MULTI-PERIOD MODEL FOR PART FAMILY/MACHINE CELL FORMATION. Objectives included in the multi-period formulation

Transcription

1 ationa Institute of Technoogy aicut Department of echanica Engineering ULTI-PERIOD ODEL FOR PART FAILY/AHIE ELL FORATIO Given a set of parts, processing requirements, and avaiabe resources The objective of the part famiy/machine ce (PF/) formation probem is to obtain a satisfactory partition of parts into famiies and machines into ces The resuting system performs we with respect to the design objectives Effectiveness of a ceuar manufacturing system is sensitive to fuctuations in product demand and product mix Generay PF/ formation modes assume that both of these factors remain constant In reaity, the demand for a product varies over its ife cyce, new products are introduced, and the production of oder products are discontinued Objectives incuded in the muti-period formuation inimize interce transfers of parts inimize dupication of machines inimize between period reconfiguration of ces Interce transfers decrease the efficiency of the ceuar system by increasing materia handing requirement, engthening fow ines, and compicating production contro The minimization of interce transfers and the minimization of machine dupication are conficting Tradeoff between interce transfers and machine dupication are aowed by incuding both objectives inimize system re-configuration is particuar to the muti-period formuation System reconfiguration is defined as changing the composition of the machine ces by moving machine from one ce to another in subsequent periods By reconfiguring the machine ces, the ceuar system can continue to operate efficienty as the product mix and demand eves change Reconfiguration is not without drawback moving machines from ce to ce requires effort and can ead to the disruption of production This suggests for a tradeoffs between system reconfiguration and increased interce transfers and/or increased machine dupication To accommodate a these objectives and to formaize the tradeoffs being made, system performance with respect to each objective is measured in monetary terms In cost terminoogy the objectives are re-written as: inimize interce materia handing costs inimize capita investment in machines, and uti-period ode 65 arch 0

2 ationa Institute of Technoogy aicut Department of echanica Engineering inimize machine reocation costs The aggregated objective of the muti-period PF/ formation probem is to minimize the system cost ixed-integer programming formuation of the muti-period PF/ formation i index of parts, j index of machines, k index of ces, index of periods, System parameters: i =,, j =,, k =,, =,, p D i - Demand (production voume) for part i in period S i - umber of processing operations for part i in period ( i r ) O,, - achine type required by the rth operation on part i in period T ij - Processing time of part i on machine type j in period j - umber of type j machines avaiabe at start of panning horizon j - apacity of machine type j P j - ost of acquiring a type j machine in period H i - Interce per unit materia handing cost for part i in period R j - ost of reocating machine type in period L inimum number of machines per ce LP inimum number of parts per famiy A a arge number Decision Variabes x ik if part i is assigned to ce k during period uti-period ode 66 arch 0

3 ationa Institute of Technoogy aicut Department of echanica Engineering y jk if machine j is assigned to ce k during period n jk - umber of type j machines assigned to ce k during period otations to simpify the expression of objective function qi - umber of interce transfers that occur during the production of part i during period b j - umber of additiona type j machines acquired at the beginning of period u j - umber of type j machines that are reocated between period (-) and period Objective Function inimize p = = = = Z = i i i j j j j i j j Where ( H D q ) + ( P b ) + ( R u ) [] qi k = Si x ik ( r= = yo i (, r, ) k yo( i, r+, ) k ) i, [] b j max 0, n jk j b js k = s= = j, [] u j = k = [ max{ 0, n n }] b j, [4] jk jk ( ) j Subjected to: onstraints to ensure that each part is assigned ony to a ce for periods in which demand exists for the part k = xik = i, [5] uti-period ode 67 arch 0

4 ationa Institute of Technoogy aicut Department of echanica Engineering Within-ce capacity constraints i= DiTij xik y jk jn jk j, k, Entire system capacity constraints [6] i= DiTij j c k = n jk j, [7] Lower imits on the number of machines per ce j= y jk L k, [8] Lower imits on the number of parts per ce i= xik LP k, [9] onstraints to ensures that the number of units of a given machine type in a ce is equa to zero uness the machine has been assigned to the ce n jk Ay jk j, k, Integraity constraints x ik [0] { 0,} i, j, k, [], y jk = n jk 0,integer i, j, k, [] QUESTIOS:. What is a muti-period ce formation probem? Write down the objective function of muti-period ce formation. Use appropriate notation for the variabes used and identify the decision variabes of the muti-period ce formation probem.. What are the costs generay considered in the objective function of muti-period ce formation? Describe the costs invoved and the tradeoff whie forming an objective function in muti-period ce formation.. Describe an objective function that is suitabe for a ceuar manufacturing system design that considers product mix and demand variation from period to period in a panning horizon. Expain the kind of tradeoff invoved in the formuation. uti-period ode 68 arch 0

5 ationa Institute of Technoogy aicut Department of echanica Engineering 4. A production system has the foowing detais: umber of machine types = 6, umber of parts = 0. The resource data and part data for the production system are given in the tabes beow. PF/ formation requirements: umber of ces =, inimum number of machine types in each ce =, inimum number of parts in each ce =. The panning horizon contains periods. Tabe. Resource data for the probem achine type Acquisition cost Reocation cost apacity (time units/period) umber avaiabe at period Part number /H * cost per unit Tabe. Part data for the probem Operation sequence machine (processing time) Period Demand Period 6 (5)-4(4)-5 (5)-(4)-(4)-(4) (5)- (4)- (5)-(4) (5)-(4)-6()-(6)-(5)-4(4) (6)- (6)-(4)-4(5)-5(4) (5)- (6)-6()-4(4)- 5 (6)-(5) (4)-(4)-4() -(6)-6(5)- (4) (5)-4(4)-(4)-() (4)-5(5)-4(4)-(5)-(4) (5)- (6)-(4)-(6) (4)-(5)-4(4)-(6) * /H ateria Handing (i) Deveop a mathematica programming formuation for PF/ formation for this probem. uti-period ode 69 arch 0

6 ationa Institute of Technoogy aicut Department of echanica Engineering (ii) Write down a feasibe soution and cacuate the objective function vaue for the soution identified. 5. A production system has the foowing detais: umber of machine types = 6, umber of parts = 9, Interce materia handing cost = Rs per unit per transfer. Tabe : apacity of machines achine type apacity (time units) Part number Tabe : Part data for the probem Operation sequence machine (processing time) 6 (5)-4(4)-5(5)-(4)-(4) 4(5)- (4)- (5)-(4) 5 (5)-(4)-6()-(6)-(5)-4(4) (6)- (6)-(4)-4(5)-5(4) (5)- (6)-6()-4(4)- 5 (6)-(5) (4)-4() -(6)-6(5)- (4) (5)-4(4)-(4)-() (4)-5(5)-4(4)-(5)-(4) (4)-(5)-4(4)-(6) Demand The management decided to have ceuar manufacturing system with the foowing detais. umber of ces =, inimum number of machine types in each ce =, aximum number of machine type in a ce =, and inimum number of parts in each ce =. A typica soution for the probem is as foows: = {, 4, 5}, PF = {4, 6, 7, 8, 9} = {,, 4}, PF = {, } = {4, 5, 6}, PF = {5, } stands for ce and PF stands for part famiy. (a) What is the interce materia handing cost for above soution? How many units of machine shoud be present in ce? (b) If a mathematica programme is used to define ce formation and part assignment and if a feasibe soution (typica soution) given above is appicabe, generate constraints that shoud ensure sufficient capacity for each machine type in a ce. Use the foowing notations for generating the constraints. uti-period ode 70 arch 0

7 ationa Institute of Technoogy aicut Department of echanica Engineering i index of parts, j index of machines, k index of ces, i =,, j =,, k =,, D i - Demand (production voume) for part i T ij - Processing time of part i on machine type j j - apacity of machine type j x ik y jk if part i isassigned toce k if machine j isassigned toce k during period n jk - umber of type j machines assigned to ce k (c) If a mathematica programme is used to define ce formation and part assignment and if the feasibe ces are = {, 4}, = {, } and = {5, 6} then define a decision variabe for the probem which shows a part assignment to a ce (part famiy). A part can be assigned to a part famiy ony. Generate a such constraints for the given probem. uti-period ode 7 arch 0