between the states s the key functonal area due to varaton n procurement and demand levels. However, n many nstances, mproper plannng and schedulng of the movement of the foodgrans results n unlfted foodgrans, excess transportaton cost and underutlzaton of avalable storage capacty, whch would n turn ncreases operatonal cost of FCI. In order to cater the need of streamlnng the process of foodgran dstrbuton, ths paper tres to answer the questons- When and how much foodgrans are to be transported from one state to other to meet the demand of foodgrans of each state. The problem consdered n the present study s closely related wth the lterature fallng under the doman of nventory dstrbuton or nventory transportaton problem (Coelho et al. 2014; Kang and Km 2010; Qu et al. 1999; Wang and Berman 2006). The problem addressed n the lterature consders transportaton of the tems between warehouse and retaler; whereas, the present study assumes transportaton of foodgrans between warehouses of dfferent states n Inda. The objectve functon and nature of the problem dffers accordngly. Our work s smlar to a study of optmzaton of storage and transportaton of wheat n Iran by Asgar et al.(2013). However, there are many dfferentatng features. They have developed a model whle consderng only one foodgran (wheat) and our proposed model s a generc mult-tem, multperod n nature. They have consdered a sngle warehouse n each state whereas multple warehouses n each of the states are consdered n our model. The other dfferences n structure of our proposed model and that of Asgar et al. (2013), arses due to dfferences n foodgran management practce n the respectve countres. Rest of the artcle s organzed as follows. Secton 2 descrbes the problem. Formulaton and descrpton of the proposed mathematcal model s gven n Secton 3. Secton 4 hghlghts the computatonal results of proposed soluton procedure and secton 5 concludes the paper. The problem background Food Corporaton of Inda (FCI) s the central nodal agency, whch undertakes the actvtes of procurement, storage and transportaton of the foodgrans for the publc dstrbuton system (PDS) n Inda. There are many foodgrans lke wheat, rce, mllet, cereals and other commodtes lke sugar, edble ol etc. whch are made avalable to the benefcares through PDS; however, the dstrbuton of wheat and rce are of the major concerns for FCI and the entre PDS. The procurement of wheat and rce s carred out by FCI wth the help of state government agences for the central pool of foodgrans (stock of the foodgrans whch are procured by central and state governments and mantaned for publc dstrbuton and varous welfare schemes) n two marketng seasons. Usually, wheat s procured n Kharf marketng season (KMS), Aprl to 2

3 July and rce s procured n Rab marketng season, October to March. Major wheat producng states are Punjab, Haryana, Madhya Pradesh and Uttar Pradesh, whle those produces rce are Punjab, Andhra Pradesh, Haryana, Chhattsgarh, Odsha, West Bengal etc. Most of these states are located n northern part of Inda. Last fve year s average procurement of wheat and rce carred out by FCI for the central pool s lakh MT and lakh MT, respectvely, whch s lakh MT n total. FCI has the major challenge of storng the procured foodgrans due to lmted storage capacty. The storage capacty avalable wth FCI can be broadly categorzed nto three types whch ncludes covered storage (warehouses, depots and slos), covered and plnth (CAP) (an ndgenous method of storng foodgrans n open covered wth plastc) and storage n open space. Due to nadequate self-owned capacty of storage space, FCI has to hre addtonal storage capacty from state governments, central warehousng corporatons (CWC), state warehousng corporatons (SWC) and prvate partes. The all-nclusve storage capacty avalable for central pool wth FCI s lakh MT ( As ths capacty s not suffcent to store the procured foodgrans (average lakh MT), foodgrans has to be stored n open whch s the promnent reason behnd the wastage of foodgrans. Due to mbalance n the levels of procurement and demand of foodgrans n the states over the tme perods and to avod the stranng of storage capactes n procurng states, FCI has to undertake movement of foodgrans from the warehouses n surplus states (stock of the foodgrans n the state s suffcent to satsfy the demand of that state) to the warehouses n defct states (stock of the foodgrans s nadequate to satsfy the demand of that state). Dependng upon the number of benefcares n each state and off-take (actual quantty of foodgrans consumed by benefcares n a state) of the last three years of the state, central government decdes the monthly allotment level of the foodgrans to the respectve state, whch s consdered as the estmated demand for the state. Then, consderng the demand and current stock level of foodgrans n each state, FCI plans movement of foodgrans from surplus state to defct states whch wll be referred as nter-state movement. More than 90% of the total nterstate movement s carred out by ralways. Road and waterways are the consdered as mode of transportaton only when ral network s not avalable. FCI coordnates and montors the monthly movement actvty wth ralway authortes to ensure the avalablty of foodgrans at defct states at the rght tme. As per avalable data ( on average of 20,00,000 bags of foodgrans each of weght 50 kg are transported every day from the procurng areas to the consumng areas, by ral, road, nland waterways etc. coverng an average dstance of 1500 km. Thus effcent transportaton of the movement s another challenge for PDS. Procurement of foodgrans takes place at varous procurement centers or mandes located n the state. The procured quantty of foodgrans durng a month needs to be allocated and stored n the warehouses located n that state. So, wthout loss of generalty, the warehouses located n the states can be assumed as the procurement ponts and as the cost ncurred n movement of the foodgrans from procurement ponts to the warehouses n a state are small as compared to the nter-state movement costs, the movement costs wthn a state are neglected n proposed model. The problem of mantanng nventores of foodgrans at varous warehouses n each state and nter-state movement of foodgrans arses manly due to followng two reasons. One reason s the demand constrant and the other s tradeoffs between nventory holdng costs and transportaton 3

4 costs. Holdng costs of foodgrans represents rents, nternal handlng charges, mantenance costs and overhead charges. Holdng costs at the FCI owned warehouses are lower compared to hred warehouses. Holdng costs at the open type of storage are consdered very hgh and treated as the penalty charges to reduce the storage of foodgrans n open to mnmze the losses. The transportaton cost represents the ral freght charges that also nclude the movement and handlng between warehouse and ralway sdng, truck rents, charges pad for temporary storages etc. In order to reach to the consumers/benefcares, the foodgrans stored at the warehouses are further channelzed to the dstrct and block levels by the respectve state governments. Therefore, warehouses located n the state represent the potental demand ponts and hence ntrastate movement s not consdered n ths model. In ths artcle, we have developed a mathematcal model for the problem of storage of the foodgrans at varous types of warehouses n each state and the nter-state movement of foodgrans between the warehouses, over a multperod tme horzon faced by FCI for the PDS n Inda. Ths model s based on followng assumptons. Assumptons 1. Demand of foodgrans n a state durng a perod s fulflled at the begnnng of the perod usng the stock avalable n the warehouse located n that state. 2. Procurement and transportaton of foodgrans s carred out at the end of the perod and procured foodgrans at the end of the perod s used to satsfy the demand for subsequent perods. 3. Demand of foodgrans n a state durng a perod s met completely wthout any shortages. 4. For the sake of convenence, several warehouses of the same type n a state are assumed as one warehouse. 5. The mode of transportaton of foodgrans s ral. It s assumed that, suffcent number of rakes and other supportng servces (labors, trucks, ralway sdngs etc.) requred for transportng the foodgrans are avalable n each perod. Mathematcal model Sets S The set of states u v W ( W ) The set of warehouses located n state uv () F The set of foodgrans T The set of perods Indces u, v Indces for states uv, S ( v) u v, j Indces for warehouses ( j ) located n state uv (), W u,( j v ) ( W v ) f Index for foodgran, f F t Index for perod, t T = {1, 2,..., T} 4

5 Input parameters h Holdng cost per perod of foodgran f n warehouse located n state u f R Transportaton cost per unt of the foodgran per unt dstance transported from ( v) j warehouse located n state u to warehouse j ( v) located n state v, R ( ) ( ) = R ( ) ( ), ft D u u v Demand of foodgran f n state u durng perod t u v v u j j cap ( u ) Storage capacty of warehouse located n state u d u v Dstance between warehouse and warehouse j ( v) located n states u and v, j respectvely. Procurement quantty of foodgran f n state u durng perod t ft Pu f,0 I ( u ) Inventory of foodgran f at warehouse located n state u at the begnnng of the f f frst perod ( t = 1), ( u,0,1 f,0 u I ) Du, f, u;, I cap, u. Decson varables ft ( u ) u u W I Inventory of foodgran f at warehouse located n state u at the end of perod t f F Q ft Quantty of foodgran ( v) f transported from warehouse to warehouse j states u and v, respectvely, durng perod t ( v) j located n ft λ ( u ) Quantty of foodgran f used to fulfll the demand of state u from warehouse located n state u durng perod t ft π ( u ) Quantty of procured foodgran f stored at warehouse located n state u durng perod t Objectve functon Mnmze h I + R d Q (1) f ft ft u v j u v j ( v) j t T u S u u W f F t T u S u u W v S v v j W f F j Constrants Flow balance equaton, I = I + π + Q Q λ ft f,( t 1) ft ft ft ft ( v) j ( v) j v Sv, u v v j W v Sv, u v v j W Demand constrant t, f, u,. (2) 5

6 ft f,( t 1) I ( u ) D + u, t 1,2,..., N 1; u, f. u u W = (3) Storage capacty I + π + Q λ cap f,( t 1) ft ft ft ( v) j f F f F f F v S, v u ( v) v j W f F, t, u,. (4) Procured quantty conservaton ft ft π ( u ) = P u, t, f, u. u W (5) Supply from each warehouse ft ft λ ( u ) = D u, u W t, f, u. (6) Non-negatvty constrant ( v) I, Q, λ, π 0, t, f, uv,,, j. (7) ft ft ft ft ( v) j Objectve functon (1) represents mnmzaton of the sum of the total holdng cost of the foodgrans stored n all the warehouses of all the states and the transportaton cost ncurred to move the foodgrans between the warehouses of dfferent states (nter-state transportaton cost), over all the perods. Constrant (2) represents the commodty flow balance equaton or flow conservaton equaton. It states that, net nventory of the foodgran n a warehouse located n a state at the end of a perod s equal to the nventory of the foodgran at the end of prevous perod plus procured quantty of the foodgran stored n the warehouse durng that perod and movement of foodgrans to the warehouse from rest of the warehouses n other states (net nflow) mnus the quantty of foodgrans used to satsfy the demand of the state n that perod and movement of foodgrans from that warehouse to the warehouses n other states (net outflow). In smple words the net nventory of a foodgran at the end of a perod n a warehouse s the result of the net nflow and net outflow on the nventory level at the end of prevous perod. Constrant (3) ensures that the total nventory of the foodgran n a state at the end of a perod s suffcent enough to satsfy the demand of next perod. Constrant (4) ensures that at any pont of tme total nventory level of foodgrans n a warehouse should not exceeds the storage capacty of the warehouse. Constrant (5) ensures that all the procured quantty of the foodgrans n a state durng a perod s stored n any of the warehouse n that state. Smlarly, constrant (6) ensures that, the demand of a state durng a perod must be satsfed from the nventory of foodgrans kept at the warehouses n that state at the begnnng of that perod. Constrant (7) mposes the non-negatvty requrements on all the decson varables. 6

7 Results and dscusson To solve the proposed model, we have bult a test problem consderng twenty three major states n Inda. Two promnent foodgrans, wheat and rce are consdered. Length of each perod s taken as one month and the problem s bult for a plannng horzon of twelve months (Aprl to March). The secondary data related to procurement and demand of the foodgrans n each month for each state s collected for twelve months (Aprl 2013 to March 2014). Data requred for other nput parameters lke storage capacty of warehouses n each state, nventory holdng costs and transportaton costs, ntal stock levels of foodgrans etc. s obtaned from varous sources lke foodgran bulletns publshed by Department of food and publc dstrbuton, Natonal PDS regster and studyng varous documents and reports related to pubc dstrbuton from FCI and state government s cvl supples sectons. Broadly three types of the warehouses are consdered n each of the state -FCI/State owned warehouse, FCI/State hred warehouse and open type of warehouse (whch has a suffcently large capacty to hold the foodgrans) representng the entre storage capacty of the state. The problem s solved usng IBM ILOG Cplex optmzaton studo, V12.5 wth Cplex OPL, on a computer wth Intel core 2 duo 2.20 GHz processor, 3.0 GB RAM. Wth the consdered test problem of two foodgrans type, 23 states wth three warehouses n each state and 12 months, the total number of varables and constrants s 119,233 and 4140, respectvely. However, Cplex solver s found to be effcent whch solves ths problem n less than one mnute. We obtan the status of nventory of wheat and rce at the end of each month n each of the warehouse of the states, movement plan of these foodgrans between the warehouses n each month, allocaton of procured quantty of foodgrans to the warehouses of the procurng state and the quantty of foodgrans suppled from a warehouse to meet the demand of a state each month. Results obtaned are partally presented n Table 1-3. Table 1- Inventory of wheat and rce at the end of each perod n the warehouses of state Punjab Inventory status Inventory status State Month warehouse Wheat Rce Total Month warehouse Wheat Rce Total Punjab Aprl w January w w w w w Total nventory at the end of Aprl Total nventory at the end of January May w February w w w w w Total nventory at the end of May Total nventory at the end of February June w March w w w w w Total nventory at the end of June Total nventory at the end of March *All fgures are n lakh MT 7

8 Table 2 - Movement plan of the foodgrans for the month of Aprl From To Foodgran type Quantty moved ( lakh MT) From To Foodgran type Quantty moved ( lakh MT) State WH* State WH State WH* State WH Punjab w1 Bhar w1 Wheat 2.50 Punjab w3 Hmachal w1 Wheat 0.04 Rce 8.89 Pradesh Rce 0.07 Uttarakhand w1 Wheat 0.32 w2 Wheat 0.20 w2 Rce 0.03 Rce 0.21 w2 Delh w2 Rce 1.36 Madhya w1 Rce 0.87 w3 Wheat 3.78 Pradesh w3 Rce Uttarakhand w3 Wheat 0.27 Jammu w3 Wheat 0.19 Kashmr w3 Maharashtra w3 Rce 6.88 Rce 1.33 Taml Nadu w1 Rce 9.63 Uttar w2 Bhar w3 Wheat 1.48 Pradesh w2 Rce 4.85 w3 Chhattsgarh w1 Rce 3.98 Gujarat w1 Wheat 5.16 Rce 0.54 Madhya w3 Maharashtra w1 Wheat Pradesh w2 Wheat 5.08 Rce 5.24 Uttar Pradesh w1 Wheat w2 Wheat 7.76 w2 Wheat 7.31 Rce 4.55 Rce 4.92 w3 Rce 0.60 *WH- warehouse Table 3 - Quantty of foodgrans suppled to satsfy demand of the state and Procurement quantty stored n the warehouses of the states for the month of Aprl State WH* Supply Procurement_storage State WH* Supply Procurement_storage Wheat Rce Wheat Rce Wheat Rce Wheat Rce Punjab w w Taml w Nadu w Maharash tra w w w West w Bengal w w w w *WH- warehouse ** All quanttes are n Lakh MT Table 1 represents the nventory status of the foodgrans at the end of each month at dfferent warehouses n one of the state, Punjab. Smlar results are obtaned for rest of twenty two states. It s an observaton that n all the states, FCI or state owned warehouses got the prorty n storng the foodgrans at the end of each month subject to the avalablty of the 8

9 storage space. In most of the cases, open type of warehouses hold zero nventores at the end of the month because of very hgh nventory holdng costs. Inventory level of a warehouse s changed n each perod due to allocaton of procured foodgrans, movement of foodgrans between the warehouses or supply from the warehouse to meet demand of the state. These all transactons are presented n Table 2 and Table 3. Table 2 represents nter-state movement of foodgrans and procurement allocaton and supply of foodgrans from warehouses to meet demand of a state s gven n Table 3 for month of Aprl Results obtaned regardng procurement allocatons, satsfyng demand for a perod and movement of foodgrans supports assumptons 1 and 2. Therefore these assumptons are requred to have mnmum total cost. Wth the gven demand and procurement data of all the states, ths model can generate the annual foodgran storage and movement plan rght at the begnnng of the year. However, ths modelng approach has certan lmtatons. As procurement of foodgrans s hghly seasonal and depends upon the producton of foodgrans whch s hghly weather dependent n Indan scenaro, the exact forecast of procurement s not possble n advance and may lead to errors. On the other hand, solvng ths problem as a seres of sngle perod problem leads to suboptmal results. We have carred out an analyss of effect of length of the plannng horzon on the total cost. It s observed that when plannng horzon s reduced from twelve months to one month, the total cost ncreases by 73.3% whch s llustrated n Fg. 1. The complexty of the problem wll ncrease sgnfcantly f the number of states and warehouse n each state s ncreased. Therefore t would be nterestng to develop a heurstcs soluton procedure to solve the resultng problem whch wll be consdered for our future study. Total Cost (n Mllon INR) months 6 months 3 months 1 month Plannng horzon Fgure 1 - Change n Total cost wth plannng horzons 9

10 Conclusons In ths paper, we have formulated a mathematcal model for storage and transportaton management of foodgrans n Inda for publc dstrbuton system. The objectve of the model s to mnmze the total cost for nventory storage and monthly nterstate movement plans. From the results obtaned t s observed that the demand of foodgrans n a state durng a perod s fulflled at the begnnng of that perod and procurement and transportaton of foodgrans s carred out at the end of the perod to acheve mnmum total cost. Ths model also represents an example of nventory dstrbuton polcy decson wth a movement of tems between the warehouses located at dfferent regons and at same herarchal level. The presented model produced good results for the test problem; however, need of a powerful heurstcs to handle the complextes of the model s hghlghted here. References Asgar, N., Farahan, R. Z., Rashd-Bajgan, H., Sajadeh, M. S Developng model-based software to optmse wheat storage and transportaton: A real-world applcaton. Appled Soft Computng 13(2): Coelho, L. C., Cordeau, J., Laporte, G Thrty Years of Inventory Routng. Transportaton Scence 48(1): Jha, R., Gaha, R., Pandey, M. K., Kacker, N Food subsdy, ncome transfer and the poor: A comparatve analyss of the publc dstrbuton system n Inda s states. Journal of Polcy Modelng 35(6): Kang, J. H., Km, Y. D Coordnaton of nventory and transportaton managements n a two-level supply chan. Internatonal Journal of Producton Economcs 123(1): Kumar, A. G., Gulat, A., Cummng Jr., R Foodgrans Polcy and Management n Inda Respondng to Today s Challenges and Opportuntes. Indra Gandh Insttute of Development Research workng paper PP-056, Mumba. Qu, W. W., Bookbnder, J. H., Iyogun, P An ntegrated nventory and transportaton system wth modfed perodc polcy for multple products. Europan Journal of Operatonal Research 115(1999): Ramaswam, B., Balakrshnan, P Food prces and the effcency of publc nterventon: The case of the publc dstrbuton system n Inda. Food Polcy 27(5-6): Saxena N. C Publc Dstrbuton System n Inda A few suggestons, Avalable at (Accessed date September 20, 2014). Wang, Q., Berman, O Inbound Logstc Plannng : Mnmzng Transportaton and Inventory Cost Inbound Logstc Plannng : Mnmzng Transportaton and Inventory Cost. Transportaton Scence 40(3): Zhou, Z., Wan, G The Publc Dstrbuton Systems of Foodgrans and Implcatons for Food Securty A Comparson of the Experences of Inda and Chna. World Insttute for development Economcs Research Research paper No. 2006/98, Unted Natons Unversty. 10