27 Congreso Nacional de Estadística e Investigación Operativa Lleida, 8-11 de abril de 2003
SOLVING THE MINSUM PRODUCT RATE VARIATION PROBLEM (PRVP) AS AN ASSIGNMENT PROBLEM N. Moreno 1, A. Corominas2 1IOC / DOE /ETSEIB UPC, 08028 Barcelona, Spain E-mail: [email protected]
2IOC / DOE / ETSEIB UPC, 08028 Barcelona, Spain E-mail: [email protected]
ABSTRACT The minsum PRVP consists in sequencing units of different types minimizing the sum of discrepancy functions between actual and ideal production rates. This problem can be reduced to an Assignment Problem (AP) with a matrix of a special structure. The efficiency of different algorithms for an AP in the minsum PRVP case has been compared. One new algorithm has been developed which uses some specific PRVP matrix properties. The work presents a computational experience with usually involved, symmetric, and new, asymmetric, objective functions where, for the first time, optimal solutions for instances of dimension up to 10.000 units have been obtained.
Keywords: Product Rate Variation Proble m, Assignment Problem AMS Classification: 90B35, 90C08.
1. Introduction The PRVP is an important produc tion problem that arises on mixed- model assembly lines. Consider that there are V variants or types to be produced on an assembly line that needs negligible time to change from one variant to another. Assume that each unit, regardless of its corresponding variant, requires the same production time (the cycle time of the line, which can be adopted as the unit of time without loss of generality).
There are U units to be produced, of which ui are of type i (i=1,...,V) with
∑u i =1
so that production rates of different product types will be kept as constant as possible. The time horizon is of U time units, where one copy of product i, i=1,...,V will be produced in each time period. The ideal production rate ri of each variant can be calculated as: ri = ui U , i = 1,...,V On other hand the problem objective can be considered as the determination of regular sequences in order to minimize the total variation in the production rates over time for different variants involved. The sequence can be described by means of x ih values (the total production of product i in time periods from 1 to h inclusive, h=1,...,U) so, in order to evaluate the deviation between actual x ih and ideal productions, discrepancy functions f i(x ih ,h) can be introduced. The total variation in product rates can be quantified minimizing the sum of discrepancy functions. We will call such a problem the minsum PRVP. The problem can be formalized as: U
min zS = ∑∑ f i ( xih , h ) h =1 i =1
The main objective to be achieved, solving the minsum PRVP, is obtaining a schedule that is “smooth” on average but not excluding the possibility that for some product units the deviations could be too high. One means of solution lies in its reduction to an Assignment Problem (AP) and the application of specific algorithms to it. This approach allows us to use a variety of objective functions. In this research the term large dimension is related to input data in the way that the instances with a large number of total product units to be produced and a large number of product types, as a secondary parameter, have been considered as the PRVP of large dimensions. The fact is that until now only computational experience of solving the minsum PRVP as an AP of small dimension has been published. But the results of solving the minsum PRVP as an AP for instances of large dimension have not yet been presented and specific algorithms that could take advantage of PRVP special characteristics, dealing with the problem as an AP, have not been developed. Moreover, there is an obstacle when the problem dimension is too large due to high time and memory requirements. Nevertheless, in industrial practice it is very usual to determine schedules that could correspond to thousands of units of different types to be sequenced. Therefore the large dimension problem is of special interest. This work presents the exploration and the development of specific algorithms taking into account that the AP matrix for the minsum PRVP has some specific properties, which allows us to obtain more efficiency of algorithms, and a computational experience of solving the aforementioned problem for instances of large dimensions (up to 10.000) with symmetric and asymmetric objective functions.
In Section 2 the current state of the problem is exposed; in Section 3 the minsum PRVP as an Assignment Problem (AP) with specific properties is solved and computational results are reported; in Section 4 some conclusions are drawn.
2. The state of the art of the problem PRVP is a particular case of the problem formulated by Monden (1983). The final objective of this problem is to find sequences where the quantity of each part used by the assembly process per unit time should be kept as constant as possible. Monden proposes a heuristic algorithm to solve this problem. Miltenburg (1989) has introduced the PRVP and formulated it as a nonlinear integer programming problem the goal being to minimize the total deviation in the production rates on a mixed-model assembly line. The objective function is defined as a sum of discrepancy functions. Miltenburg presents an exact algorithm with large time requirements; which is why he also proposes two heuristic algorithms. Inman and Bulfin (1991) have solved a version of the problem in which the objective function to be minimized is the sum of convex functions of differences between actual and “ideal” completion times of production for each unit, us ing due dates which correspond to "ideal", but not feasible (being generally non- integer), completion times of the individual products. Algorithms based on dynamic programming can be applied to obtain the exact solution of the problem. The algorithm proposed by Miltenburg et al. (1990) only permits solving small dimension instances, since memory and time requirements grow exponentially with the increase of a total number of units to be sequenced and even more with that of a type number. Bautista et al. (1996) present an algorithm based on bounded dynamic programming (BDP) which solves the problem for instances of larger dimensions in a short time due to the usage of bounds and their comparison with a value of the solution obtained solving the problem with a heuristic algorithm. Ding and Cheng (1993) develop a new sequencing heuristic procedure that solves the problem faster than Miltenburg's algorithm, according to their computational results. Also various efficient heuristic algorithms have been presented by Bautista et al.(2002). Kubiak (1993) has introduced the term PRVP. Kubiak and Sethi (1991,1994) explore the minsum PRVP and demonstrate that the objective function can be computed as a sum of penalties for deviation of actual instant of one unit productio n from the ideal instant. Their very important contribution is to show that, when the functions that compose the objective function are nonnegative and convex, the minsum PRVP can be reduced to an assignment problem (AP). Such a reduction makes possible the calculation of the optimal sequence for the problem in polynomial time in the total number of units produced, applying algorithms for the assignment problem. The intermediate step between the formulated problem and its reduction to an assignment problem is the reformulation of the problem to an equivalent one introducing the ideal position for each unit. Bautista et al. (1997) reintroduce the minsum PRVP and establish a general procedure of matrix calculation for the assignment problem, avoiding the calculation of an ideal
position. Moreover, they introduce a wide range of objective functions and show relationships between them. To sum up, different exact and heuristic algorithms exist to solve the PRVP. The re are some results presented by Boix and Sebastián (1995). They solve the PRVP as an assignment problem limited in total demand by 60. Korkmazel and Meral (2001) publish a small computational experience of solving the problem that is limited in total demand by 340 due to the executable program size restrictions. The latest published computational study has been made by Kovalyov et al. (2001). They have restricted their computational experience to dimension of 200 and solved in total 100.000 instances. Nevertheless there is no computational experience available of the minsum PRVP solution as an AP of large dimensions. The present work presents procedures, suitable with reference to computational time and flexible in relation to a variety of objective functions, to solve the minsum PRVP of large dimensions as an AP.
3. The minsum PRVP as an AP 3.1 Algorithms for the AP Exact approaches proposed for the solution of AP can be grouped into three classes: primal-dual algorithms (based on the identification of shortest paths), primal algorithms and dua l algorithms. An excellent synthesis of the current state of the AP has been presented by Dell’Amico and Toth (1998). From their computational experience the authors have concluded that the most efficient sequential codes correspond to primaldual and dual algorithms. Jonker and Volgenant (1987) develop the algorithm LAPJV based on the shortest path approach. In an adapted form, LAPJV can also be used on sparse problems. Volgenant (1996) has developed a new version of the algorithm LAPJV (LAPJVsp), which can solve very large problems within short computing time due to the idea to work just with the part of matrix selecting its special core. Amatller (1999) compiles solution methods of AP, problem instances, codes, data sets and presents a code of one primal algorithm for the AP (BARR). Barr’s primal algorithm (1977) considered as one of the best in the class of primal algorithms, is initially chosen. But he has come to the conclusion that this algorithm is more time consuming than the primal-dual and dual algorithms. Four available codes have been used in this research. They are: BARR and three codes based on identification of the shortest path, APC (Carpaneto et al. (1988)), NAUC (Bertsekas (1991)) and LAPJV (Jonker and Volgenant (1987)).
3.2 The minsum PRVP as a particular case of the AP The minsum PRVP considered as an AP has some specific properties presented below: 1. It is easy to obtain a good quality solution by the application of heuristic procedures. 4
2. Consider the matrix for AP in the way that the rows correspond to product units and the columns correspond to positions. In each row we have one (or, in the case of tie, two) ideal positions. The matrix values increase to the right and to the left from these positions. 3. The minsum PRVP regarded as AP also allows interpreting it as a transportation problem that offers a possibility of developing some new heuristic procedures. From this it can be inferred that: 1. A priori it is possible to think that primal algorithms could be more efficient than in the case of AP instances without specific properties, due to the availability of a good initial solution. 2. In general, optimal solutions contain only matrix elements situated not far from the ideal positions. From this, it seems reasonable to use the algorithm that only adopts the more promising elements with later checking of optimality of the obtained solution (if this proves negative the new elements must be included and the algorithm repeated). Bautista et al. (1997) show that the AP matrix for the minsum PRVP is composed of values that can be computed as follows : U
ϕik ( t ) = ∑ [ f i( k , h ) − f i ( k − 1,h)] h =t
The reduction of the minsum PRVP to an AP enables us to use a wide variety of objective functions to evaluate the regularity of a sequence, including appropriate different functions for each considered product type. The choice of the discrepancy functions depends on the purpose to be attained. For example, if we have to penalize underproduction more than overproduction or to give more importance to some specific product type it is possible to consider new objective functions by introducing asymmetric discrepancy functions or different weights, in the way that this new function takes into account desired features. For solving the problem with symmetric linear and symmetric quadratic discrepancy functions we have used the functions proposed by Miltenburg (1989): 2 xih 1 f i ( xih , h ) = − ri h
f i 2 ( xih , h ) = ( xih − ri ⋅ h ) f i 3 ( xih ,h) =
xih − ri h
f i 4 ( xih , h ) = xih − ri ⋅ h Solving the problem with asymmetric linear discrepancy functions we have defined them as follows: f i ( xih , h ) = α 1 ⋅ max(0, xih − ri ⋅ h ) + α 2 ⋅ max(0, ri ⋅ h − x ih )
The coefficients α 1 and α 2 can be chosen taking into account the objective to be reached. With condition α 2 > α 1 the underproduction is more penalized than the overproduction. In the case of asymmetric non-linear discrepancy functions we have defined them as: f i ( xih , h ) = α 1 ⋅ max(0, xih − ri ⋅ h ) + α1 ⋅ max(0, ri ⋅ h − xih ) + β ⋅ [ max(0,ri ⋅ h − x ih ) ] This function type allows penalizing of underproduc tion still more than the asymmetric linear function, because to the right from two minimal elements of the row it grows very quickly, if m is a large number, and its values are larger than to the left. Another type of functions we have used are weighted discrepancy functions. They have been defined as: f ( xih , h ) = wi ⋅ xih − ri ⋅ h and f ( xih , h ) = wi ⋅ ( xih − ri ⋅ h )2 where wi is a m
weight given to product type i ( i = 1,...,V ) .The weights wi have been defined in three ways: wi = ri , wi = 1 / ri , wi = ran(n) , where ran(n) is a random number uniformly distributed in [0,1]. The value of the objective function z S (introduced in Section 1) is given by the sum: V
z S = ∑∑ f i( 0 , h ) + ∑∑ ϕik ( t ik ) i =1 h =1 V
i = 1 k= 1
Finally, objective function ζ S = ∑∑ g i ( δ ik ) (Inman and Bulfin (1991), Bautista et i = 1 k= 1
al.(1997)) has been considered. This objective function is based on discrepancies δ ik between the actual values tik (the instant in which unit k of variant i is sequenced) and ideal values d ik withδ ik = t ik − dik . This type of function has been used only to reach a conclusion about algorithms performance in this case. Usually we have dealt with the objective function composed of f i ( xih , h ) discrepancy functions.
3.3 Computational experience This section presents computational results of solving the minsum PRVP as an AP for instances, that are generated by fixing the total number of units to be produced and the product type number, and randomly selecting the product number of each type, which is uniformly distributed. All available codes from the literature and codes that we have additionally implemented are in FORTRAN. The algorithms have been executed on a SUN 450 Ultra SPARC2 with a processor of 250 Mhz and 512 of RAM. Due to available capacity of memory we have dealt with a complete matrix for dimensions up to 10.000 but in the case of too large dimensions, as for U greater than 10.000, computing of the matrix should be adapted, calculating its element values whe n the algorithm required them. The initial computational experience includes the solution of one hundred full density integer cost matrix instances with discrepancy functions f i 2 and f i 4 (Miltenburg (1989)). The first computational results have shown that between available algorithms better results with regard to computational time are always given by LAPJV and APC, that closely follows LAPJV (Figure 1). 6
18 16 14 CPU(sec) average time
4 2 0 100
Figure 1: Time comparison for four available AP codes The first computational experiments have shown that for BARR (primal) increasing the problem dimension the computational time was so high that this algorithm could not compete with others. In order to reduce its computational time the new initialization with a good heuristic solution was tried instead of Barr’s original initialization, nevertheless algorithm performance has not been improved. We have decided to continue the study dealing with the best performing algorithm LAPJV. Later we have added a modified version LAPJVsp, which deals with the sparse matrices. Two modifications ha ve been introduced in LAPJVsp (LAPJVsp*) in order to improve computational time making use of some specific AP matrix properties in the minsum PRVP case. They are: different way of the appropriate matrix “core” choice and a new procedure of the optimality check. Regarding the efficiency of LAPJVsp* our computational experience has shown that the “quota” band (those elements which are situated in positions that correspond to deviation of actual production from ideal less than one unit) is the most suitable set of elements, which allows us to achieve the optimal solution of the minsum PRVP quickly and without its extension in the majority of cases. The computational experience has been carried out with dense double precision cost matrix for instances with dimension up to U=10.000. In total 1.200 instances with quadratic discrepancy functions have been solved. Moreover other different types of discrepancy functions have been tried. The influence of the total number of units, of number of product types and standard deviation σ of ui on computing time has been studied. The behavior of both algorithms for the same set of ins tances has been compared (Figure 2). Computational experience with a total number of units equal to 500 in all cases and with different product type numbers has been carried out. The increase of the number of types especially influences on the computing times of LAPJVsp* algorithm, since the number of elements of "sparse" matrix tend to come closer to those of the complete matrix when there are many product types with a small number of units. So a simple rule of the appropriate algorithm choice has been obtained which consists in the calculation of the value V/U. In the case if V/U