A Multi Objective DSS For Optimization Of Ferro- Alloy .
International Journal of Technical Research(IJTR)Vol 1,Issue 1,Mar-Apr 2012A Multi Objective DSS for Optimization of FerroAlloy Cost Using LIPSOLSwarnambuj Suman#1, Rajeev Kumar Singh*2, Sanjay Kumar Jha#3 , Vijay Kumar Jha##4#Department of Production Engineering, Birla Institute of Technology, MesraRanchi, Jharkhand, IS, SAIL, Ranchi, Jharkhand, India2##vinitraj3@gmail.comDepartment of Information Technology, Birla Institute of Technology, MesraRanchi, Jharkhand, India4vkjha@bitmesra.ac.inABSTRACTThe steel industries in the modern scenario is striving hard tomaintain the high quality standard and at the same timekeeping the production cost low. The above scenario alsodemands an efficient decision support system to take rightdecision at right time. The current work proposes amathematical model and solves it using different software toachieve both the quality of steel i.e. the required chemistry andminimize the cost of production simultaneously. The developedmodel is also validated using data from the steel plants. Takingthe right decision requires highly experienced individuals. Dueto lack of highly experienced and capable experts the task ofdecision making in steel industries have become tough. Thealready existing decision support systems lack in efficientreasoning mechanism on which the success of the decisionsupport system Programming, MS Excel Solver, Ore BlendingI. INTRODUCTIONThe modern world of globalised economy has rendered thesteel industries to face a tough task to not only continuouslyimprove their productivity but also maintain a defined qualitylevel in their product. This highly competitive scenario hasled to the requirement of dedicated decision support system.Taking right decisions require highly experienced individualswhich is very rare in the modern world. This is because theexpertise which is required to take decisions uses knowledgeearned in an extensive manner through long experience. Theproblem all ready existing becomes more acute if thedecision which is to be taken requires experts from differentfield as bringing the experts from different fields together isvery demanding and is very difficult especially for smallersteel manufacturer. This has motivated us to work towardsthe development of a decision support system for costminimization in steel industry.Steel is a Ferro-alloy mainly consisting of carbon and ofcourse iron. The carbon contents vary from 0.2 % to 2.1% byweight, depending upon the different grade of steel. TheFerro-alloys also include some elements like sulphur,phosphorus, aluminium, chromium, silicon etc which areadder in order to achieve desired physical properties in thesteel produced. Steel is also distinguishable from wroughtiron, which can contain a small amount of carbon included inthe form of slag inclusions.Multi objective optimization problems usually have morethan one objective which is conflicting with each other. Inorder to simplify the problem, it is considered as if there isonly one objective and the other objectives are considered asconstraints. These problems with several objectives, are alsocalled vector optimization problems, and were originallystudied in the context of economics before being applied to35
International Journal of Technical Research(IJTR)Vol 1,Issue 1,Mar-Apr 2012the field of industrial engineering. The mathematical form ofa multi objective problem isMinimize m[f1(x), f2(x) fi(x)]where i 1,2,3Subject togj(x) 0where j 1,2,3 phk(x) 0where k 1,2,3 qIn order to simplify the solution the conflicting functions[f1(x), f2(x) fi(x) ] are replaced by a overriding functionF(x) and the other functions are considered as constraints. Itis very rare to find a single solution for a multi objectiveoptimization problem hence the concept of Pareto optimum isused to arrive at a near true solution.and manufacturing, military command and control, medicine,and law. A successful decision-support system aims atassisting rather than replacing the human decision maker.The objective of the present work can be elaborated as todevelop a mathematical model to (i) optimize the ratio inwhich different ores having different cost and compositionmust be added so that the production cost of Ferro-alloy isleast and (ii) achieve the required grade chemistry of Ferroalloy while different ores having different chemicalcompositions are mixed. The work also includes the casestudies from steel industries to validate the model developed.II. LITERATURE SURVEYA number of scholars have done research regardingdevelopment of multi objective knowledge based decisionsupport system and its application in various fields. Thesurvey is typically divided into different segments as follows:II.A. Research works related to development of decisionsupport system.Fig 1: Pareto optimaIf any of the components of F(x) are competing, there is nounique solution to this problem. The concept of noninferiority (also called Pareto optimality) and must be used tocharacterize the objectives. In the fig above A and B areclearly non-inferior solution points because an improvementin one objective, F1, requires a degradation in the otherobjective, F2, i.e., F1B F1A, F2B F2A. Non-inferior solutionsare also called Pareto optima. A general goal in multiobjective optimization is constructing the Pareto optima.Decision Support Systems (DSS) are a specific class ofinformation system that supports business and organizationaldecision-making activities. DSSs include knowledge-basedsystems. A properly designed DSS is an interactive softwarebased system intended to help decision makers compileuseful information from a combination of raw data,documents, personal knowledge, or business models toidentify and solve problems and make decisions. Today, suchcomplex systems are commonly found in many planning,operational, and management areas in a diverse range offields such as transportation, economics, engineering designRuijun Zhang, et.al [1] developed a cost optimization basedmodel and implemented it in a multi role knowledge baseddecision support system. A DSS having mechanism to extractrequisite data from the developed database and a reasoningmechanism of an expert system was developed. P.A.Manohar et. al [2] proposed a system integrating knowledgebased system and mathematical modeling for fabricating andecision support system with aim of optimizing schedules ofsteel rolling while producing different composition of steel tomeet customer demands. Magnus Frohling, Otto Rentz[3]formulated a decision support model using MILP(mixedinteger linear programming ) for the computation of thechemical compositions of the intermediate and output flowsalong with the specific raw material consumption duringrecycling of ferrous wastes in blast furnaces. Juan R.González et.al [4] worked to develop a generic anddynamically extensible optimization based DSS. Accordingto the authors an optimization-based decision supportsystems can be divided into three stages: Formulation,Solution and Analysis. Authors further concentrated onbuilding optimization-based DSSs that focuses on theSolution stage by providing powerful and dynamicalgorithmic solvers. The paper authored by Sebastien Gebus ,Kauko Leiviska[5] describes how defect-related knowledge35
International Journal of Technical Research(IJTR)Vol 1,Issue 1,Mar-Apr 2012on an electronic assembly line can be integrated in thedecision making process at an operational and organizationallevel. W. Wen, I.C. Chen [6] developed a knowledge baseddecision support system to measure the performance of theenterprise. The developed system not only served theorganization in measuring performance but also helped inpredicting the future sales.etc. In their manuscript, theauthors Dursun Delen, David B. Pratt [7] made efforts todevelop an advanced information systems infrastructure inthe form of decision support systems to help managers makebetter decisions for increasingly more complicatedmanufacturing problem scenarios. The proposed work aimsto get a larger user base who would provide the much neededfeedback to improve the knowledge bases (making themliving knowledge repositories of the domain), model typesand graphical user Interface besides providing better accessto the tool by decision makers. This is achieved by using theSimulation Data eXchange (SDX).II.B Research works related to multi objective optimization.A model was suggested by Rajib K Mohapatra[8] et al. withan objective of minimizing the cost of material handling insteel plants by optimizing the material flow. The objectivesof the developed models were to a) minimization of handlingcost and b) optimize the quantity to be handled. The modelwas developed using mathematical (operation research) toolsand validated using case studies. Nizar Bel Hadj Ali et al.[9]considered a design optimization problem of steel structures,where minimizing multi-stage production cost is taken as theobjective and the constraints are implemented according tostatutory codes of standards. The methodology used foranalysis of the problem is genetic algorithm, GA. Eachdesign variable is coded in to a substring and these substringsare merged to form a chromosome string representing adesign solution. C.J. Rick and M. Engholm[10] discussed thevarious factors that affect the design, selection and utilizationof ferro-alloys with an objective of optimizing the productioncost. UTCAS is used to develop process model forsimulations. It is an advanced computer system speciallydesigned for the convertor process management whichincludes an effective real time process control as well as toolfor process design and production evaluation. In the articleby Debashis Mohanty et.al [11] an attempt has been made tosimultaneously optimize the two fundamental expectationsfrom an industrial rotary kiln: the daily production quantityand the associated quality measured in terms of its ironcontent(which are conflicting in nature). ANN is used tomake a relationship between the input and the output of thesystem. The objective functions captured this way are thensubjected to a multi-objective optimization using PredatorPrey Genetic Algorithm.II.C Research work related to optimization in ore blending.The researchers Rui Bai[12] et.al had an objective ofoptimizing operation control so as to make the quality indicesof raw slurry extent within their target value ranges. Themodel controls as well as optimizes the ore blending processby combining the technologic computation model and fuzzyreasoning approach. The model is having two layers namelythe setting layer and the loop control layer to control andoptimize the synthesis characteristics and quality indicesrespectively. The paper by Gang Yu et.al[13] presents anonlinear multi-objective programming model for a mineralprocessing production planning (MPPP) for optimizing fiveproduction indices, including its iron concentrate output, theconcentrate grade, the concentration ratio, the metal recovery,and the production cost. Two evolutionary algorithms namedthe gradient-based NSGA-II (G-NSGA-II) and the gradientbased SPEA2 (G-SPEA2) are proposed for the multiobjective (MPPP) optimization. The paper by Liang LiTiejun Xu [14] is based on the fuzzy question of the multiobjective ore blending. An algorithm was developed toaddress the sensitivity and flexibility related to ore blendingprocess. Yong he et al [15] used soft computing tools likegenetic algorithm and neural network to simulate arelationship between the mining variables in order tooptimize the cut-off grade and grade of crude ore. Thesuggested two layered system integrates genetic algorithmand neural network to fabricate a genetic neural model. Thegenetic algorithm is used to find the optimal cut-off gradeand grade of crude ore globally by representing them bychromosomes of population for evolution. The self adaptiveneural network is used to determine the local connectionbetween the revenue and the chromosomes i.e. neuralnetwork helps in calculating the fitness function. Finally,genetic algorithms are used globally to search the optimalcut-off grade and grade of crude ore to maximize fitnessfunction.III. METHODOLOGY35
International Journal of Technical Research(IJTR)Vol 1,Issue 1,Mar-Apr 2012A function is linear if it can be expressed in the formf(x1,x2,x3 xn) c1x1 c2x2 c3x3 . cnxn where the ci areconstants. If the objective function and all the constraintfunctions are linear functions, then the model is called alinear programming model. The objective function as shownin the developed mathematical model is linear in nature. Theconstraint inequalities are also linear in nature. Hence thealgorithms capable of solving the linear problem should beselected. The current work uses linear interior point solveralgorithm for solving the linear problems of optimization.The user interface for input of required data and displayingthe output is developed by using MS EXCEL spreadsheet.The MS EXCEL has also been used for post optimizationcalculations.III.A Linear interior point solver (LIPSOL)LIPSOL stands for Linear programming Interior-PointSolvers. It is MATLAB-based software for solving linearprograms by interior-Point methods. LIPSOL is designed tosolve relatively large problems. In order to apply LIPSOL toany linear problem the following condition must be fulfilled.The problem must be largeThe requisite matrices must be sparse.The requisite problem matrix must be definitepositive. The current problem satisfies all the above pre requisites forthe application of the LIPSOL method. LIPSOL basicallyuses mehrotra’s predictor corrector algorithm which is aprimal dual interior point method. Firstly the problem isconverted to a formIII.B Microsoft excelThe Microsoft excel solver is yet another software tool usedfor the user interface i.e. to receive input from the user anddisplay the results obtained. The front end calculations arealso performed using the MS EXCEL spread sheetsIV. MATHEMATICAL MODELINGA standard operational research model consists of decisionvariables, objective functions and constraints. In thefollowing model the objective function is the f(x) i.e. the costequation and the decision variables are the amount in whichthe ores must be blended i.e. x1, x2, x3, x4, x5, x6, x7, x8, x9, x10x11 and x12. The cost optimization model of ore blending ispursuant to the lowest unit cost of the materials under thepremise of controlling the chemistry i.e. the quantity (part perpart) of each element.Let the initial bath weight be X unitsLet the initial composition of the bath be:ElementsQuantity (%)Ca1Mna2Sa3Pa4Sia5Cra6Moa7Nia8Ala9For a given grade of steel let the minimum and maximumlimits of different elements of steel are:ElementsminTxf xSuch thatA. x b0 x uWhere u is the upper boundary included in the constraintmatrix A. Further the slack variables are added into theproblem. The primal variable i.e. x and the primal slackvariable i.e. s are used to transform the existing problem intoa dual interior point problem. They both are solvedsimultaneously to arrive at the feasible solution.CMnSPSiCrMoNiAlComposition 935
International Journal of Technical Research(IJTR)Vol 1,Issue 1,Mar-Apr 2012Let the composition of different alloys in percent be asfollows:Carbon balancingInput carbon a1x d1x1 e1x2 f1x3 g1x4 h1x5 i1x6 j1x7 k1x8 l1x9 m1x10 n1x11 o1x12Minimum amount of carbon for given grade of steel (x x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) b1Maximum amount of carbon for given grade of steel (x x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) c1Then for satisfying the grade chemistry, the followingcondition must be satisfiedLet the quantity of ore 1, ore 2, ore 3, ore 4, ore 5, ore 6, ore7, ore 8, ore 9, ore 10, ore 11and ore 12 to be added to getthe desired chemistry be X1, X2, X3, X4, X5, X6, X7, X8, X9,X10, X11 and X12 respectively.Let the per unit cost of ore 1, ore 2, ore 3, ore 4, ore 5, ore 6,ore 7, ore 8, ore 9, ore 10, ore 11, and ore 12 be cost1, cost2,cost3, cost4, cost5, cost6, cost7, cost8, cost9, cost10, cost11 andcost12 respectively .The first objective is to find the optimum ratio in which theores must be blended so that the production cost of steel isthe least. Hence the function that needs to be optimized isF (x) Cost1x1 Cost2x2 Cost3x3 Cost4x4 Cost5x5 Cost6x6 Cost7x7 Cost8x8 Cost9x9 Cost10x10 Cost11x11 Cost12x12(x x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12)b1 (a1x d1x1 e1x2 f1x3 g1x4 h1x5 i1x6 j1x7 k1x8 l1x9 m1x10 n1x11 o1x12 ) (x x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) c1From the above inequalities the following can be deduced:(b1- a1)x (b1- d1)x1 (b1- e1)x2 (b1- e1)x3 (b1- g1)x4 (b1h1)x5 (b1- i1)x6 (b1- j1)x7 (b1- k1)x8 (b1- l1)x9 (b1m1)x10 (b1- n1)x11 (b1- o1)x12 0(a1- c1)x (d1- c1)x1 (e1- c1)x2 (f1- c1)x3 (g1- c1)x4 (h1c1)x5 (i1- c1)x6 (j1- c1)x7 (k1- c1)x8 (l1- c1)x9 (m1c1)x10 (n1- c1)x11 (o1- c1)x12 0Similarly the chemistry balancing inequalities weredeveloped for different elements viz. manganese, sulphur,phosphorus, silicon, molybdenum, aluminium, chromium andmolybdenum. Hence the linear mathematical model wasdeveloped as follows:Minimize : F(x) Cost1x1 Cost2x2 Cost3x3 Cost4x4 Cost5x5 Cost6x6 Cost7x7 Cost8x8 Cost9x9 Cost10x10 Cost11x11 Cost12x12Subject to:(b1- a1)x (b1- d1)x1 (b1- e1)x2 (b1- e1)x3 (b1- g1)x4 (b1- h1)x5 (b1- i1)x6 (b1- j1)x7 (b1- k1)x8 (b1- l1)x9 (b1- m1)x10 (b1- n1)x11 (b1- o1)x12 0(a1- c1)x (d1- c1)x1 (e1- c1)x2 (f1- c1)x3 (g1- c1)x4 (h1- c1)x5 (i1- c1)x6 (j1- c1)x7 (k1- c1)x8 (l1- c1)x9 (m1- c1)x10 (n1- c1)x11 (o1- c1)x12 0(b2- a2)x (b2- d2)x1 (b2- e2)x2 (b2- e2)x3 (b2- g2)x4 (b2- h2)x5 (b2- i2)x6 (b2- j2)x7 (b2- k2)x8 (b2- l2)x9 (b2- m2)x10 (b2- n2)x11 (b2- o2)x12 0(a2- c2)x (d2- c2)x1 (e2- c2)x2 (f2- c2)x3 (g2- c2)x4 (h2- c2)x5 (i2- c2)x6 (j2- c2)x7 (k2- c2)x8 (l2- c2)x9 (m2- c2)x10 (n2- c2)x11 (o2- c2)x12 0(b3- a3)x (b3- d3)x1 (b3- e3)x2 (b3- e3)x3 (b3- g3)x4 (b3- h3)x5 (b3- i3)x6 (b3- j3)x7 (b3- k3)x8 (b3- l3)x9 (b3- m3)x10 (b3- n3)x11 (b3- o3)x12 035
International Journal of Technical Research(IJTR)Vol 1,Issue 1,Mar-Apr 2012(a3- c3)x (d3- c3)x1 (e3- c3)x2 (f3- c3)x3 (g3- c3)x4 (h3- c3)x5 (i3- c3)x6 (j3- c3)x7 (k3- c3)x8 (l3- c3)x9 (m3- c3)x10 (n3- c3)x11 (o3- c3)x12 0(b4- a4)x (b4- d4)x1 (b4- e4)x2 (b4- e4)x3 (b4- g4)x4 (b4- h4)x5 (b4- i4)x6 (b4- j4)x7 (b4- k4)x8 (b4- l4)x9 (b4- m4)x10 (b4- n4)x11 (b4- o4)x12 0(a4- c4)x (d4- c4)x1 (e4- c4)x2 (f4- c4)x3 (g4- c4)x4 (h4- c4)x5 (i4- c4)x6 (j4- c4)x7 (k4- c4)x8 (l4- c4)x9 (m4- c4)x10 (n4- c4)x11 (o4- c4)x12 0(b5- a5)x (b5- d5)x1 (b5- e5)x2 (b5- e5)x3 (b5- g5)x4 (b5- h5)x5 (b5- i5)x6 (b5- j5)x7 (b5- k5)x8 (b5- l5)x9 (b5- m5)x10 (b5- n5)x11 (b5- o5)x12 0(a5- c5)x (d5- c5)x1 (e5- c5)x2 (f5- c5)x3 (g5- c5)x4 (h5- c5)x5 (i5- c5)x6 (j5- c5)x7 (k5- c5)x8 (l5- c5)x9 (m5- c5)x10 (n5- c5)x11 (o5- c5)x12 0(b6- a6)x (b6- d6)x1 (b6- e6)x2 (b6- e6)x3 (b6- g6)x4 (b6- h6)x5 (b6- i6)x6 (b6- j6)x7 (b6- k6)x8 (b6- l6)x9 (b6- m6)x10 (b6- n6)x11 (b6- o6)x12 0(a6- c6)x (d6- c6)x1 (e6- c6)x2 (f6- c6)x3 (g6- c6)x4 (h6- c6)x5 (i6- c6)x6 (j6- c6)x7 (k6- c6)x8 (l6- c6)x9 (m6- c6)x10 (n6- c6)x11 (o6- c6)x12 0(b7- a7)x (b7- d7)x1 (b7- e7)x2 (b7- e7)x3 (b7- g7)x4 (b7- h7)x5 (b7- i7)x6 (b7- j7)x7 (b7- k7)x8 (b7- l7)x9 (b7- m7)x10 (b7- n7)x11 (b7- o7)x12 0(a7- c7)x (d7- c7)x1 (e7- c7)x2 (f7- c7)x3 (g7- c7)x4 (h7- c7)x5 (i7- c7)x6 (j7- c7)x7 (k7- c7)x8 (l7- c7)x9 (m7- c7)x10 (n7- c7)x11 (o7- c7)x12 0(b8- a8)x (b8- d8)x1 (b8- e8)x2 (b8- e8)x3 (b8- g8)x
Steel is a Ferro-alloy mainly consisting of carbon and of course iron. . must be added so that the production cost of Ferro-alloy is least and (ii) achieve the required grade chemistry of Ferro-
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