Applications of Mathematics and Computer Engineering
Using Nonlinear Mixed Integer Optimization in Printed Circuit Board Assembly Stefan Emet Department of Mathematics University of Turku Turku 20014 Finland
[email protected]
Olli S. Nevalainen Dept. of Information Technology University of Turku Turku 20014 Finland
Timo Knuutila Dept. of Information Technology University of Turku Turku 20014 Finland
Abstract: The Printed Circuit Board (PCB) assembly business is a fast paced field of industry where the manufacturers must quickly adapt their production to meet the customer requirements. The goal of the production scheduling is to prioritize jobs and optimize the usage of the production lines by allocating the jobs optimally to different lines. For the efficient usage of the production lines the workload balancing between the machines must be done properly. Balancing is usually a challenging operation and doing it consists of multiple calculations that solve the optimal placement time for each machine for different groups of components. In the present paper, the problem is modeled and solved using Mixed Integer Nonlinear Programming (MINLP) techniques. A pseudo-convex objective function for optimizing the production planning is presented. Different convexification techniques of non-linear functions are presented. The convexified model guarantees, in theory, that the global optimal solution will be found. A set of test problems are solved using the CPLEX-software. The presented techniques can easily be applied in the design of new industrial systems, and, to improve the performance of already existing ones. Key–Words: Mixed Integer Non-Linear Programming (MINLP), Printed Circuit Board (PCB), Assembly optimization
1 Introduction
much profit is obtained per time unit. There are many possibilities and open questions in the modeling of the price function P (t). However, when using gradientbased optimization methods, it is preferable to formulate functions that can be expressed explicitely and as convex as possible. Most optimization algorithms perform well when the problems are convex, that is, all functions are convex.
The mixed integer optimization methods provides efficient tools, with negligible costs, that can be applied in the solving of complicated technical problems in different industries. The objective in many industries is to achieve greater profits and/or smaller costs. The costs might be of any kind: raw material, equipment, salaries, energy, waste costs etc. Reduced energy consumption and/or a reduced amount of waste are also important environmental issues that can, in some cases, be tackled using mathematical optimization. In many industrial production planning problems the task is to decide which decisions should be made at what times in order to maximize the profit. The objective may, in theory, be formulated as the following: τ 1 P (t)dt (1) max τ 0
Definition 1 Assume that f : C → IR is differentiable. The function f is pseudo-convex if ∇f (x)T (y − x) ≥ 0 ⇒ f (y) ≥ f (x), ∀x, y ∈ C or equivalently f (y) < f (x) ⇒ ∇f (x)T (y − x) < 0, ∀x, y ∈ C. From the definition above it can be noted that all local minimas of a pseudo-convex function (that is, for all x such that ∇f (x) = 0), are also global ones. The following result regarding fractional functions is given in [9]:
where τ describes the length of the current period. The function P (t) returns, for example, the profit of the produced products at the time t. Note, that the same decisions are made repeatedly in periods of the length τ . By dividing the profit by the length of the period, the objective gives a comparable measure of how
ISBN: 978-960-474-270-7
Theorem 2 Assume that f (x) is differentiable and a real valued convex function and g(x) is a positive linear function over a convex set C. Then f (x)/g(x) is a pseudo-convex function over C.
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2 Convexification techniques
hence force the variable s to its lower bound. In analogy, if the objective is to maximize terms containing products of the form (6), it is enough to include the constraint (8) in the linear model. Functions of the form xy (10)
In the following some linearisation and/or convexification techniques are presented. These can be applied in problems containing non-convex funtions in order to rewrite the problem into an equivalent convex problem [10]. In many engineering problems the objective is to minimize a sum of terms containing so-called Chebyshev distances. In two-dimensions, the Chebyshev distance of two points (xi , yi )) and (xj , yj ), also known as the maximum metric, is defined as follows:
where x is a continuous variable and y is a binary variable are so-called bilinear functions [10]. These can also be rewritten in a linear form using an additional variable s as follows: x − xub (1 − y) ≤ s ≤ x − xlb (1 − y) xlb y ≤ s ≤ xub y
Definition 3 The Chebyshev-distance, dij , between two points (xi , yi ) ∈ IR2 and (xj , yj ) ∈ IR2 is dij = max{|xi − xj |, |yi − yj |}
where xlb and xub are lower- and upper bounds on the variable x. If the objective is to maximize the bilinear term (10), only the right-hand-side of the inequalities (11) and (12) are needed. Discontinuous functions can also be described linearly using these techniques. Consider the function f that is a function of the binary variable y: c1 , if y = 1, f (y) = c2 , otherwise.
(2)
The objective in many assembling and/or scheduling problems is to minimize the total length of the movements for different production units. Assume, that the objective is to minimize distances of the form (2): min max{|xi − xj |, |yi − yj |}
(3)
The absolute value function in (2) and (3) is nondifferentiable, but the minimization problem (3) can however be written in an equivalent linear form as follows min dij subject to ⎧ dij ≥ xi − xj ⎪ ⎪ (4) ⎨ dij ≥ −xi + xj d ≥ yi − yj ⎪ ⎪ ⎩ ij dij ≥ −yi + yj
Hence, f can be written linearly as follows f = c1 y + c2 (1 − y) = (c1 − c2 )y + c2
3 Printed circuit board assembly 3.1 A pseudo-convex problem In [1], [4] and [7] the production planning of a Printed Circuit Board (PCB) assembly line is considered and modeled as a Mixed Integer Linear Programming (MILP) problem. The main issue is the choice of components: which components should be assembled on the respective machines in order to minimize the production cycle time. In the following, an alternative approach using a pseudo-convex objective function is given.
Other logical formulations that are often used in planning problems are, for example, bilinear functions of the form: (5) y1 y2 where y1 and y2 are binary decision variables. More generally, logical functions of the form y1 y2 · · · yn
(6)
where yi ∈ {0, 1}, for i = 1, . . . , n, can be rewritten linearly using the additional variable s as follows: s ≥ 1 + (y1 + y2 + · · · + yn ) − n 1 s ≤ (y1 + y2 + · · · + yn ) n s ∈ {0, 1}
max subject to
(7) (8) (9)
Note, that if the objective is to minimize terms containing products of the form (6), it is enough to include the constraint (7) in the linear model. The minimization will force the constraint (7) to be active, and,
ISBN: 978-960-474-270-7
(11) (12)
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1 τ
K
k=1
ck Yk
⎧ I M ⎪ ⎪ tik zikm ⎪ ⎪ ⎪ i=1 m=1 ⎪ ⎪ M K ⎪ ⎪ ⎪ z ⎪ ⎪ ⎨ k=1 m=1 ikm zikm − di yikm ⎪ ⎪ ⎪ ⎪ yikm − zikm ⎪ ⎪ M ⎪ ⎪ ⎪ yikm ⎪ ⎪ ⎪ m=1 ⎩ yikm − Yk
≤τ = di ≤0 ≤0 ≤1 ≤0
(13)
Applications of Mathematics and Computer Engineering
In problem (13) the indexes i = 1, . . . , I, j = 1, . . . , C, k = 1, . . . , K and m = 1, . . . , M . The variables Yk and yikm are binary variables and zikm are integer ones. The continuous variable τ denotes the assembly time of the slowest machine in the production line. In (13) all constraints are linear ones, the only nonlinear function is the objective function that is of the form (1). The problem formulation (13) includes so-called Special Ordered Sets of order one (SOS1) that allows an efficient type of Branch and Bound (BB) algorithm. The pseudo-convex problem (13) can be solved using the Extended Cutting Plane (ECP) method [11] which is an extension of Kelley’s Cutting Plane (CP) method [6]. The (ECP) method that has been proven efficient on many complex engineering problems [3].
subject to ⎧ αijrs − M (1 − βijrs ) ⎪ ⎪ ⎪ ⎪ xij + yti r + ytj s − 2 ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ xij ⎪ ⎪ ⎪ i=0 ⎪ ⎪ n ⎪ ⎨ x j=0
ij
≤ γijrs ≥ βijrs = 1, = 1,
(17)
⎪ ⎪ ⎪ ui − uj + (n + 1)xij ≤ n, ⎪ ⎪ μ ⎪ ⎪ ⎪ ⎪ yur ≤ 1, ⎪ ⎪ ⎪ u=0 ⎪ μ ⎪ ⎪ ⎪ ⎩ yur = 1, u=0
where βijrs are binary variables and γijrs continuous ones. The variables αijrs are obtained from the cijrs [8]. The problem formulation (16) and (17) is hence a MILP-problem that can be solved to global optimality.
3.2 A convexified problem
4 Numerical examples In [8] a heuristic approach for optimizing the production of a PCB is given. The following MINLPproblem is considered in [8]:
min
n m
m
n
cijrs xij yti r ytj s
A test set of pseudo-convex PCB-problems (13) were generated using parameters given in [7], the dimensions of these problems are shown in Table 1. The cpu-times in the table indicate the computing time for solving corresponding problem. The problems were solved using the ECP-method and the CPLEXsoftware [5].
(14)
r=0 s=0 i=0 j=0
Table 1: Characteristics of example problems (13). subject to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
n
i=0 n
j=0
Variables 1 1 1 1
xij = 1,
ui − uj + (n + 1)xij ≤ n, ⎪ μ ⎪ ⎪ ⎪ yur ≤ 1, ⎪ ⎪ ⎪ u=0 ⎪ ⎪ μ ⎪ ⎪ ⎪ yur = 1, ⎩
(15)
where the variables xij and y are binary variables, the indexes i = 0, . . . , n, j = 0, . . . , n, r = 0, . . . , m, and u = 0, . . . , μ. The objective function (14) contains terms of Chebyshew-distances, denoted by cijrs , and products of the binary variables. More detailes and background of the problem (14) is given in [8]. Using the formulations in (4), (7), (11) and (12) the problem (14) can be written in a linear form: n m
m
n
γijrs
360 332 720 652 1800 1612 3600 3424
-
0.03 3.33 2.72 5.52
5 Conclusion In the present paper techniques for optimizing the production planning of a PCB were presented. First, an approach using a pseudo-convex objective function was presented. Such problems can be solved using, for example, the ECP-method. Convexification techniques for problems containing bilinear terms were presented and applied on a PCB-problem. The latter approach results in a linear formulation of the PCBproblem that can be evaluated rapidly, but the size of the problem may still be challenging.
(16)
r=0 s=0 i=0 j=0
ISBN: 978-960-474-270-7
cpu
It is shown in Table 1 that relatively large problems can be solved rapidly as these (PCB)-problems contain no nonlinear inequalities.
u=0
min
Constraints
continuous integer linear nonlinear [sec]
xij = 1,
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Applications of Mathematics and Computer Engineering
References: [1] J. C. Ammons, M. Carlyle, L. Cranmer, G. DePuy, K. Ellis, L. F. McGinnis, C. A. Tovey and H. Xu H, Component allocation to balance workload in printed circuit card assembly systems. IIE Transactions 29, 1997, pp. 265-275. [2] M. Avriel, Nonlinear Programming: Analysis and Methods. Prentice-Hall Inc., Englewood Cliffs, New Yersey, 1976. [3] I. Castillo, J. Westerlund, S. Emet and T. Westerlund, Optimization of Block Layout Design Problems with Unequal Areas: A Comparison of MILP and MINLP optimization methods. Computers & Chemical Engineering 30, 2005, pp. 54-69. [4] S. Emet, T. Knuutila, E. Alhoniemi, M. Maier, M. Johnsson and O. S. Nevalainen Workload balancing in printed circuit board assembly Int Journal Adv Manuf Techn 50, 2010, pp. 11751182. [5] ILOG, ILOG CPLEX 10.0 User’s manual, ILOG Inc. [6] J. Kelley, The cutting-plane method for solving convex programs, Journal of SIAM 8, 1960, pp. 703-712. [7] D. M. Kodek and K. Krisper, Optimal algorithm for minimizing production cycle time of a printed circuit board assembly line. Int. J. Prod. Res. 23, 2004, pp. 5031-5048. [8] T. Leipala and O. S. Nevalainen, Optimization of the movements of a component placement machine. Eur. J. Oper. Res. 38, 1989, pp. 167-177. [9] M. L. Pinedo, Planning and Scheduling in Manufacturing and Services. Springer, New York, 2005. [10] R. Porn, K-M. Bjork and T. Westerlund, Global solution of Optimization Problems with Signomial Parts. Discrete Optimization 5 , 2008, pp. 108-120. [11] T. Westerlund and R. P¨orn, Solving PseudoConvex Mixed Integer Problems by Cutting Plane Techniques. Optimization and Engineering 3, 2002, pp. 253–280.
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