UNIVERSITY OF ILLINOIS LIBRARY AT URbANA-CHAMPAIGN BOOKSTACKS CENTRAL CIRCULATION BOOKSTACKS The person eharging th.s mattm U « for disciplinary action and m-V '"»" the University. „„„„- r c N T E n 333-8400 When renewing by phone, write new due da,e below previous due date. Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/productassignmen92177rama Faculty Working Paper 92-0177 330 B3S5 1992:177 COPY 2 STX \" y& 9) •$ * o, Product Assignment in Flexible Multi-Lines The Case of Single Stage with Demand Splitting Narayan Raman Department of Business Administration University of Illinois Udatta. S. Palekar Department of Mechanical and Industrial Engineering University of Illinois Bureau of Economic and Business Research College of Commerce and Business Administration University of Illinois at Urbana-Champaign BEBR FACULTY WORKING PAPER NO. 92-0177 College of Commerce and Business Administration University of Illinois at Grbana-Champaign November 1992 Product Assignment in Flexible Multi-Lines The Case of Single Stage with Demand Splitting Narayan Raman Department of Business Administration Gdatta. S. Palekar Department of Mechanical and Industrial Engineering Product Assignment in Flexible Mult i- Lines The Case of Single Stage with Demand Splitting Narayan Raman Department of Business Administration University of Illinois at Urbana- Champaign Champaign, Illinois Udatta. S. Palekar Department of Mechanical and Industrial Engineering University of Illinois at Urbana-Champaign Urbana, Illinois November 1992 ABSTRACT This study deals with the Flexible Multi-line Design problem in a serial manufacturing system. Such systems process a variety of products in large volumes with stable demand rates. These products have similar processing requirements in that they visit the various manufacturing stages in the same sequence. Each stage on any line comprises multiple identical CNC machines which perform a set of predetermined tasks on the products assigned to that line. Given the fixed cost of providing a line, and the fixed cost of each workcenter at each stage, the objective of the flexible multi-line design problem is to simultaneously determine the number of lines required as well as find the product-to-line allocation such that the total investment in lines and workcenters is minimized. In this paper, we consider the special case of a single-stage system in which a product can be assigned to multiple lines. This special case arises as an important subproblem in the general multi-stage problem. However, it merits independent consideration for systems in which the same stage is the bottleneck for all products; for such systems, the multi-stage FMD problem reduces to a single-stage problem. In this paper we permit overlapping product partitions, the demand of any product can then be spread across several lines. We develop some characteristics of the optimal solution; in particular, we show that it must satisfy the sequential assignment property which renders it solvable in polynomial time using a dynamic programming algorithm. However, we develop an alternative, enumerative solution method that results in a much smaller average running time by making effective use of an imbedded greedy algorithm. This study considers the problem of designing a flexible multi-line in a serial manufacturing system with multiple stages. Such systems process a variety of products in large volumes with stable demand rates. These products have similar processing requirements in that they visit the various manufacturing stages in the same sequence. Each stage on any line comprises multiple identical CNC machines which perform a set of predetermined tasks on the products assigned to that line. While these tasks require similar processing capabilities, the actual tasks done and their processing times are product-specific. The flexible CNC machines can switch from one product to another with negligible changeover time. The adjacent stages are tightly coupled with minimal buffer storage space in between. Each line is paced, and therefore, its cycle time is constrained by the maximum processing time across all stages required by any product assigned to it. Given the fixed cost of providing a line, and the fixed cost of each workcenter at each stage, the objective of the flexible multi-line design (FMD) problem is to partition the set of products such that each subset is assigned to exactly one line, and the total investment in lines and workcenters is minimized. This problem is motivated by the manufacturing facility of one of the major auto com- panies that produces fuel-supply systems. This facility produces a number of different components that go through a number of forming operations during fabrication. At any given stage, the compo- nent is subjected to a specific type of forming operation. The tight coupling of the individual stages allows limited in-process buffer so that the entire line is forced to operate in a paced fashion. This problem also arises in printed circuit board manufacture (Farber et al. 1988). Indeed, the FMD problem arises naturally in many systems in the context of implementing a just-in-time approach within cellular manufacture. Given a set of products with their individual demands and processing requirements, FMD determines the optimal set of families, as well as the optimal configuration of the various cells that need to be formed. Additionally, it can be used at periodic intervals to evaluate the need for a system redesign in the face of changing product demands and processing needs. The problems most closely related to the FMD problem are the mixed-model line balancing problem (Wester and Kilbridge 1964; Thomopolous 1967, 1970; MacAskill 1972; Dar-El 1978; Okamura and Yamashita (1979; and Yano and Rachamadugu 1991) and the line segmentation problem (Ahmadi and Matsuo 1991). Much of the previous work on mixed-model line balancing problem addresses the assignment of tasks required for assembling a number of products to operators stationed along an assembly line. The tasks are general enough in that they can be assigned to any operator on the line as long as the precedence relations among them are satisfied. It is easy to see that in such tandem systems, the cycle time and the overall output are constrained by the total processing time required at the bottleneck station. Consequently, the bulk of the research on this problem has considered the objective of smoothing workload assignments across all stations. Because of the variety of products assembled on this line, the amount of processing required at any station varies from one cycle to another, and workload balancing is based on the average processing time per cycle at each station. Work overloads are relieved by permitting limited operator movement upstream and downstream of the assigned station (Dar-El and Cucuy 1977, Dar-El 1978), or through the use of utility workers (Yano and Rachamadugu 1991). One of the major thrust of this research is on determining the appropriate sequence in which the various models should be processed at each station in order to minimize such overloads. Okamura and Yamashita (1979) address the objective of minimizing the maximum distance that any worker will have to move away from his workstation in order to complete all tasks assigned to him; as Yano and Rachamadugu (1991) note, this objective is similar to minimizing the maximum work overload at any station. Yano and Rachamadugu deal with the objective of minimizing the average work overload given that the overload at any station can be met through the use of utility workers. An alternative line of research involving mixed-model lines addresses sequencing the various prod- ucts for the objective of smoothing the rate of parts usage in assembling the final products. This problem was proposed by Monden (1983) in the context of just-in- time manufacture. Miltenberg (1989) considers the problem in which all final products require the same number and mix of parts. Under this assumption, smoothing part usage rate reduces to minimizing the sum of differences between the cumulative actual production and cumulative actual demands across all products. Miltenberg proposes nonlinear integer programming formulations, and proposes heuristic solution methods. Kubiak and Sethi (1991) relax Miltenberg's assumption, and also consider a more gen- eral form of the objective function; more importantly, they show that the resulting problem can be formulated as an assignment problem. Similar problems are studied by Miltenberg and Sinnamon (1989) and Inman and Bulfin (1991). The FMD problem is similar to mixed-model line balancing in that it considers a paced flow line producing multiple products. In addition, the objective of minimizing total investment in lines and workcenters leads to workload balancing. However, these two problems differ in significant ways. First, the assignment of tasks to stations (stages) is not an issue here because any given task can be done only at a predetermined stage. Second, the stages are "manned" by stationary CNC machines. Consequently, there can be no variation in the time spent at any station from one cycle to another, and the sequence in which the different models are run is immaterial. Workload balance in our context is achieved purely by the formation of parallel lines and grouping products with similar processing times on a line. While there are economic incentives in having multiple lines in order to reduce idle time, the benefits of doing so need to be traded off against the fixed cost of providing the line. Ahmadi and Matsuo (1991) consider the line segmentation problem (LSP); for a given number of machines at each stage, and a given partition of products into families such that each family is assigned to one line, the objective of LSP is to allocate machines at each stage to individual lines such that the overall makespan is minimized. They present several heuristics for solving LSP and show their efficacy with respect to valid lower bounds. The FMD problem differs from LSP in two important ways. First, LSP considers a multi-model situation in which the entire (daily) demand of any product is produced in one batch before the line changes over to produce the next product. In our mixed-model approach, each product is allowed to be produced as often as desired subject to the overall demand constraints. Second, FMD addresses a problem in which product- to-line allocation is done jointly with the determination of the number of lines and the number of workstations required at each stage for each line. This paper is the first of two papers that together address our research on the FMD problem. In both papers, we consider the special case in which there is only one stage. This special case arises as an important subproblem while solving the general multi-stage problem. However, this case merits independent consideration for systems in which the same stage is the bottleneck for all products; for such systems, the multi-stage FMD problem reduces to a single-stage problem. Furthermore, we have encountered several systems that have only one stage. The two papers differ in that this paper considers overlapping product partitions; the demand of any product can then be spread across several lines. In the companion paper (Palekar and Raman 1992), we address the case in which each product is constrained to be produced on only one line. This paper is organized as follows. The problem formulation is given in §1. We develop some dominance properties in §2 that result in an efficient graph representation of the FMD problem. This representation is used in §3 to generate the optimal solution based on a dynamic programming approach. We also develop an alternative polynomial-time algorithm that makes repeated use of a greedy heuristic algorithm. Both the graph representation and the greedy algorithm play important roles in generating strong bounds and efficient solution methods for the no-demand-splitting case considered in the companion paper. We conclude in §5 with a summary of the main results of this paper. 1 PROBLEM DESCRIPTION In this section, we present a mixed integer programming formulation of the flexible multiline design problem. However, first we give the notation used in the paper. Af = the set of products, and \Af\ = N F\ = fixed cost per line 2*2 = fixed cost per machine Pj = processing time of product j, j G Af Ji = set of products with processing times greater than or equal to i, {j\pj > Pi, j £ Af} dj = per period demand of product j, j £ Af A = available time per period on any machine 77 = cycle time of line / Ci = the set of products assigned to line / In any feasible solution, the cycle time rj of line / equals the processing time of its pivot, i. e., the product with the longest processing time that is assigned to that line. Let tt(/) denote the pivot of line /, X(j) denote the line for which product j is the pivot, and uj denote the number of workcenters required at line A(j'). Then, the cycle time of any line / with pivot j is 77 = p : and its capacity is An 3 /p y We assume that A > p ; , Vj € Af so that [A/pj\ « A/pj. The flexible multi-line design problem is stated as FMD1 N Minimize Z x = ^{F x y 3 + F 2 nj) (1) subject to 5>/.- = l,;i€.V (2) Pi ( Yl d * x J> p^j'ii^ ( 3 ) xjio,;i,jeM (5) yj € {0, 1}; rij > 0, ; integer, ; j £ M (6) where x Jt is the fraction of product i's demand assigned to line X(j) and {1, if a line is opened with pivot j 0, otherwise. Equation (2) insures that the demand of each product is fully assigned, and a product is assigned only to lines with cycle times no less than the processing time of the product. Constraint (3) requires that all product-to-line assignments be capacity feasible. Constraint (4) insures that the fixed cost of opening a line is accounted for. Finally, constraints (5) and (6) specify the nature of the variables. The total number of machines required on any line / with pivot j is A and the total idle time on this line / is An j-Pi \Yl d * x ]i\ It is clear that the idle time on this line is reduced by assiging to it products which have processing times close to pj. Thus, the FMD problem aims at balancing processing times as compared to workloads that is done in a mixed-model line balancing problem. It is also seen that the idle time is unaffected by the sequence in which the various products are processed. 2 DOMINANCE PROPERTIES In this section, we develop dominance properties, and construct an efficient graph representation of problem FMD1. Proposition 1. There exists an optimal solution to FMDl with pivot set V = {j\j 6 N, yj = 1} such that pk ^ pi for k,l £V, k ^ I. Proof: For any optimal solution a to FMDl that does not have the above property, we construct an alternative solution a' from a by merging line X(k) with line A(/) while the assignments on other lines remain unchanged. Then Z 2 {a) - Z 2 {o') = 2F l +F 2 -F x - F 2 > Pi £*€£, d tX« + Pk T,t(EC k d t x tk A Pi (Et€£i ^ X " + £t€£ fc d t X *) where the inequality follows from F\ > 0, pk = pi, and the known inequality \a + b]<\a] + \b]. (7) Hence, if a is optimal, then so is a' and the proof is complete. D Proposition 2. There exists an optimal solution to FMDl in which i) if i is not a pivot product, then it is assigned to exactly one line, i.e., z ut 6 {0, 1} for all iEAf\V andue V C\ J { . ii) if i is a pivot product, then it is assigned to at most two lines. Proof As before, we show that any solution a that is optimal to FMDl and that does not have the stated property can be modified to yield an alternative optimal solution that does so. Without loss of generality, we assume that a satisfies proposition 1. Let L be the total number of lines in a. Renumber these lines so that n > r 2 > ...> t l . (8) Let Di = Yli€AfdiX V( ni denote the total quantity assigned to line /, / = 1,2, ...,L. Construct another solution a' from a in the following manner. Rank all products in Af in the nonincreasing order of their processing times. Starting from line 1, assign products from the top of this list, such that the total quantity assigned to line / is D\. If this results in any product being partially assigned to a line, then allocate the remaining quantity to the subsequent line. Let r/ and £' h respectively, denote the cycle time of line / and the set of products assigned to line /, / = 1, 2, . . . , L. Note that a' satisfies property i) given above. Also note that r[ = n . (9) Lemma 1. rf < T\, V/. Proof: Consider the following disjunctive cases: a) min ieC ,{pi} < r, +1 , I = 1,2, ...,X - 1 From the construction of a', T/'+i < min i^c\{?i) < r /+i for / = 1, 2, . . . , L — 1. The result follows from (9). b) min i€C , t {pj} > r t+u forsomete {1,2, ... ,X - 1 Because 77 = max q £c t {Pq} f° r an y H ne U it follows from (8) that L n+i > pi, v« g |J c k k=t+l Consequently, together with the fact that the total quantity allocated to each line is the same in both a and a\ it must be true that in this case, U Ck= U 4 /t=i k=i and r' t+l = r t+1 to yield the desired result. D Now Z 2 {a')~Z 2 {a) = £ j; fr/AMl " £ £ fa D { /A] ie/fi=i «€.V/=i < and a' is optimal. If it satisfies property ii) as well, the proof is complete. Otherwise, merge all those lines which have the same pivot to construct another solution a" that satisfies both i) and ii) and, from proposition 1, is optimal as well. □ In the rest of the paper as well, we assume that the products are numbered such that if i < j, i,j G JV, then pi > pj. We also assume that they satisfy proposition 1 which can now restated as Remark 1. If F\ > 0, then in any optimal solution, pk > pi for any k,l G "P such that k > I. We now the give the central result of this section. Proposition 3. (Sequential Assignment Property) There exists an optimal solution to FMDl with the property that if x Jt > 1, then Xj q = 1 for q = j + l,j + 2, . . ., i — 1. Proof: As before, let a be an optimal solution to FMDl that does not have this property. Then there has to be at least one product t,j < t < i - 1 that is produced on line x(k),k ^ j. Note that k < t < i. If k < j, then construct a' from a by shifting 6 = d t Xk t , the demand of t currently allocated to line X(k) to line A(j), and replacing these units on line X(k) by products currently assigned to line A(,;') considered in the increasing order of their index starting with j. If 6 < d : Xjj, then j continues to remain a pivot, otherwise its entire demand is absorbed by line X(k) and j is replaced as a pivot by some q,q > j, with p q < p r In either case, ^(o - ') < Z2(cr), anc ^ therefore, a' is optimal. If k > j, then construct a' by shifting b — Y?q=t+\ dq units of demand corresponding to products t + 1 through i from line X(j) to line X(k), and replace these units on line X(j) with products currently assigned to line A(A:) considered in the increasing order of their index starting with k. As before, it follows that ^(o - ') < £2(0"), an< ^ therefore, a" is optimal. Repeating these steps whenever required yields the solution a' that is optimal to FMDl and that satisfies the condition stated in the proposition. D Hereafter, we deal only with those solutions that satisfy the sequential assignment property. An immediate consequence of the above propositions is that in an optimal solution, if j, j > 1 is a pivot in an optimal solution, then it is assigned to at most two lines, namely X(j) — 1 and A^'), i.e., x q j > 0,only if g G {x{X(j)-l),j}. Furthermore,^ = 1 for all u, u = j+l,j+2, . . .,tt(X(j)+1)-1. Problem FMD1 can now be represented on graph Q = (V,£) shown in Figure 1. In this graph, node V{j, which is depicted as ij in the figure, represents an assignment in which product j is produced on line with pivot i. Note that node V{j is feasible only if pj < p,-; hence, the upper triangular nature of this graph. Let Ef? denote the arc leading from V{j to V uv . [As we discuss later, each arc joining two nodes in Figure 1 actually represents a set of arcs.] We can append a dummy sink node T at the end to denote an artificial product. The optimal solution to FMDl then corresponds to the shortest path from V u to T. INSERT FIGURE 1 HERE We partition arc-set £ into disjoint subsets H, B and T where H is the set of horizontal arcs (h-arcs) of the form E\' 3+l , while B is the set of backward arcs (b-arcs) of the form E™ where u < i. T comprises the forward arcs (f-arcs). From the sequential assignment property (SAP), it follows that any path that includes a b-arc is not dominant. Furthermore, in an optimal solution, a pivot must be (at least partially) assigned to its own line (otherwise it is being produced at a higher than required cycle time). Consequently, if V{j lies in an optimal path, then so must Va. Together with SAP, this implies that V{j is reachable only via node Va along the path comprising the h-arcs E*- t+1 - #£,•+] - ... - £,-j_i- Therefore, we need consider only those f-arcs that are incident on a pivot, i.e., arcs of the form £/ ,J Clearly, product 1 must be the pivot for line 1 in any feasible solution. Consider a path II in which i and j are adjacent pivots, j > i; i.e, II passes through V u , . . . , Vi )t _i, Va, . . ., Vij-i, Vjj, . . .. Let Mjk denote the number of machines required on line X(j) corresponding to node V 3 k- Then the number of machines required at line 1 in II is _ Pi Y2u~=\ d u The capacity remaining, hereafter the remnant, at line 1 after this assignment is n,t-i = y, "«• Clearly, if r lit _i > 0, then it is optimal to use this capacity for (partially) meeting the demand of product t, so that xi,- > 0, and in general, the remnant available at any line will be used for producing the pivot product of the next line. Consequently, the number of machines required at line (j) corresponding to node Vjk is M jk = -1 L and rik = ir ~ (i d " ~ r ' j -) ■ (10) Note that rjf. < Ajp^ hence, it is strictly less than one machine's capacity on line A(j), and because Pj > Pki for k > j, it is less than one machine's capacity on subsequent lines as well. We now determine the cost of each arc in Q. The cost of f-arc Ejjf 1 ' is c jk - ti + t 2 Pk+\(d k +i - Tjk) (11) From (10), it follows that r jk and therefore, c-jjj" ' +1 depend upon i, the pivot for the line imme- diately preceding A(j). This in turn implies by induction that they depend upon the path selected to reach V ]k - While this suggests that the number of arcs in Q is exponential in the number of products, note that the costs of all f-arcs incident upon any node differ by no more than F 2 . For example, there are k f-arcs, of the form E^ 1, ^yE^ 1 ' +1 t . . ,,E^ 1,k+1 that are incident upon Vjt + i t jt+i- However, because, r uk ,r v k < A/pk+\, for for any u, v < k, we have k2 u+l - 4 +u+1 i < ft (12) This indicates that if two paths reaching the same node differ in cost by F 2 or more, then the longer path is dominated. It is clearly also true that if these paths had the same cost, then the path with the smaller remnant is dominated. Therefore, the set of undominated paths Vk+i,k+i reaching node Vfc+i t jt+i comprises only those arcs that are within F 2 of each other in cost but have distinct Tjk values. Because there can be no more that A/pk+i such values, V'fc+i.jfc+i = |^it+i,A;+i| <• A/pk+i- And in general, for any node Vjj, we have tfijj = \^j]\ < A/min{^{pi} = ift. Thus each f-arc shown in Figure 1 represents a set of arcs whose cardinality is less than or equal to ij) and whose costs differ by at most F 2 . The remnant at node Vk+i,jt+i is A\Mk+i,k+i\ ax (ii\ r k+i,k+i = "Jfc+1 + r jk- K i6 ) Pk+l 10 Note that rjt+i^+i depends upon the arc selected to reach this node, and it can take A/pjr+i values. Now consider the h-arc E J - k . The marginal increase in the number of machines required on line (j) for producing the (k + l)th product, given that products j + 1, J + 2, . . . , k are assigned to this line, is Mj, jt+i - Mjk = Pi (ESj d u - r M -i) Pi (dk+i - rjk) Pi (Eu=j d u - r t ,j-i) where t is the pivot for the line immediately preceding A(j). The cost of h-arc E J -' k +1 is (14) From the above discussion, it follows that this cost also depends upon the path selected to reach Vjk, and h-arc E J jk + in effect represents a set of undominated arcs whose cardinality is no more than ifi. 3 Solution Algorithms for FMD1 FMDl can be formulated as a dynamic program in which the stages are the consecutively numbered products, and the states at a given stage i are given by the combination of the pivots, j, j < i, to which product i can be assigned, and the remnant available at the pivot. Define g*(t) to be the minimum cost of reaching stage t,t = 2, . . .,T. The resulting shortest path problem can be stated as where and where, Z* = min g*(T) g*{t) = min,< t {g a jt }, s 2, select arc Ej 13 ^ ,\_i- For reaching any other pivot node, select the f-arc with the minimum cost as given by (17). In other words, II* is the solution to the dynamic program Z* = min g*(T) where and g*(t) = minj< t {gj t }, *jt where q is the pivot immediately preceding j in V*. Lemma 2. Let z*(t) denote the cost of reaching stage t, along path IT* in Q* \ Then, g G {t) 3, because there is only one f-arc leading to node V22- Let t = r be the first stage where the f-arcs differ. Then g G (t) = g m (t) < z*(t), l r r+l,T+l (23) C T+lir+1 (t/ ) < C T+lr+1 (C/ ). (24) Therefore, 9? +2 ,r + 2 < Awi + <+£K (^) < ^ + « + * + C^S? «T). (25) and 9? + l,r + 2 < 9? + l,r + l + C3l#i (^) < ^(T + 1) + F 2 + C^J? «T). ( 26 ) This implies that g G (r + 2)< g G +hT+2 < z*(t + 2) + F 3 (27) On the other hand, if the remnant difference r* T - r G T does not result in any machine saving, then it is carried forward to node V^+i, T +i» and g G (T + 2) r*- t for any node Vj t on path II*, if follows that g G (t)) Similar to the above argument, we can then consider the two cases in which IT* passes through either node V" T / +liT > +1 or node V T ' )T '+i to show that g G (t) < z*{t) + F 2 for all t > t'. This completes the proof of the lemma. □ The result stated in the proposition follows immediately from the lemma if we substitute N for t, and note that the cost of all arcs leading to node T is zero. □ The following result is derived similar to Proposition 4. Corollary 1. Let Z* and Z G be the optimal cost and the cost under Greedy to reach node V^ from Vn. Then, ZJ > Z G - F 2 . 3.2 An Exact Algorithm We construct an improvement algorithm for solving FMDl exactly that combines corollary 1 with the Greedy solution. Suppose that the optimal solution is given by path IT* with pivots Jidh---tJl an( * solution value Z* . Let the Greedy solution is given by path II G with pivots j G ,j G , ■ • -,3li wi^ solution value Z G . Consider subgraph Q G . Assume that II G ^ IT*. Note that Z G < g G l>N < Z* + F 2 (29) where the last inequality follows from corollary 1. From Z* < Z G , we then have gf l Let rrikN = <>kN + fkN where ikN {fkN) is the integer (fractional) part of rrikN- If A: = j£, then the path II*. from V\\ to Vkk in the optimal solution must be different from the shortest path II G between these two nodes in Q G , and hence, these two paths must differ in at least one f-arc. Traversing Q G backwards, let j + 1 be the first stage where II* and IT G differ. Suppose that the f-arcs leading to Vj+ij+i in paths II* and II G are £/ J +1,J+1 and ££ + j* J+1 , respectively. Clearly, V XJ 6 Tj. Let A tJ = r tJ - r hj be the remnant difference achieved at Vj+ij+i, and K{ 3 = g G . — g G ■ be the cost penalty incurred if arc E{? 1,i+1 is selected instead of arc E{ + )' j+1 . If AijPk/A > fkN, then the additional remnant provided by arc E^ ,]+1 is large enough to absorb the fractional part of the machine required at line (k). Consequently, this switch saves a machine, and the cost of the resulting solution is g G N + «tj — -^2- After completing the switch, the fractional part of machine remaining at (k) is l — (&ijPk/A— fkN)- On the other hand, if AijPk/A < fkN, then no machine saving is effected; the fractional part of machine required at line (k) is fkN — A XJ pk/A, and the cost of the resulting solution is g G N + k,j. A vertex v is fathomed if its solution value 4> v > UB + i*2- Note that in this case, any completion of that vertex can be no better than the incumbent solution. Backtracking along each V{ 3 G Tj until Vn is reached, and pricing each vertex in the manner shown above will eventually lead to the evaluation of all candidate f-arcs that constitute the difference between II G and II*.. II*. is clearly the best among all candidate paths that have been enumerated. Repeating this exercise for each Vjn € IV will clearly determine the optimal path IT*.. We now give a detailed description of the algorithm. Initial Solution and Pre-Processing Step 1 i) Solve FMD1 using Greedy with solution value Z G . Set the current upper bound UB = Z G '. Record the Greedy solution as the current incumbent. ii) For j = 2, 3, . . . , N, and i < j, determine r,j and compute Ay = Tij - r hjj \ and n io = g$ - g G }j where hj is defined by (17). 16 iii) Determine IV- Go to Step 2. Branching, Updating and Fathoming Step 2 Construct a search tree S rooted at T by generating a vertex at level 1 in S corresponding to each node in T^. For each such vertex v that corresponds to (say) node Vjn in Q G , set V = fjNi h = Pj/A, and v = gf N . Determine Tj and generate vertices at the second level corresponding to nodes in Tj, and similarly generate vertices at other levels in S such that the descendants of any unfathomed vertex that corresponds to (say) node Vkk are vertices corresponding to nodes in r^-i- For any vertex v at level 2 or below that is a descendant of vertex u, set 8 V = 6 U , and z v = AikS v where v and u correspond, respectively, to nodes V t k and V^+i^+i in Q G . \iz v > 9 U , then set 4> v = ^u+^ik—F^. If v < UB, then set UB = 0„, and compute V = l — {z v -9 u ). Record v as the incumbent. If z v < 6 U , then set v = 4> u + K,jt. Fathom v if v > UB + F2. Else, set 6 V — U — z v . When the entire tree is generated, go to Step 3. Generation of the Optimal Solution Step 3 At the end of the procedure, let v be the incumbent vertex, which corresponds to (say) node V{j. Trace the path leading from v to the root vertex in S, and find the nodes in G corresponding to each vertex in this path. These nodes determine the corresponding path in Q G from V tJ to T. Find the shortest path from V n to V{j using Greedy to complete the solution. 17 3.3 An Example Problem We illustrate the above algorithm with the following 5-product example Fi = 50; F 2 = 100; A = 800; d x = 60; d 2 = 280; d 3 = 180; d 4 = 1000; d 5 = 1900; pi = 5.0; p 2 = 2.5; p 3 = 2.0; p 4 = 1.0; p 5 = 0.45. At the end of step 1, the Greedy solution is V n - V 22 - V 23 - V 44 - V 55 with a value z G = 800. The gf t and K Jt values are shown in Table 1, while the Tj t and the A Jt values are given in Table 2. INSERT TABLES 1 AND 2 HERE From Table 1, it can be seen that T 5 = {V 45 ,V 55 };T 4 = {F 44 };r 3 = {Vi3,V 2 3,V 3 3};r 2 = {fii,Vaa};ri = {^11} The enumeration tree is shown in Figure 3, and the details for this tree are given in Table 3. Paths corresponding to the Greedy and the optimal solutions are shown on graph Q G in Figure 4. The optimal path is V\\ - V 22 - V33 - V44 - V55 with a value z* = 750. The arcs shared by both paths are shown with double lines while the arcs exclusive to the optimal path are shown in thin bold lines. 4 Conclusion This paper addresses the flexible multi-line design problem in a single-stage manufacturing system. For a given fixed cost of providing a line, and the fixed cost of each workcenter, the objective of the flexible multi-line design problem is to simultaneously determine the number of lines required as well as find the product-to-line allocation such that the total investment in lines and workcenters is minimized. In this paper, we consider the case in which a product can be assigned to multiple lines. We show that in this case, the optimal solution must satisfy the sequential assignment property, i. e., the products assigned to any line must be consecutively ordered in their processing times. We give a dynamic programming algorithm that solves the problem in polynomial time. We also construct an alternative, enumerative algorithm that results in a much smaller average running time by making use of an imbedded greedy algorithm. 18 REFERENCES 1. Ahmadi, R. H. and H. Matsuo (1991), "The Line Segmentation Problem," Operations Re- search, Vol. 39, 42-55. 2. Dar-El E. M. (1978), "Mixed- Model Assembly Line Sequencing Problems," Omega, Vol. 6, 317-323. 3. Dar-El, E. M. and S. Cucuy (1977), "Optimal Mixed- Model Sequencing for Balanced Assem- bly Lines," Omega, Vol. 5, 333- -341. 4. Farber, M., H. Luss and C.-S. 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Rachamadugu (1991), "Sequencing to Minimize Work Overload in Assembly Lines with Product Options," Management Science, Vol. 37, 572-586. 20 TABLE 1 Values of gQ and Kj t in the Example Problem 3 1 gft at t = ftjt 3»t Z — 1 2 3 4 5 l 2 3 4 5 150 350 450 1050 2250 50 50 500 1450 2 - 300 400 700 1300 - 150 500 3 - - 450 650 1150 - - 50 100 350 4 - - - 550 850 - - 50 5 - - - - 800 - - - - 9 G (t) 150 300 400 550 800 tit 1 2 2 4 5 TABLE 2 Values of r Jt and Ajt in the Example Problem i l Tjt at t = Ajt at t = 1 2 3 4 5 1 * 5 4 5 100 140 120 80 100 -160 -1634 2 - 140 280 240 260 - 160 -1474 3 - - 360 160 260 - - 80 80 -1474 4 - - - 80 580 - - - -1154 5 - - - - 1734 - - - - 21 TABLE 3 Details of the Enumeration Tree Vertex v Corresponding Node in Q G ^u s v 4>v Zv Remarks T 1 V 45 0.275 0.001250 850 2 V55 0.025 0.000562 800 Incumbent, UB = 800 3 V44 0.025 0.000562 800 0.000 4 Vis 0.475 0.001250 900 -0.200 Fathomed, v > UB + F2 5 V23 0.275 0.001250 850 0.000 6 V33 0.175 0.001250 900 0.100 Fathomed, v > UB + F2 7 Via 0.034 0.000562 850 -0.090 8 V23 0.025 0.000562 800 0.000 9 V33 0.980 0.000562 750 0.045 Current incumbent Revised UB = 750 10 v 12 0.275 0.001250 900 0.000 Fathomed,

UB + F 2 11 V 22 0.275 0.001250 850 0.000 Fathomed, 4> V >UB + F 2 12 Vl2 0.034 0.000562 900 0.000 Fathomed, 4> V >UB + F 2 13 ^22 0.034 0.000562 850 0.000 Fathomed, UB + F 2 14 V12 0.025 0.000562 850 0.000 Fathomed, (p v >UB + F 2 15 V22 0.025 0.000562 800 0.000 Fathomed,

UB + F 2 16 V12 0.980 0.000562 800 0.000 Fathomed, v >UB + F 2 17 V22 0.980 0.000562 750 0.000 22 © »"***"? • 1 1 s ' » * .' ■>•. wv» , o> o p 23 .\YsV*"»X*\V» • v*» CO O ex o o © bJO 24 3 o x ca 0) (-1 H (3 o CO 3 bC 25 Figure 4 - Paths 11* and IP in the Example Problem 26 HECKMAN IXI BINDERY INC. |a| JUN95 ., U .-T,,P^ ^MANCHESTER.