Single Stage Plant-Warehouse Location Problem (SSPWLP): A State of Art Formulation

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(1977) first considered this problem (SSPWLP). Goods flow from plants to warehouses to markets. Here we need to locate plants and warehouses of appropriate capacities so that sum total of location cost of plants and warehouses and distribution cost of goods from plant to warehouses to markets is minimized. They considered normalized decision variables (see Sharma and Muralidhar (2009)). However they used the formulation style of Geoffrion and Graves (1974). We use normalized variables but use the variable style of Sharma (1991) and Sharma and Berry (2007) that reduces the number of variables. We give several strong linking constraints by drawing from the works of Sharma and Namdeo (2005) and Sharma and Berry (2007). Below we give the formulation in brief. Details can be seen in the body of the paper. Variables names are self explanatory and makes understanding the model easier. New Formulation of SSPWLP: Min sum(i,j), cpw(i,j)*xpw(i,j) + sum(j,k), cwm(j,k)*xwm(j,k) Sum(i), yp(i)*fp(i) + sum(j), yw(j)*fw(j) (0) Sum(i,j), xpw(i,j) = 1 (0a) Sum(j,k), xwm(j,k) = 1 (0b) Sum(j), xwm(j,k) >= d(k) for all k (0c) Sum(j), xpw(i,j) <= capp(i) for all i (1) Sum(j), xpw(i,j) <= capp(i)*yp(i) for all i (2) Sum(i), xpw(i,j) <= capw(j) for all j (3) Sum(i), xpw(i,j) <= capw(j)*yw(j) for all j (4) xpw(i,j) <= yp(i)*capw(j) for all i, j (5) xpw(i,j) <= yw(j)*capp(i) for all i,j (6) xwm(j,k) <= d(k)*yw(j) for all j,k (7) Sum(j), xpw(i,j) <= yp(i) for all i (8) Sum(i), xpw(i,j) = 1 (11) Sum(j), capw(j)*yw(j) >= 1 (12) xpw(i,j) >= 0 for all i,j; xwm(j,k) >= 0 for all j,k (13) yp(i) = (0,1) for all i and yw(j) = (0,1) for all j (14) This (the above formulation of SSPWLP) is the best formulation of SSPWLP that is amenable to solution by LP relaxation and attendant branch and bound and/or branch and cut solution procedure. In literature the distribution phase between plant and warehouse (where plant and warehouse are to be located) is solved by Sharma and Agarwal (2014) as MID_CPLP were Lagrangian Relaxation (LR) was deployed to get RHS_CPLP and LHS_CPLP (different classes of capacitated plant location problems) that were attempted by well known relaxations (LR and Linear Programming) already available in literature (see Priyanka Verma and Sharma (2011)). Computational investigation is underway to determine efficacy of different new linking constraints given in this paper. Operations Research Single Stage Plant Warehouse Location Problem Location-Distribution Problem Locating Plant and Warehouses Simultaneously Simple Plant Location Problem and Capacitated Plant Location Problem 1. Introduction Location-distribution problems have lot of varieties: SPLP (simple plant location problem), CPLP (capacitated plant location problem), single stage capacitated warehouse location problem (SSCWLP), and two stage warehouse location problems (see Sharma and Agarwal ( 2014 ) for latest references TSCWLP). In most of these problems plants and warehouses were not located simultaneously. Closest to problem considered in this paper is the problem of 2-stage warehouse location problem where warehouses are located in successive stages (but plant was located already, see Sharma and Namdeo ( 2005 ) and Sharma and Pritee Agarwal et. al. (2012, 2013, 2014, 2016)). For latest in plant-warehouse location theory refer to Sharma ( 2019 , 2019 , 2020 , 2020 , 2021 and 2022 ). Here we describe SPLP, CPLP, SSCWLP and TSCWLP in brief. Problem SPLP is to locate plants of unlimited capacities so that sum total of fixed cost of plants plus distribution cost of plants to markets (so that their demand is met) is minimized. Here plants supply goods directly to markets. Problem CPLP is similar to SPLP but plants have finite capacities. Ordinary decision variables are X(i,k) which denoted the absolute quantity transported from plants to markets. If demand at a market k is D(k) then we have x(i,k) = X(i,k)/sum(k), D(k) and x(i,k) is referred to as normalized distribution variable and it is between 0 and 1. It has several advantages, see Sharma and Muralidhar ( 2009 ) for details. In literature SPLP and CPLP were for single commodity and single period only. In SSCWLP, plants are already located, we need to locate warehouses in between plants and markets. Here goods flow from plants to warehouses to markets. Here if we consider single commodity then ‘normalized’ distribution variables are useful, but for multi commodity case ‘normalized’ distribution variables are not useful. Geoffrion and Graves used the distribution variable Y(i,j,k) to denote absolute quantity of goods transported from plant ‘i’ to warehouse ‘j’ to market ‘k’. this style did not require flow balance constraints (see Geoffrion and Graves ( 1974 ). However Sharma ( 1991 ) used transportation variables as XPW(i,j) (quantity transported from plant ‘i’ to warehouse ‘j’) and XWM(j,k) (quantity transported from warehouse ‘j’ to market ‘k’. it is easy to see that Sharma ( 1991 ) formulation has less decision variables compared to number of decision variables in Geoffrion and Graves ( 1974 ) but require additional flow balance constraints at each of the warehouses. Sharma and Berry ( 2007 ) who gave several ‘strong’ constraints (both supply and demand side) and showed that Sharma ( 1991 ) type formulation was significantly superior to formulation style of Geoffrion and Graves ( 1974 ). Sharma and Berry ( 2007 ) considered only a single product and single period problem; whereas Sharma ( 1991 ) and Geoffrion and Grave (1974) developed models that were capable of considering multi-products and multi-period cases. Geoffrion and Graves ( 1974 ) assumed that whatever came in at a warehouse was moved out and there was no provision of keeping inventory at warehouses in their model. Sharma ( 1991 ) considered a model where inventory was allowed to kept at warehouses (but not at markets) and shortages were not allowed. Sharma ( 2019 , 2019 , 2020 , 2020 , 2021 and 2022 ) have developed models that allow inventory at markets and warehouses and allow shortages at markets. Sharma and Agarwal ( 2014 ) and Sharma and Verma (2011) relaxed flow balance constraints in SSCWLP at warehouses to initiate Lagrangian Relaxation procedure and solved RHS and LHS CPLPs that resulted. In this paper we apply all the advances listed above to the Single Stage Plant Warehouse Location Problem (SSPWLP) and give its most modern formulation below. 2. Problem Formulation of SSPWLP It has been well established in literature (see Sharma and Berry ( 2007 ) that formulation style of Geoffrion and Graves ( 1974 ) is not efficient; and hence that is not given in this paper. We use the style of Sharma ( 1991 ) that was demonstrated to be efficient by Sharma and Berry ( 2007 ). Since it is the case of single commodity, we use the normalized decision variables (see Sharma and Muralidhar ( 2009 ) and Sharma and Berry ( 2007 ) and Kauffman et. al (1977). Index : ‘i’ for plant, ‘j’ for warehouse and ‘k’ for market. Constants of the Problem capp(i) is capacity of plant ‘i’ as a fraction of total market demand; capw(j) is capacity of warehouse ‘j’ as a fraction of total market demand; d(k) demand at market ‘k’ as a fraction of total market demand (d(k) = D(k)/sum(k), D(k)); cpw(i,j) is the cost of transporting sum of all market demand from plant ‘i’ to warehouse ‘j’ and cwm(j,k) is the cost of transporting sum of all market demand from warehouse ‘j’ to market ‘m’. Variables of the Problem xpw(i,j) is the quantity transported from plant ‘i’ to warehouse ‘j’ as a fraction of total market demand (xpw(i,j) = XPW(i,j)/sum(k), D(k)); xwm(j,k) is the quantity transported from warehouse ‘j’ to market ‘k’ as a fraction of total market demand (xwm(j,m) = XWM(j,m)/sum(k), D(k)); yp(i) is location variable for plant ‘i’, it is equal to 1 if plant is located at ‘i’ and 0 otherwise; and yw(j) is location variable for warehouse ‘j’, it is equal to 1 if warehouse is located at ‘j’ and 0 otherwise. Formulated differently, (with normalized xpw(i,j) and xwm(j,k)) Min sum(i,j), cpw(i,j)*xpw(i,j) + sum(j,k), cwm(j,k)*xwm(j,k) + Sum(i), yp(i)*fp(i) + sum(j), yw(j)*fw(j) (0) Sum(i,j), xpw(i,j) = 1 (0a) Sum(j,k), xwm(j,k) = 1 (0b) Sum(j), xwm(j,k) > = d(k) for all k (0c) Sum(j), xpw(i,j) < = capp(i) for all i (1) Sum(j), xpw(i,j) < = capp(i)*yp(i) for all i (2) Sum(i), xpw(i,j) < = capw(j) for all j (3) Sum(i), xpw(i,j) < = capw(j)*yw(j) for all j (4) xpw(i,j) < = yp(i)*capw(j) for all i, j (5) xpw(i,j) < = yw(j)*capp(i) for all i,j (6) xwm(j,k) < = d(k)*yw(j) for all j,k (7) Sum(j), xpw(i,j) < = yp(i) for all i (8) Sum(i), xpw(i,j) = 1 (11) Sum(j), capw(j)*yw(j) > = 1 (12) xpw(i,j) > = 0 for all i,j; xwm(j,k) > = 0 for all j,k (13) yp(i) = (0,1) for all i and yw(j) = (0,1) for all j (14) Equation (0) is sum of cost of location and distribution. Equations (0a), (0b) and (0c) ensure that demand is met at all markets. Eq. (1) ensures flow out of a plant to be within its capacity, Eq. (2) is a strong linking constraint, Eq. (3) ensures that inflow at a warehouse is within its capacity, and Eq. (4) is again a strong linking constraint. Equations (5 ) and (6) are new linking constraints to SSPWLP but are borrowed from Sharma and Namdeo ( 2005 ). Again, equations (7) are strong linking constraints borrowed from literature (Sharma and Berry ( 2007 )). Eqs. (8) and (9) are weak linking constraints, (10) is flow balance constraint at each of the warehouses. Equations (1 1) and (12) ensure that we have an additional constraint to ensure a feasible solution to problem SSPWLP. These are missed in Kaufman et. al. (1977) and Sharma and Berry ( 2007 ); but Priyank Dubey ( 2020 ) established its efficacy. Equations (1 3) force non negativity restriction on transportation variables and equations (14) ensure that location variables are (0,1). Few comments are in order here. Sharma and Namdeo ( 2005 ) did not give constraints (0a), (0b), (11) and (12) in their formulation for 2-stage warehouse location problem; and Sharma and Berry ( 2007 ) forgot to include constraint (12) in their formulation of SSCWLP (single stage capacitated warehouse location problem) and Sharma, Jha and Priyank (2023) (a paper submitted to IEOM 2023 Houston conference) showed that efficacy of (12) in problem SSCWLP was highly significant. We put all these classes of constraints for SSPWLP to give its state-of-art formulation. It is also important to note here that despite stronger LP relaxation bound given by strong constraints ((2) and (7)) compared to LP relaxation bounds given by weak constraints (8) and (9), SSUWLP (single stage un-capacitated warehouse location problem) ran faster with weak constraints than the strong constraints (see Sharma and Verma ( 2012 )). This encouraged Sharma and Verma ( 2012 ) to develop a hybrid formulation (here weak constraints were augmented by few promising strong constraints) whose performance was better than that of weak and strong formulation of SSCWLP. Thus determination of efficacy of each of the linking constraints is important an experimental investigation is underway to determine this. 3. Discussion We give the usefulness of the formulation given above in the form of a table. Table 1 Advantages of New Formulation of SSPWLP given in this paper. SN Advantage 1 New linking constraints (5) and (6) are given that is expected to boost the performance of optimizing algorithms. 2 New feasibility constraints (11) and (12) are expected to boost the performance of optimizing algorithms. 3 We ensure that all strong linking constraints given in Sharma and Berry ( 2007 ) (4 and 7) are included in this formulation. 4 New linking constraint (2) is added to formulation that is expected to boost the performance of optimizing algorithms. We define following models: P1: without (5), (6), (11) and (12). P2: without (11) and (12) And model P3 with all constraints (0a), (0b), (0c) and (1) to (14). An experimental investigation carried out to establish the efficacy of new constraints added to SSPWLP problem in this paper is given below. We solved problems for 50 plants and 50 warehouses. In problem set 1 we solved 25 problems with overcapacity between 125–150%. In problem set 2 we solved 25 problems with overcapacity of about 200%. These problems were solved on Intel(R) Core(TM) i5-7200U CPU @ 2.50GHz processor. 4. Computational Results PROBLEM SET 1 (overcapacity of plant and warehouse between 125%-150%) Salient Result for P1-P2 : Criterion µ p1 µ p2 | t –value | Iterations 9197.12 6161.64 1.553 No. of Nodes 926.76 625.16 1.528 Root Relaxation Solution Time 0.0328 0.0536 7.076 Objective Fn. 40220.2836 40512.5464 0.337 Execution time 0.0236 0.02872 1.059 Salient Result for P2-P3 : Criterion µ p2 µ p3 | t –value | Iterations 6161.64 1792.80 4.704 No. of Nodes 625.16 74.68 5.112 Root Relaxation Solution Time 0.0536 0.0388 5.220 Objective Fn. 40512.5464 40540.9728 0.039 Execution time 0.02872 0.0312 0.18 Salient Result for P1-P3 : Criterion µ p1 µ p3 | t –value | Iterations 9197.12 1792.80 3.695 No. of Nodes 926.76 74.68 4.287 Root Relaxation Solution Time 0.0328 0.0388 1.964 Objective Fn. 40220.2836 40540.9728 0.384 Execution time 0.0236 0.0312 2.69* • Sig at 0.0064 From above table we can say that there is no significant difference in objective function values of models P1, P2 and P3; but P1 takes significantly less execution time compared to P3. Therefore we can solve problem first by using model P1 and then use this advanced start to improve further by using model P3 (GAMS offers such a capability). PROBLEM SET 2 (overcapacity of plant and warehouse 200%) Salient Result for P1-P2 : Criterion µ p1 µ p2 | t –value | Iterations 7464.68 7456.60 0.004 No. of Nodes 631.76 568.80 0.383 Root Relaxation Solution Time 0.0292 0.0556 7.066 Objective Fn. 23365.4344 22921.6592 0.685 Execution time 0.02876 0.02816 1.105 From above table we can say that between P1 and P2 there is no significant difference in terms of number of iterations and no of nodes processed; so we can say that P1 is as good as P2. Salient Result for P2-P3 : Criterion µ p2 µ p3 | t –value | Iterations 7456.60 1274.36 3.352 No. of Nodes 568.80 11.72 4.034 Root Relaxation Solution Time 0.0556 0.0552 0.110 Objective Fn. 22921.6592 22422.4352 0.848 Execution time 0.02816 0.03952 1.551* • Sig at 0.067 Salient Result for P1-P3: Criterion µ p1 µ p3 | t –value | Iterations 7464.68 1274.36 4.577 No. of Nodes 631.76 11.72 5.662 Root Relaxation Solution Time 0.0292 0.0552 8.362 Objective Fn. 23365.4344 22422.4352 1.491 Execution time 0.02876 0.03952 1.551* • Sig at 0.067 From above table we can say that there is no significant difference in objective function values of models P1, P2 and P3 (except that P3 gives sig better objective functions than P1); but P1 takes significantly less execution time compared to P3. Therefore we can solve problem first by using model P1 and then use this advanced start to improve further by using model P3 (GAMS offers such a capability). As more constraints are added (5, 6, 11 and 12) progressively in model P2 and P3 we get better objective function values and at the expense of higher execution time. 5. Conclusion Priyank Dubey ( 2020 ) has established the importance of feasibility constraints such as (11) and (12). Sharma and Verma () have established the importance of Hybrid formulations (weak constraints + most promising strong constraints). Thus we may construct a model P4 (min (0), s.t. (0a) to (4); (8) and (9); (10) to (14) and most promising constraints associated with strong linking constraints such as (5) to (7). It is expected that this model may give best results for large sized problem instances of SSPWLP. This is a topic of future research. Declarations None of the authors received any financial grants for conducting this research. All authors have no competing and no conflict of interest to get this work published. Biographies Prof. RRK Sharma: He is B.E. (mechanical engineering) from VNIT Nagpur India, and PhD in management from I.I.M., Ahmedabad, INDIA. He has nearly three years of experience in automotive companies in India (Tata Motors and TVS-Suzuki). He has 33 years of teaching and research experience at the Department of Industrial and Management Engineering, I.I.T., Kanpur, 208016 INDIA. To date he has written 1217 papers (peer-reviewed (402) /under review (34) / working papers 781 (not referred)). He has developed over ten software products. To date, he has guided 68 M TECH and 23 Ph D theses at I.I.T. Kanpur. He has been Sanjay Mittal Chair Professor at IIT KANPUR (15.09.2015 to 14.09.2018) and is currently a H.A.G. scale professor at I.I.T. Kanpur. In 2015, he received “Membership Award” given by IABE USA (International Academy of Business and Economics). In 2016 he received the “Distinguished Educator Award” from IEOM (Industrial Engineering and Operations Management) Society, U.S.A. In 2021, he received IEOM Distinguished Service Award. In 2019, 2020, 2021 and 2022 he was invited by the Ministry of Human Resources Department, India, to participate in the NIRF rankings survey for management schools in India. In 2019, 2020, 2021, 2022 and 2023 he was invited to participate in the Q.S. ranking exercise for ranking management schools in South Asia in 2019, 2020, 2022 and 2023. He was invited to participate in Times Higher Education Academic survey for world university rankings in 2023. Dr. Vinay Singh: He has earned his Bachelor Degree in engineering (Computer Science and Engineering) from RBS College Agra, Masters in Human Resource Development and Management from IIT Kharagpur and PhD in Management from IIT Kanpur. Currently he is working as Assistant Professor in the department of Management at ABV-Indian Institute of Information Technology and Management Gwalior, India since Nov 2012. So far he has 26 publications in peer review journals to his credit. He has supervised 92 Masters Students and guided 02 PhD theses. He has also earned two national patents in embedded products design and has developed three software packages. He has received 03 research project grants from prestigious agencies of India. Mr. Pushkar Awasthi: He is second year MTECH student in the department of Industrial and Management Engineering, IIT Kanpur 208016 India. References Geoffrion AM and Graves GW, “Multi commodity distribution system design by using Bender’s decomposition”, Management Science, 1974, 20(5), pp. 82-114. Kaufman L, et. al., “A Plant Warehouse Location Problem”, Operational Research Quarterly, 1977, V 28 (3), Part I, pp. 547-554. Namdeo, S., “Developing new strong constraints for the two stage warehouse location problem – With and without restrictions on the arc flow”, Department of Industrial and Management Engineering, Indian Institute of Technology, Kanpur 208016 (completed 2005). Priyank Dubey, ‘Efficacy of Feasibility Constraints in the Context of SSCWLP (Single Stage Capacitated Warehouse Location Problem)’, Department of Industrial and Management Engineering, Indian Institute of Technology, Kanpur 208016 (completed 2020). Thesis supervisor: Prof. RRK Sharma. Sharma, R.R.K., “Modeling a Fertilizer Distribution System”, European Journal of Operational Research, 51, 1991, pp. 24-34. RRK Sharma, “Working Paper Series: Lecture Notes in Management Science: Vol 1”, A collection of 148 working papers, (All Authored by Prof. RRK Sharma). EXCEL PUBLISHERS NEW DELHI, April 2019; p. 149. ISBN: 9-789-388-237116. RRK Sharma, “Working Paper Series: Lecture Notes in Management Science: Vol 2”, A collection of 295 working papers, (All Authored by Prof. RRK Sharma); EXCEL PUBLISHERS NEW DELHI, 2019. Aug 2019; p. 234. ISBN: 9-789-388-237796. RRK Sharma, “Working Paper Series: Lecture Notes in Management Science: Vol 3”, (150 articles are written: All Authored by Prof. RRK Sharma); FEB 2020. ISBN: 978-93-89947-08-3; Mar 2020; p. 156. RRK Sharma, “Working Paper Series: Lecture Notes in Management Science: Vol 4”, (It has 01 article + 4 software: All Authored by Prof. RRK Sharma); (Oct 2020); p. 192. ISBN: 9789389947212. RRK Sharma, “Working Paper Series: Lecture Notes in Management Science: Vol 5”, (It has 139 articles are written: All Authored by Prof. RRK Sharma); ISBN: 978-93-89947-31-1; Jan 2021. RRK Sharma, “Working Paper Series: Lecture Notes in Management Science: Vol 6”, (It has 048 articles are written so far: All Authored by Prof. RRK Sharma); ISBN: 978-93-91355-65-4; May 2022. RRK Sharma, Ankita M, Vimal kr, Vinay Singh and Pritee Agarwal, “Developing modified Benders decomposition method for single stage multi commodity multi period warehouse location problem”, American J of Operations Research, V6; 2016, pp. 245-259. Sharma, RRK, Agarwal, Pritee and Vinay Singh, “BENDERS’ DECOMPOSITION FOR DIFFERENT FORMULATIONS OF SINGLE STAGE CAPACITATED WAREHOUSE LOCATION PROBLEM (SSCWLP): A BRIEF THEORETICAL AND EMPIRICAL INVESTIGATION”, International Journal of Business Research, V 12(1), 2012, pp. 43-50. Sharma, R.R.K. and Berry, V., “Developing New Formulations and Relaxations of Single Stage Capacitated Warehouse Location Problem (SSCWLP): Empirical Investigation for Assessing Relative Strengths and Computational Effort”, European Journal of Operational Research, 2007, V 177, pp. 803-812. IF: 4.2; Cite Score: 8.0+ Sharma, R.R.K. and Muralidhar, A., “A new formulation and relaxation of the simple plant location problem”, Asia Pacific Journal of Operational Research, V 26(1), Feb 2009; pp. 1-11. Sharma, R.R.K. and Namdeo, S., “Two stage capacitated warehouse location problem: Developing new strong constraints”, Proceedings, Fifth International Conference on Operational Research for Development : ICORD V”, held at Jamshedpur, INDIA during Dec. 19-21, 2005, pp. 330-333. RRK Sharma and Pritee Agarwal, “Solving Single Stage Capacitated Warehouse Location Problem (SSCWLP) by Branch and Bound and Benders’ Decomposition Methods: A Comparative Study”, International J of Operations and Quantitative Management, V 19 (3); Sep. 2013; pp. 147-156. Sharma, RRK and Pritee Agarwal, “Solving SSCWLP using Benders’ decomposition: Theoretical and Computational Study for Different Formulations”, International J of Strategic management, V 14 (1); 2014; pp. 35-44. RRK Sharma and Pritee Agarwal, “Approaches to solve MID_CPLP problem: Theoretical results and empirical investigation”, American J of Operational Research, 4, 2014, pp. 142-154. RRK Sharma and Priyanka Verma, “Hybrid Formulations of single stage uncapacitated warehouse location problem: Few theoretical and empirical results”, International Journal of Operations and Quantitative Management, V 18 (1), Mar 2012, pp. 53-69. Priyanka Verma and RRK Sharma, “Vertical Decomposition Approach to solve Single Stage Capacitated Warehouse Location Problem”, American Journal of Operational Research, V 1 (3), 2011, pp. 1-18. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3039168","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":208062653,"identity":"d7775097-6aea-4f86-9490-55bb7ce36bb2","order_by":0,"name":"pushkar awasthi","email":"","orcid":"","institution":"IIT Kanpur","correspondingAuthor":false,"prefix":"","firstName":"pushkar","middleName":"","lastName":"awasthi","suffix":""},{"id":208063280,"identity":"04b7e742-e0a2-4d74-b7b4-7f7f0ac4b196","order_by":1,"name":"rrk sharma","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqElEQVRIiWNgGAWjYFACNgaGDwYSDBJgjgGRWhhnGEhIkKaFmYeBAaqFGGA++1jyZ5sCizrJBuaHHxgK7hDWInMu7Zh0DtBh0gxsxhIMBs8Ia5HgYW9jBmmRY2AwA/rlMFFamj9bgLWwfyNWC9sBaQaww3iItoUtTbLHQEJyZjNPsUQCkVqMP/z4U8cvcbx944cPf4jQggDMQJxAioZRMApGwSgYBbgBAMRLJ+HmiCXfAAAAAElFTkSuQmCC","orcid":"","institution":"IIT Kanpur","correspondingAuthor":true,"prefix":"","firstName":"rrk","middleName":"","lastName":"sharma","suffix":""},{"id":208063281,"identity":"7d753ec7-bdfe-441d-840c-df8b9e40b398","order_by":2,"name":"vinay singh","email":"","orcid":"","institution":"ABV-IIITM Gwalior","correspondingAuthor":false,"prefix":"","firstName":"vinay","middleName":"","lastName":"singh","suffix":""}],"badges":[],"createdAt":"2023-06-08 13:25:03","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":true,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true,"coiExplicitlySet":false},"doi":"10.21203/rs.3.rs-3039168/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3039168/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":38441963,"identity":"992d3256-314a-4e9f-a376-abe1e61092ad","added_by":"auto","created_at":"2023-06-13 06:18:45","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":352730,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3039168/v1/be897f2b-9919-47e9-9327-b556c85307da.pdf"}],"financialInterests":"","formattedTitle":"\u003cp\u003e\u003cstrong\u003eSingle Stage Plant-Warehouse Location Problem (SSPWLP): A State of Art Formulation\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eLocation-distribution problems have lot of varieties: SPLP (simple plant location problem), CPLP (capacitated plant location problem), single stage capacitated warehouse location problem (SSCWLP), and two stage warehouse location problems (see Sharma and Agarwal (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) for latest references TSCWLP). In most of these problems plants and warehouses were not located simultaneously. Closest to problem considered in this paper is the problem of 2-stage warehouse location problem where warehouses are located in successive stages (but plant was located already, see Sharma and Namdeo (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2005\u003c/span\u003e) and Sharma and Pritee Agarwal et. al. (2012, 2013, 2014, 2016)). For latest in plant-warehouse location theory refer to Sharma (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2019\u003c/span\u003e, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2019\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2020\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2020\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2021\u003c/span\u003e and \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eHere we describe SPLP, CPLP, SSCWLP and TSCWLP in brief.\u003c/p\u003e \u003cp\u003eProblem SPLP is to locate plants of unlimited capacities so that sum total of fixed cost of plants plus distribution cost of plants to markets (so that their demand is met) is minimized. Here plants supply goods directly to markets. Problem CPLP is similar to SPLP but plants have finite capacities. Ordinary decision variables are X(i,k) which denoted the absolute quantity transported from plants to markets. If demand at a market k is D(k) then we have x(i,k)\u0026thinsp;=\u0026thinsp;X(i,k)/sum(k), D(k) and x(i,k) is referred to as normalized distribution variable and it is between 0 and 1. It has several advantages, see Sharma and Muralidhar (\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2009\u003c/span\u003e) for details. In literature SPLP and CPLP were for single commodity and single period only.\u003c/p\u003e \u003cp\u003eIn SSCWLP, plants are already located, we need to locate warehouses in between plants and markets. Here goods flow from plants to warehouses to markets. Here if we consider single commodity then \u0026lsquo;normalized\u0026rsquo; distribution variables are useful, but for multi commodity case \u0026lsquo;normalized\u0026rsquo; distribution variables are not useful. Geoffrion and Graves used the distribution variable Y(i,j,k) to denote absolute quantity of goods transported from plant \u0026lsquo;i\u0026rsquo; to warehouse \u0026lsquo;j\u0026rsquo; to market \u0026lsquo;k\u0026rsquo;. this style did not require flow balance constraints (see Geoffrion and Graves (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1974\u003c/span\u003e). However Sharma (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1991\u003c/span\u003e) used transportation variables as XPW(i,j) (quantity transported from plant \u0026lsquo;i\u0026rsquo; to warehouse \u0026lsquo;j\u0026rsquo;) and XWM(j,k) (quantity transported from warehouse \u0026lsquo;j\u0026rsquo; to market \u0026lsquo;k\u0026rsquo;. it is easy to see that Sharma (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1991\u003c/span\u003e) formulation has less decision variables compared to number of decision variables in Geoffrion and Graves (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1974\u003c/span\u003e) but require additional flow balance constraints at each of the warehouses. Sharma and Berry (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e) who gave several \u0026lsquo;strong\u0026rsquo; constraints (both supply and demand side) and showed that Sharma (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1991\u003c/span\u003e) type formulation was significantly superior to formulation style of Geoffrion and Graves (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1974\u003c/span\u003e). Sharma and Berry (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e) considered only a single product and single period problem; whereas Sharma (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1991\u003c/span\u003e) and Geoffrion and Grave (1974) developed models that were capable of considering multi-products and multi-period cases. Geoffrion and Graves (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1974\u003c/span\u003e) assumed that whatever came in at a warehouse was moved out and there was no provision of keeping inventory at warehouses in their model. Sharma (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1991\u003c/span\u003e) considered a model where inventory was allowed to kept at warehouses (but not at markets) and shortages were not allowed. Sharma (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2019\u003c/span\u003e, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2019\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2020\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2020\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2021\u003c/span\u003e and \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) have developed models that allow inventory at markets and warehouses and allow shortages at markets.\u003c/p\u003e \u003cp\u003eSharma and Agarwal (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) and Sharma and Verma (2011) relaxed flow balance constraints in SSCWLP at warehouses to initiate Lagrangian Relaxation procedure and solved RHS and LHS CPLPs that resulted.\u003c/p\u003e \u003cp\u003eIn this paper we apply all the advances listed above to the Single Stage Plant Warehouse Location Problem (SSPWLP) and give its most modern formulation below.\u003c/p\u003e"},{"header":"2. Problem Formulation of SSPWLP","content":"\u003cp\u003eIt has been well established in literature (see Sharma and Berry (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e) that formulation style of Geoffrion and Graves (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1974\u003c/span\u003e) is not efficient; and hence that is not given in this paper. We use the style of Sharma (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1991\u003c/span\u003e) that was demonstrated to be efficient by Sharma and Berry (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). Since it is the case of single commodity, we use the normalized decision variables (see Sharma and Muralidhar (\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2009\u003c/span\u003e) and Sharma and Berry (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e) and Kauffman et. al (1977).\u003c/p\u003e \u003cp\u003e \u003cb\u003eIndex\u003c/b\u003e:\u003c/p\u003e \u003cp\u003e\u0026lsquo;i\u0026rsquo; for plant, \u0026lsquo;j\u0026rsquo; for warehouse and \u0026lsquo;k\u0026rsquo; for market.\u003c/p\u003e \u003cp\u003e \u003cb\u003eConstants of the Problem\u003c/b\u003e \u003c/p\u003e \u003cp\u003ecapp(i) is capacity of plant \u0026lsquo;i\u0026rsquo; as a fraction of total market demand; capw(j) is capacity of warehouse \u0026lsquo;j\u0026rsquo; as a fraction of total market demand; d(k) demand at market \u0026lsquo;k\u0026rsquo; as a fraction of total market demand (d(k)\u0026thinsp;=\u0026thinsp;D(k)/sum(k), D(k)); cpw(i,j) is the cost of transporting sum of all market demand from plant \u0026lsquo;i\u0026rsquo; to warehouse \u0026lsquo;j\u0026rsquo; and cwm(j,k) is the cost of transporting sum of all market demand from warehouse \u0026lsquo;j\u0026rsquo; to market \u0026lsquo;m\u0026rsquo;.\u003c/p\u003e \u003cp\u003e \u003cb\u003eVariables of the Problem\u003c/b\u003e \u003c/p\u003e \u003cp\u003expw(i,j) is the quantity transported from plant \u0026lsquo;i\u0026rsquo; to warehouse \u0026lsquo;j\u0026rsquo; as a fraction of total market demand (xpw(i,j)\u0026thinsp;=\u0026thinsp;XPW(i,j)/sum(k), D(k)); xwm(j,k) is the quantity transported from warehouse \u0026lsquo;j\u0026rsquo; to market \u0026lsquo;k\u0026rsquo; as a fraction of total market demand (xwm(j,m)\u0026thinsp;=\u0026thinsp;XWM(j,m)/sum(k), D(k)); yp(i) is location variable for plant \u0026lsquo;i\u0026rsquo;, it is equal to 1 if plant is located at \u0026lsquo;i\u0026rsquo; and 0 otherwise; and yw(j) is location variable for warehouse \u0026lsquo;j\u0026rsquo;, it is equal to 1 if warehouse is located at \u0026lsquo;j\u0026rsquo; and 0 otherwise.\u003c/p\u003e \u003cp\u003eFormulated differently, (with normalized xpw(i,j) and xwm(j,k))\u003c/p\u003e \u003cp\u003eMin sum(i,j), cpw(i,j)*xpw(i,j)\u0026thinsp;+\u0026thinsp;sum(j,k), cwm(j,k)*xwm(j,k) +\u003c/p\u003e \u003cp\u003eSum(i), yp(i)*fp(i)\u0026thinsp;+\u0026thinsp;sum(j), yw(j)*fw(j) (0)\u003c/p\u003e \u003cp\u003eSum(i,j), xpw(i,j)\u0026thinsp;=\u0026thinsp;1 (0a)\u003c/p\u003e \u003cp\u003eSum(j,k), xwm(j,k)\u0026thinsp;=\u0026thinsp;1 (0b)\u003c/p\u003e \u003cp\u003eSum(j), xwm(j,k)\u0026thinsp;\u0026gt;\u0026thinsp;=\u0026thinsp;d(k) for all k (0c)\u003c/p\u003e \u003cp\u003eSum(j), xpw(i,j)\u0026thinsp;\u0026lt;\u0026thinsp;=\u0026thinsp;capp(i) for all i (1)\u003c/p\u003e \u003cp\u003eSum(j), xpw(i,j)\u0026thinsp;\u0026lt;\u0026thinsp;=\u0026thinsp;capp(i)*yp(i) for all i (2)\u003c/p\u003e \u003cp\u003eSum(i), xpw(i,j)\u0026thinsp;\u0026lt;\u0026thinsp;=\u0026thinsp;capw(j) for all j (3)\u003c/p\u003e \u003cp\u003eSum(i), xpw(i,j)\u0026thinsp;\u0026lt;\u0026thinsp;=\u0026thinsp;capw(j)*yw(j) for all j (4)\u003c/p\u003e \u003cp\u003expw(i,j)\u0026thinsp;\u0026lt;\u0026thinsp;=\u0026thinsp;yp(i)*capw(j) for all i, j (5)\u003c/p\u003e \u003cp\u003expw(i,j)\u0026thinsp;\u0026lt;\u0026thinsp;=\u0026thinsp;yw(j)*capp(i) for all i,j (6)\u003c/p\u003e \u003cp\u003exwm(j,k)\u0026thinsp;\u0026lt;\u0026thinsp;=\u0026thinsp;d(k)*yw(j) for all j,k (7)\u003c/p\u003e \u003cp\u003eSum(j), xpw(i,j)\u0026thinsp;\u0026lt;\u0026thinsp;=\u0026thinsp;yp(i) for all i (8)\u003c/p\u003e \u003cp\u003eSum(i), xpw(i,j)\u0026thinsp;\u0026lt;\u0026thinsp;=\u0026thinsp;yw(j) for all j (9)\u003c/p\u003e \u003cp\u003eFlow balance constraint:\u003c/p\u003e \u003cp\u003eSum(i), xpw(i,j)\u0026thinsp;=\u0026thinsp;sum(k), xwm(j,k) for all j (10)\u003c/p\u003e \u003cp\u003eSum(i), capp(i)*yp(i)\u0026thinsp;\u0026gt;\u0026thinsp;=\u0026thinsp;1 (11)\u003c/p\u003e \u003cp\u003eSum(j), capw(j)*yw(j)\u0026thinsp;\u0026gt;\u0026thinsp;=\u0026thinsp;1 (12)\u003c/p\u003e \u003cp\u003expw(i,j)\u0026thinsp;\u0026gt;\u0026thinsp;=\u0026thinsp;0 for all i,j; xwm(j,k)\u0026thinsp;\u0026gt;\u0026thinsp;=\u0026thinsp;0 for all j,k (13)\u003c/p\u003e \u003cp\u003eyp(i) = (0,1) for all i and yw(j) = (0,1) for all j (14)\u003c/p\u003e \u003cp\u003eEquation (0) is sum of cost of location and distribution. Equations\u0026nbsp;(0a), (0b) and (0c) ensure that demand is met at all markets. Eq.\u0026nbsp;(1) ensures flow out of a plant to be within its capacity, Eq.\u0026nbsp;(2) is a strong linking constraint, Eq.\u0026nbsp;(3) ensures that inflow at a warehouse is within its capacity, and Eq.\u0026nbsp;(4) is again a strong linking constraint. Equations\u0026nbsp;(5\u003cb\u003e) and (6) are new linking constraints to SSPWLP but are borrowed from\u003c/b\u003e Sharma and Namdeo (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2005\u003c/span\u003e\u003cb\u003e).\u003c/b\u003e Again, equations (7) are strong linking constraints borrowed from literature (Sharma and Berry (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e)). Eqs.\u0026nbsp;(8) and (9) are weak linking constraints, (10) is flow balance constraint at each of the warehouses. Equations\u0026nbsp;(1\u003cb\u003e1) and (12) ensure that we have an additional constraint to ensure a feasible solution to problem SSPWLP. These are missed in Kaufman et. al. (1977) and\u003c/b\u003e Sharma and Berry (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e\u003cb\u003e); but\u003c/b\u003e Priyank Dubey (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2020\u003c/span\u003e\u003cb\u003e) established its efficacy. Equations\u0026nbsp;(1\u003c/b\u003e3) force non negativity restriction on transportation variables and equations (14) ensure that location variables are (0,1).\u003c/p\u003e \u003cp\u003eFew comments are in order here. Sharma and Namdeo (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2005\u003c/span\u003e) did not give constraints (0a), (0b), (11) and (12) in their formulation for 2-stage warehouse location problem; and Sharma and Berry (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e) forgot to include constraint (12) in their formulation of SSCWLP (single stage capacitated warehouse location problem) and Sharma, Jha and Priyank (2023) (a paper submitted to IEOM 2023 Houston conference) showed that efficacy of (12) in problem SSCWLP was highly significant. We put all these classes of constraints for SSPWLP to give its state-of-art formulation. It is also important to note here that despite stronger LP relaxation bound given by strong constraints ((2) and (7)) compared to LP relaxation bounds given by weak constraints (8) and (9), SSUWLP (single stage un-capacitated warehouse location problem) ran faster with weak constraints than the strong constraints (see Sharma and Verma (\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2012\u003c/span\u003e)). This encouraged Sharma and Verma (\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2012\u003c/span\u003e) to develop a hybrid formulation (here weak constraints were augmented by few promising strong constraints) whose performance was better than that of weak and strong formulation of SSCWLP. Thus determination of efficacy of each of the linking constraints is important an experimental investigation is underway to determine this.\u003c/p\u003e"},{"header":"3. Discussion","content":"\u003cp\u003eWe give the usefulness of the formulation given above in the form of a table.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eAdvantages of New Formulation of SSPWLP given in this paper.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSN\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAdvantage\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNew linking constraints (5) and (6) are given that is expected to boost the performance of optimizing algorithms.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNew feasibility constraints (11) and (12) are expected to boost the performance of optimizing algorithms.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWe ensure that all strong linking constraints given in Sharma and Berry (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e) (4 and 7) are included in this formulation.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNew linking constraint (2) is added to formulation that is expected to boost the performance of optimizing algorithms.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eWe define following models:\u003c/p\u003e \u003cp\u003eP1: without (5), (6), (11) and (12).\u003c/p\u003e \u003cp\u003eP2: without (11) and (12)\u003c/p\u003e \u003cp\u003eAnd model P3 with all constraints (0a), (0b), (0c) and (1) to (14).\u003c/p\u003e \u003cp\u003eAn experimental investigation carried out to establish the efficacy of new constraints added to SSPWLP problem in this paper is given below. We solved problems for 50 plants and 50 warehouses. In problem set 1 we solved 25 problems with overcapacity between 125\u0026ndash;150%. In problem set 2 we solved 25 problems with overcapacity of about 200%. These problems were solved on Intel(R) Core(TM) i5-7200U CPU @ 2.50GHz processor.\u003c/p\u003e"},{"header":"4. Computational Results","content":"\u003cp\u003e\u003cstrong\u003ePROBLEM SET 1 (overcapacity of plant and warehouse between 125%-150%)\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSalient Result for P1-P2\u003c/strong\u003e:\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Taba\" border=\"1\"\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCriterion\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep1\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep2\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e| t \u0026ndash;value |\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eIterations\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e9197.12\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e6161.64\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.553\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eNo. of Nodes\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e926.76\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e625.16\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.528\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eRoot Relaxation Solution Time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0328\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0536\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e7.076\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eObjective Fn.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e40220.2836\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e40512.5464\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.337\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eExecution time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0236\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.02872\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.059\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eSalient Result for P2-P3\u003c/strong\u003e:\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tabb\" border=\"1\"\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCriterion\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep2\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep3\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e| t \u0026ndash;value |\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eIterations\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e6161.64\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1792.80\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.704\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eNo. of Nodes\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e625.16\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e74.68\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e5.112\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eRoot Relaxation Solution Time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0536\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0388\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e5.220\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eObjective Fn.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e40512.5464\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e40540.9728\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.039\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eExecution time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.02872\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0312\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.18\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eSalient Result for P1-P3\u003c/strong\u003e:\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tabc\" border=\"1\"\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCriterion\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep1\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep3\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e| t \u0026ndash;value |\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eIterations\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e9197.12\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1792.80\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e3.695\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eNo. of Nodes\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e926.76\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e74.68\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.287\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eRoot Relaxation Solution Time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0328\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0388\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.964\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eObjective Fn.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e40220.2836\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e40540.9728\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.384\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eExecution time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0236\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0312\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.69*\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003ctfoot\u003e\n\u003ctr\u003e\n\u003ctd colspan=\"4\"\u003e\u0026bull; \u003cstrong\u003eSig at 0.0064\u003c/strong\u003e\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tfoot\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003eFrom above table we can say that there is no significant difference in objective function values of models P1, P2 and P3; but P1 takes significantly less execution time compared to P3. Therefore we can solve problem first by using model P1 and then use this advanced start to improve further by using model P3 (GAMS offers such a capability).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003ePROBLEM SET 2 (overcapacity of plant and warehouse 200%)\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSalient Result for P1-P2\u003c/strong\u003e:\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tabd\" border=\"1\"\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCriterion\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep1\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep2\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e| t \u0026ndash;value |\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eIterations\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e7464.68\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e7456.60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.004\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eNo. of Nodes\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e631.76\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e568.80\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.383\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eRoot Relaxation Solution Time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0292\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0556\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e7.066\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eObjective Fn.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e23365.4344\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e22921.6592\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.685\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eExecution time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.02876\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.02816\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.105\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003eFrom above table we can say that between P1 and P2 there is no significant difference in terms of number of iterations and no of nodes processed; so we can say that P1 is as good as P2.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSalient Result for P2-P3\u003c/strong\u003e:\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tabe\" border=\"1\"\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCriterion\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep2\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep3\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e| t \u0026ndash;value |\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eIterations\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e7456.60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1274.36\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e3.352\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eNo. of Nodes\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e568.80\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e11.72\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.034\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eRoot Relaxation Solution Time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0556\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0552\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.110\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eObjective Fn.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e22921.6592\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e22422.4352\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.848\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eExecution time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.02816\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.03952\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.551*\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003ctfoot\u003e\n\u003ctr\u003e\n\u003ctd colspan=\"4\"\u003e\u0026bull; Sig at 0.067\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tfoot\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSalient Result for P1-P3:\u003c/strong\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tabf\" border=\"1\"\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCriterion\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep1\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e\u0026micro;\u003csub\u003ep3\u003c/sub\u003e\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e| t \u0026ndash;value |\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eIterations\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e7464.68\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1274.36\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.577\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eNo. of Nodes\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e631.76\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e11.72\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e5.662\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eRoot Relaxation Solution Time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0292\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.0552\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e8.362\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eObjective Fn.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e23365.4344\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e22422.4352\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.491\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eExecution time\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.02876\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.03952\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.551*\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003ctfoot\u003e\n\u003ctr\u003e\n\u003ctd colspan=\"4\"\u003e\u0026bull; \u003cstrong\u003eSig at 0.067\u003c/strong\u003e\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tfoot\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003eFrom above table we can say that there is no significant difference in objective function values of models P1, P2 and P3 (except that P3 gives sig better objective functions than P1); but P1 takes significantly less execution time compared to P3. Therefore we can solve problem first by using model P1 and then use this advanced start to improve further by using model P3 (GAMS offers such a capability).\u003c/p\u003e\n\u003cp\u003eAs more constraints are added (5, 6, 11 and 12) progressively in model P2 and P3 we get better objective function values and at the expense of higher execution time.\u003c/p\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003ePriyank Dubey (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) has established the importance of feasibility constraints such as (11) and (12). Sharma and Verma () have established the importance of Hybrid formulations (weak constraints\u0026thinsp;+\u0026thinsp;most promising strong constraints). Thus we may construct a model P4 (min (0), s.t. (0a) to (4); (8) and (9); (10) to (14) and most promising constraints associated with strong linking constraints such as (5) to (7). It is expected that this model may give best results for large sized problem instances of SSPWLP. This is a topic of future research.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eNone of the authors received any financial grants for conducting this research. All authors have no competing and no conflict of interest to get this work published.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eBiographies\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eProf. RRK Sharma:\u0026nbsp;\u003c/strong\u003eHe is B.E. (mechanical engineering) from VNIT Nagpur India, and PhD in management from I.I.M., Ahmedabad, INDIA. He has nearly three years of experience in automotive companies in India (Tata Motors and TVS-Suzuki). He has 33 years of teaching and research experience at the Department of Industrial and Management Engineering, I.I.T., Kanpur, 208016 INDIA. To date he has written 1217 papers (peer-reviewed (402) /under review (34) / working papers 781 (not referred)). He has developed over ten software products. To date, he has guided 68 M TECH and 23 Ph D theses at I.I.T. Kanpur. He has been Sanjay Mittal Chair Professor at IIT KANPUR (15.09.2015 to 14.09.2018) and is currently a H.A.G. scale professor at I.I.T. Kanpur. In 2015, he received \u0026ldquo;Membership Award\u0026rdquo; given by IABE USA (International Academy of Business and Economics). In 2016 he received the \u0026ldquo;Distinguished Educator Award\u0026rdquo; from IEOM (Industrial Engineering and Operations Management) Society, U.S.A. In 2021, he received IEOM Distinguished Service Award. In 2019, 2020, 2021 and 2022 he was invited by the Ministry of Human Resources Department, India, to participate in the NIRF rankings survey for management schools in India. In 2019, 2020, 2021, 2022 and 2023 he was invited to participate in the Q.S. ranking exercise for ranking management schools in South Asia in 2019, 2020, 2022 and 2023. He was invited to participate in Times Higher Education Academic survey for world university rankings in 2023.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDr. Vinay Singh:\u003c/strong\u003e He has earned his Bachelor Degree in engineering (Computer Science and Engineering) from RBS College Agra, Masters in Human Resource Development and Management from IIT Kharagpur and PhD in Management from IIT Kanpur. Currently he is working as Assistant Professor in the department of Management at ABV-Indian Institute of Information Technology and Management Gwalior, India since Nov 2012. So far he has 26 publications in peer review journals to his credit. He has supervised 92 Masters Students and guided 02 PhD theses. He has also earned two national patents in embedded products design and has developed three software packages. He has received 03 research project grants from prestigious agencies of India.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMr. Pushkar Awasthi:\u003c/strong\u003e He is second year MTECH student in the department of Industrial and Management Engineering, IIT Kanpur 208016 India.\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eGeoffrion AM and Graves GW, \u0026ldquo;Multi commodity distribution system design by using Bender\u0026rsquo;s decomposition\u0026rdquo;, Management Science, 1974, 20(5), pp. 82-114. \u003c/li\u003e\n\u003cli\u003eKaufman L, et. al., \u0026ldquo;A Plant Warehouse Location Problem\u0026rdquo;, Operational Research Quarterly, 1977, V 28 (3), Part I, pp. 547-554. \u003c/li\u003e\n\u003cli\u003eNamdeo, S., \u0026ldquo;Developing new strong constraints for the two stage warehouse location problem \u0026ndash; With and without restrictions on the arc flow\u0026rdquo;, Department of Industrial and Management Engineering, Indian Institute of Technology, Kanpur 208016 (completed 2005).\u003c/li\u003e\n\u003cli\u003ePriyank Dubey, \u0026lsquo;Efficacy of Feasibility Constraints in the Context of SSCWLP (Single Stage Capacitated Warehouse Location Problem)\u0026rsquo;, Department of Industrial and Management Engineering, Indian Institute of Technology, Kanpur 208016 (completed 2020). Thesis supervisor: Prof. RRK Sharma. \u003c/li\u003e\n\u003cli\u003eSharma, R.R.K., \u0026ldquo;Modeling a Fertilizer Distribution System\u0026rdquo;, European Journal of Operational Research, 51, 1991, pp. 24-34. \u003c/li\u003e\n\u003cli\u003eRRK Sharma, \u0026ldquo;Working Paper Series: Lecture Notes in Management Science: Vol 1\u0026rdquo;, A collection of 148 working papers, (All Authored by Prof. RRK Sharma). EXCEL PUBLISHERS NEW DELHI, April 2019; p. 149. ISBN: 9-789-388-237116. \u003c/li\u003e\n\u003cli\u003eRRK Sharma, \u0026ldquo;Working Paper Series: Lecture Notes in Management Science: Vol 2\u0026rdquo;, A collection of 295 working papers, (All Authored by Prof. RRK Sharma); EXCEL PUBLISHERS NEW DELHI, 2019. Aug 2019; p. 234. ISBN: 9-789-388-237796. \u003c/li\u003e\n\u003cli\u003eRRK Sharma, \u0026ldquo;Working Paper Series: Lecture Notes in Management Science: Vol 3\u0026rdquo;, (150 articles are written: All Authored by Prof. RRK Sharma); FEB 2020. ISBN: \u003cstrong\u003e978-93-89947-08-3; Mar 2020; p. 156. \u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003eRRK Sharma, \u0026ldquo;Working Paper Series: Lecture Notes in Management Science: Vol 4\u0026rdquo;, (It has 01 article + 4 software: All Authored by Prof. RRK Sharma); (Oct 2020); p. 192. ISBN: 9789389947212. \u003c/li\u003e\n\u003cli\u003eRRK Sharma, \u0026ldquo;Working Paper Series: Lecture Notes in Management Science: Vol 5\u0026rdquo;, (It has 139 articles are written: All Authored by Prof. RRK Sharma); ISBN: 978-93-89947-31-1; Jan 2021. \u003c/li\u003e\n\u003cli\u003eRRK Sharma, \u0026ldquo;Working Paper Series: Lecture Notes in Management Science: Vol 6\u0026rdquo;, (It has 048 articles are written so far: All Authored by Prof. RRK Sharma); ISBN: 978-93-91355-65-4; May 2022. \u003c/li\u003e\n\u003cli\u003eRRK Sharma, Ankita M, Vimal kr, Vinay Singh and Pritee Agarwal, \u0026ldquo;Developing modified Benders decomposition method for single stage multi commodity multi period warehouse location problem\u0026rdquo;, American J of Operations Research, V6; 2016, pp. 245-259. \u003c/li\u003e\n\u003cli\u003eSharma, RRK, Agarwal, Pritee and Vinay Singh, \u0026ldquo;BENDERS\u0026rsquo; DECOMPOSITION FOR DIFFERENT FORMULATIONS OF SINGLE STAGE CAPACITATED WAREHOUSE LOCATION PROBLEM (SSCWLP): A BRIEF THEORETICAL AND EMPIRICAL INVESTIGATION\u0026rdquo;, International Journal of Business Research, V 12(1), 2012, pp. 43-50.\u003c/li\u003e\n\u003cli\u003eSharma, R.R.K. and Berry, V., \u0026ldquo;Developing New Formulations and Relaxations of Single Stage Capacitated Warehouse Location Problem (SSCWLP): Empirical Investigation for Assessing Relative Strengths and Computational Effort\u0026rdquo;, European Journal of Operational Research, 2007, V 177, pp. 803-812. IF: 4.2; Cite Score: 8.0+\u003c/li\u003e\n\u003cli\u003eSharma, R.R.K. and Muralidhar, A., \u0026ldquo;A new formulation and relaxation of the simple plant location problem\u0026rdquo;, Asia Pacific Journal of Operational Research, V 26(1), Feb 2009; pp. 1-11. \u003c/li\u003e\n\u003cli\u003eSharma, R.R.K. and Namdeo, S., \u0026ldquo;Two stage capacitated warehouse location problem: Developing new strong constraints\u0026rdquo;, Proceedings, Fifth International Conference on Operational Research for Development : ICORD V\u0026rdquo;, held at Jamshedpur, INDIA during Dec. 19-21, 2005, pp. 330-333. \u003c/li\u003e\n\u003cli\u003eRRK Sharma and Pritee Agarwal, \u0026ldquo;Solving Single Stage Capacitated Warehouse Location Problem (SSCWLP) by Branch and Bound and Benders\u0026rsquo; Decomposition Methods: A Comparative Study\u0026rdquo;, International J of Operations and Quantitative Management, V 19 (3); Sep. 2013; pp. 147-156. \u003c/li\u003e\n\u003cli\u003eSharma, RRK and Pritee Agarwal, \u0026ldquo;Solving SSCWLP using Benders\u0026rsquo; decomposition: Theoretical and Computational Study for Different Formulations\u0026rdquo;, International J of Strategic management, V 14 (1); 2014; pp. 35-44. \u003c/li\u003e\n\u003cli\u003eRRK Sharma and Pritee Agarwal, \u0026ldquo;Approaches to solve MID_CPLP problem: Theoretical results and empirical investigation\u0026rdquo;, American J of Operational Research, 4, 2014, pp. 142-154. \u003c/li\u003e\n\u003cli\u003eRRK Sharma and Priyanka Verma, \u0026ldquo;Hybrid Formulations of single stage uncapacitated warehouse location problem: Few theoretical and empirical results\u0026rdquo;, International Journal of Operations and Quantitative Management, V 18 (1), Mar 2012, pp. 53-69. \u003c/li\u003e\n\u003cli\u003ePriyanka Verma and RRK Sharma, \u0026ldquo;Vertical Decomposition Approach to solve Single Stage Capacitated Warehouse Location Problem\u0026rdquo;, American Journal of Operational Research, V 1 (3), 2011, pp. 1-18. \u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[{"identity":"5cdc140d-6266-4729-8d70-b122698f1335","identifier":"10.13039/501100004541","name":"Ministry of Human Resource Development","awardNumber":"0000-0000-0000-0000","order_by":0}],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Indian Institute of Technology Kanpur","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Single Stage Plant Warehouse Location Problem, Location-Distribution Problem, Locating Plant and Warehouses Simultaneously, Simple Plant Location Problem and Capacitated Plant Location Problem","lastPublishedDoi":"10.21203/rs.3.rs-3039168/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3039168/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eKaufman et. al. (1977) first considered this problem (SSPWLP). Goods flow from plants to warehouses to markets. Here we need to locate plants and warehouses of appropriate capacities so that sum total of location cost of plants and warehouses and distribution cost of goods from plant to warehouses to markets is minimized. They considered normalized decision variables (see Sharma and Muralidhar (2009)). However they used the formulation style of Geoffrion and Graves (1974). We use normalized variables but use the variable style of Sharma (1991) and Sharma and Berry (2007) that reduces the number of variables. We give several strong linking constraints by drawing from the works of Sharma and Namdeo (2005) and Sharma and Berry (2007). Below we give the formulation in brief. Details can be seen in the body of the paper. Variables names are self explanatory and makes understanding the model easier.\u003c/p\u003e\n\u003cp\u003eNew Formulation of SSPWLP:\u003c/p\u003e\n\u003cp\u003eMin sum(i,j), cpw(i,j)*xpw(i,j) + sum(j,k), cwm(j,k)*xwm(j,k)\u003c/p\u003e\n\u003cp\u003eSum(i), yp(i)*fp(i) + sum(j), yw(j)*fw(j) (0)\u003c/p\u003e\n\u003cp\u003eSum(i,j), xpw(i,j) = 1 (0a)\u003c/p\u003e\n\u003cp\u003eSum(j,k), xwm(j,k) = 1 \u0026nbsp;(0b)\u003c/p\u003e\n\u003cp\u003eSum(j), xwm(j,k) \u0026gt;= d(k) for all k (0c)\u003c/p\u003e\n\u003cp\u003eSum(j), xpw(i,j) \u0026lt;= capp(i) for all i (1)\u003c/p\u003e\n\u003cp\u003eSum(j), xpw(i,j) \u0026lt;= capp(i)*yp(i) for all i \u0026nbsp;(2)\u003c/p\u003e\n\u003cp\u003eSum(i), xpw(i,j) \u0026lt;= capw(j) for all j (3)\u003c/p\u003e\n\u003cp\u003eSum(i), xpw(i,j) \u0026lt;= capw(j)*yw(j) for all j (4)\u003c/p\u003e\n\u003cp\u003expw(i,j) \u0026lt;= yp(i)*capw(j) for all i, j (5)\u003c/p\u003e\n\u003cp\u003expw(i,j) \u0026lt;= yw(j)*capp(i) for all i,j (6)\u003c/p\u003e\n\u003cp\u003exwm(j,k) \u0026lt;= d(k)*yw(j) for all j,k \u0026nbsp;(7)\u003c/p\u003e\n\u003cp\u003eSum(j), xpw(i,j) \u0026lt;= yp(i) for all i (8)\u003c/p\u003e\n\u003cp\u003eSum(i), xpw(i,j) \u0026lt;= yw(j) for all j (9)\u003c/p\u003e\n\u003cp\u003eFlow balance constraint:\u003c/p\u003e\n\u003cp\u003eSum(i), xpw(i,j) = sum(k), xwm(j,k) for all j (10)\u003c/p\u003e\n\u003cp\u003eSum(i), capp(i)*yp(i) \u0026gt;= 1 (11)\u003c/p\u003e\n\u003cp\u003eSum(j), capw(j)*yw(j) \u0026gt;= 1 \u0026nbsp;(12)\u003c/p\u003e\n\u003cp\u003expw(i,j) \u0026gt;= 0 for all i,j; xwm(j,k) \u0026gt;= 0 for all j,k (13)\u003c/p\u003e\n\u003cp\u003eyp(i) = (0,1) for all i and yw(j) = (0,1) for all j (14)\u003c/p\u003e\n\u003cp\u003eThis (the above formulation of SSPWLP) is the best formulation of SSPWLP that is amenable to solution by LP relaxation and attendant branch and bound and/or branch and cut solution procedure. In literature the distribution phase between plant and warehouse (where plant and warehouse are to be located) is solved by Sharma and Agarwal (2014) as MID_CPLP were Lagrangian Relaxation (LR) was deployed to get RHS_CPLP and LHS_CPLP (different classes of capacitated plant location problems) that were attempted by well known relaxations (LR and Linear Programming) already available in literature (see Priyanka Verma and Sharma (2011)). Computational investigation is underway to determine efficacy of different new linking constraints given in this paper.\u003c/p\u003e","manuscriptTitle":"Single Stage Plant-Warehouse Location Problem (SSPWLP): A State of Art Formulation","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-06-13 06:18:38","doi":"10.21203/rs.3.rs-3039168/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"9ed69aa5-0b8d-4547-874f-b813e1fb0996","owner":[],"postedDate":"June 13th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":22229038,"name":"Operations Research"}],"tags":[],"updatedAt":"2023-06-13T06:18:38+00:00","versionOfRecord":[],"versionCreatedAt":"2023-06-13 06:18:38","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3039168","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3039168","identity":"rs-3039168","version":["v1"]},"buildId":"_2-kVJe1T_tPrBINL-cwx","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}