Selection, Acquisition, and Allocation of ... - Semantic Scholar
Selection, Acquisition, and Allocation of Manufacturing Technology in a Multi-Period Environment Shabbir Ahmed School of Industrial & Systems Engineering Georgia Institute of Technology
Nikolaos V. Sahinidis∗ Department of Chemical and Biomolecular Engineering University of Illinois at Urbana-Champaign June 20, 2000. Revised: July 1, 2002.
Abstract This paper addresses a multi-period investment problem for the selection, acquisition, and allocation of alternative technology choices to meet the demand of a number of product families over a long-range planning horizon. We first show that the problem is N P-hard. We then present a solution strategy based upon perturbing the linear programming relaxation solution of a multiperiod mixed-integer linear programming ∗
Corresponding author: [email protected]
1
formulation. With mild assumptions on the problem parameters, we carry a probabilistic analysis which proves that the proposed solution approach is asymptotically optimal almost surely.
1
Introduction
The evolution of manufacturing technology in recent years has made available a wide variety of technological options for the manufacturing of products. Increased competition along with a dynamic demand environment, characterized by short product life cycles and changing product mixes, have placed technology adoption decisions among the key strategic concerns for a firm. The type of technology to adopt depends, for example, on the flexibility afforded and the investment and operating cost structures. Typically, flexible technology that is able to manufacture a wide variety of products is more expensive to acquire and operate than dedicated technology. The technology selection decision then has to be made by evaluating the tradeoff between “economies of scale” of the investment decisions and the “economies of scope” afforded by the ability to meet the demand of a variety of products. In addition to the selection of the technology type, the manufacturing firm also has to decide when and how much of each technology type to acquire. These decisions are governed by the forecasted available technology and investment costs. The strategic technology adoption decisions have to be made in conjunction with operational level decisions of how the available capacities are to be allocated to meet the demands of the various products. It is often necessary to model strategic technology adoption problems over planning horizons with a very large number of time periods. For instance, integer programs that arise in the modeling of petroleum production involve quarterly discretization of planning horizons that span several decades (Haugland, Hallefjord & Asheim 1988, Nygreen, Christiansen, Haugen, Bjorkvoll & Kristiansen 1998, van den Heever & Grossmann 2000). The size of these
2
models, as well as others that arise in long-range capacity planning of electric utilities (Levin, Tishler & Zahavi 1983, Wang, Jaraiedi & Torreis 1996), is predominantly determined by the temporal rather than the spatial component of the problem. The purpose of the current paper is to present an optimization framework to aid technology adoption decisions problems with a large number of time periods. In particular, we develop a fast, linear programming-based, approximation scheme that is asymptotically optimal “with probability 1” as the number of time periods increases for a hard combinatorial optimization problem of selection, acquisition, and allocation of several types of technology choices for the production of multiple product families over a multi-period planning horizon. There exists a wide body of literature on optimization approaches to multifacility capacity expansion and evaluation of technology investments. For surveys of the early literature on capacity planning the reader is referred to Freidenfelds (1981) and Luss (1982). Fong & Srinivasan (1981) considered the problem of capacity expansion of multiple facilities to satisfy the demand in different regions to minimize combined capacity expansion and transportation costs. The authors developed an efficient heuristic for the resulting mixed-integer linear programming (MILP) based upon interchanging capacities of two facilities. Klincewicz, Luss & Yu (1988) addressed a large-scale capacity planning model for opening, expanding, and closing multiple facilities to satisfy demands of different regions. The authors developed a heuristic based upon network optimization techniques for this problem. Fine & Freund (1990) formulated the problem of acquisition and allocation of dedicated and flexible technology under demand uncertainty as a two-stage stochastic program. Linearity of the investment costs was a key assumption in their development, thus providing no ability to model economies of scale. Sahinidis & Grossmann (1992) considered the problem of multiperiod capacity expansion of chemical processing networks. Using a number of MILP reformulation techniques, the authors developed exact algorithms for this problem. Li & Tirupati (1994) considered the problem of selection, acquisition, and allocation of dedicated and flexible technology to 3
satisfy the demand of a number of product families. Under certain assumptions, the authors showed that the constraint matrix of the resulting formulation is totally unimodular, thus resulting into integral solutions for concave cost functions. This and other structural properties were exploited to develop two heuristics for the problem. Rajagopalan (1994) studied the problem of capacity acquisition for a number of alternative technologies to satisfy the demand of a single product family. Assuming non-decreasing demand, specific cost structures, and fixed charges to model economies of scale, the author used variable disaggregation to develop an MILP formulation whose linear programming relaxation has integral solutions. Recently, Rajagopalan, Singh & Morton (1998) presented a stochastic multiperiod capacity expansion and replacement model where technological capacity becomes available only at certain time periods. The authors exploited the structure of the optimal solution to develop a dynamic programming strategy. Because of the combinatorial complexity of most of the above-mentioned problems, the use of heuristic strategies is prevalent in the technology selection and capacity expansion literature (Fong & Srinivasan 1981, Klincewicz et al. 1988, Li & Tirupati 1994). However, the qualitative justification of these heuristics has been based upon empirical evidence. In a recent line of work, Liu & Sahinidis (1997) and Ahmed & Sahinidis (2000) proposed LP-based approximation schemes for a capacity expansion problem in the chemical process industries. Using probabilistic analysis tools, the schemes were theoretically proven to be asymptotically optimal in the number of time periods. The principal aim of probabilistic analysis of an algorithm for an optimization problem is to characterize the probable performance of the algorithm on a “typical” instance of the problem. At the onset of the analysis, a typical problem instance is described by assuming probability distributions on the problem parameters. Tools from probability theory are then used to draw inferences regarding the distributions of various measures of algorithmic performance. As pointed out by Coffman & Lueker (1991), complete knowledge of these distributions is usually out of reach and we must often settle for 4
weaker results such as asymptotic distributions. That is, probabilistic inferences about the algorithm’s performance can only be quantified when the problem size is extremely large. Beginning with the seminal works of Beardwood, Hamilton & Hammersley (1959) and Karp (1976), we have witnessed a wide body of literature on the probabilistic analysis of algorithms for classical combinatorial optimization problems such traveling salesman problems (Karp & Steele 1985), knapsack problems (Dyer & Frieze 1989), bin-packing problems (Coffman & Lueker 1991), vehicle routing problems (Bramel & Simchi-Levi 1997), and generalized assignment problems (Romeijn & Piersma 2000), only to name a few. Reviews of this topic appear in Karp, Lenstra, McDiarmad & Kan (1985) and Bramel & Simchi-Levi (1997). The remainder of the paper is organized as follows. In Section 2, we model a very general problem of selection, acquisition, and allocation of several types of technology choices for the production of multiple product families in a dynamic demand environment. The technology choices are classified according to their acquisition and allocation costs as well as the degree of flexibility afforded. We consider discrete time periods and fixed charge economies of scale to formulate the problem as a multiperiod MILP. The developed model is quite general in terms of the existing literature. We do not require non-decreasing demand patterns and allow several technology types that are capable of producing subsets of the product families. Furthermore, we allow for limited availability of procurable capacity of the various technology types along the planning horizon. We show that the problem under consideration is N Phard. This complexity characterization motivates the need for efficient heuristic methods for the problem. In Section 3, we present a polynomial time heuristic method that is guaranteed to produce feasible solutions for the problem under study. By means of probabilistic analysis tools, in Section 4, we prove that the relative error between the heuristic and the exact MILP solutions almost surely vanishes asymptotically as the problem size increases. Computational results are presented in Section 5 demonstrating that the heuristic produces good quality solutions for small- as well as large-scale problems. The computations also demonstrate 5
that, whereas the optimality gap of a myopic heuristic worsens with increasing number of time periods, the optimality gap of the proposed heuristic improves in accordance to our analytical results.
2
Model Formulation and Complexity
In this section, we present an MILP formulation for the problem of determining an optimal strategy for the selection, acquisition, and allocation of various technology types to satisfy the demands of a number of product families over a multiperiod planning horizon. The investment and allocation costs for the various technology types, as well as the demands of the various product families are assumed to be known over the planning horizon. The objective is to time and size technology acquisitions, and allocate the available capacity to various product families in a way that minimizes the total discounted investment and allocation cost for the entire planning horizon. A fixed-charge cost model captures the economies of scale in the investment costs. The following notation is used to describe the model.
Sets and indices: I set of technology choices; i
index for technology choices (i ∈ I);
J
set of product families;
j
index for product families (j ∈ J );
Ij
set of technology types capable of producing product family j;
Ji
set of product families that can be produced with technology i;
n
number of time periods in the planning horizon;
t
index for time periods (t = 1, . . . , n).
Parameters:
6
αit
variable cost of acquiring one unit of capacity of technology type i in period t;
βit
fixed cost associated with acquiring capacity of technology type i in period t;
δijt
cost associated with allocating a unit of capacity of technology type i to product family j in period t;
µij
yield rate of product family j per unit capacity of technology type i;
djt
demand of product family j in period t;
Uit
capacity of technology type i available for acquisition in time period t;
Xi0
initial capacity of technology type i.
Variables: Wijt amount of capacity of technology type i allocated to product family j in period t; Xit Yit
amount of capacity addition of technology type i in period t; binary variables indicating whether capacity of technology type i is acquired in period t (Yit = 1) or not (Yit = 0).
Without loss of generality, we assume that ∪i∈I Ji = J and ∪j∈J Ij = I, so that there is a technology available for each product family and I consists of only those technologies that can produce one or more of the product families in J . A technology i is dedicated if Ji is a singleton and flexible if |Ji | > 1. We also assume that all cost parameters are appropriately discounted. Following Rajagopalan (1994) and Li & Tirupati (1994), we ignore inventory fluctuations and do not permit disposal of excess capacity. With the above assumptions, the technology selection, acquisition, and allocation problem is formulated as follows.
Problem (P):
n
min
αit Xit + βit Yit +
t=1 i∈I
j∈Ji
Wijt ≤ Xi0 +
δijt Wijt
(1)
j∈Ji
Xit ≤ Uit Yit
s.t.
t
τ =1
7
t = 1, . . . , n; i ∈ I
(2)
Xiτ t = 1, . . . , n; i ∈ I
(3)
i∈Ij
µij Wijt = djt
Wijt , Xit ≥ 0 Yit ∈ {0, 1}
t = 1, . . . , n; j ∈ J
(4)
t = 1, . . . , n; i ∈ Ij ; j ∈ J
(5)
t = 1, . . . , n; i ∈ I
(6)
In the formulation above, the objective (1) minimizes the sum of investment and allocation costs over the planning horizon. The variable upper bound (2) ensures that the level of technology acquired in any period does not exceed the capacity available for acquisition in that period. Constraint (3) enforces the condition that, for any technology, the total capacity allocated to the product families does not exceed the installed capacity. Constraint (4) ensures that the allocated capacities will meet product demand. Replacing this constraint by a two-sided inequality or by any other system of linear equalities and inequalities in terms of the allocation variables (Wijt ) does not affect the analysis to follow. The non-negativity and binary restrictions of the variables are enforced through (5) and (6). The formulation presented above is fairly general with respect to the similar models in the existing literature. Unlike Rajagopalan (1994), we do not make assumptions on the structure of the cost components and demand with respect to time. Furthermore, we are allowing flexible technology types capable of producing several product families. An essential difference with the single flexible technology model of Li & Tirupati (1994) is that we are accommodating a wide variety of flexible technologies that are capable of producing different subsets of product families, and part of the decision process includes selecting among these various technology types. We also allow for limited availability of procurable capacity of the various technology choices along the planning horizon. Finally, the possibility of unequal yield rates (µij ) between any two technology types as well as the possibility of unavailability of a particular technology type in a particular period makes our model more general than that of Fong & Srinivasan (1981). These modeling generalizations introduce substantial complexity into the problem. Next, we prove that the problem under study is N P-hard. 8
Lee & Luss (1987) addressed the complexity of multi-facility capacity expansion problems with concave costs under the assumption that infinite capacities are available for expansion. They proved that, although the complexity is exponential in the number of facilities, the problem can be solved in polynomial effort with respect to the number of time periods when the set of facilities is fixed. Owing to the similarity between multi-facility capacity expansion problems and problem (P), it might appear that the analysis of Lee and Luss is applicable to (P). However, the presence of bounds (Uit ) on the procurable capacity makes problem (P) difficult even in the case of a single technology choice as discussed next. Consider first the n-period capacitated lot-sizing problem with zero holding costs (Bitran & Yanasse 1982): Problem (CLSP):
min
n
(pt xt + st yt )
t=1
s.t.
It−1 + xt = rt + It
t = 1, . . . , n
x t ≤ Kt y t
t = 1, . . . , n
It , x t ≥ 0
t = 1, . . . , n
yt ∈ {0, 1}
t = 1, . . . , n
An instance of (CLSP) is defined by the number of time periods n, the costs coefficients pt and st , the production capacities Kt , the demands rt and the initial inventory I0 . Given any such instance of the (CLSP) we can construct a single technology instance of (P) as follows. - The set of technology types is a singleton, i.e., I = {i}. - There are n time periods. - The set of product families is also a singleton J := {j}. - The yield rate is µij = 1. - The allocation costs are all zero, i.e., δijt = 0 for all t = 1, . . . , n. 9
- The investment cost components are αit = pt and βit = st for all t = 1, . . . , n. - The available levels of technology are given by Uit = Kt for all t = 1, . . . , n. - The initial available capacity is Xi0 = I0 . - The demand of the product family j is given by djt =
t
τ =1 rτ .
With the above data, problem (P) reduces to: Problem (P1):
min
n
(pt Xt + st Yt )
t=1
s.t.
Xt ≤ Yt Kt t = 1, . . . , n Wt ≤ I0 +
t
Xτ t = 1, . . . , n
τ =1 t
Wt =
τ =1
rτ t = 1, . . . , n
Xt , Wt ≥ 0 t = 1, . . . , n Yt ∈ {0, 1} t = 1, . . . , n
Note that the indices i and j have been dropped since there is a single technology choice and a single product. Lemma 2.1 Problems (P1) and (CLSP) are equivalent. 2
Proof. See Appendix I.
The (CLSP) with zero holding costs has been shown to be N P-hard for the following parameter structures with respect to time periods (Bitran & Yanasse 1982): (i) Constant st , non-increasing pt , non-increasing Kt , (ii) Constant st , non-decreasing pt , non-decreasing Kt , (iii) Non-decreasing st , zero pt , non-decreasing Kt , 10
(iv) Non-increasing st , zero pt , non-increasing Kt . Since for any given instance of (CLSP) we can construct an equivalent single technology instance of (P), we have the following result: Theorem 1 Problem (P) with one of the above combinations of investment cost structure and the procurement bound is N P-hard.
3
A Temporal Capacity Shifting Heuristic
The complexity characterization of problem (P) in the previous section motivates the need for efficient heuristic strategies. In this section, we develop a capacity shifting heuristic that provides a theoretically sound means for obtaining solutions to problems with long planning horizons. The motivation for the development of this heuristic and subsequent analysis comes from the empirical observations made by Chang & Gavish (1995) and Liu & Sahinidis (1995) on the effect of the length of planning horizon on the quality of relaxation gaps in MILP models of capacity expansion problems. While investigating lower bounding procedures for telecommunication network expansion problems, Chang & Gavish (1995) considered a capacity expansion problem similar to (P) as one of their subproblems (Chang & Gavish 1995, problem (SP2) in pg. 48). The authors conducted computational studies to study the effect of the length of the planning horizon on the quality of the relative gap of the Lagrangian relaxation of (SP2), and made the observation that the gap quality did not deteriorate as the number of planning periods grew. Liu & Sahinidis (1995) actually observed decreasing relative LP relaxation gaps with the increase in the length of the planning horizon in chemical process network expansion problems. These observations appear to be counter-intuitive since a larger number of time periods implies a larger number of binary variables in the
11
integer program and increased complexity per Corollary 1. In the next section, we provide a mathematical explanation of this phenomenon using the heuristic strategy developed next. The empirical evidence cited above suggests the possibility of construction of good quality solutions from the LP relaxation solution for instances of (P) with large planning horizons. Note that simply rounding up the values of the binary variables (Yit ) in the LP relaxation of (P) results in a feasible solution. However, such a naive rounding strategy might result in very poor solutions, possibly requiring technology adoption to be carried out in all periods. We propose to perturb the LP relaxation solution in a way that keeps the number of technology acquisitions small. Since fixed costs of technological capacity addition are typically high, we wish to acquire as much capacity as possible whenever the procurement decision is taken. It is easy to see that, if the investment costs are equal in all periods, there is an optimal solution to (P) where capacity acquisitions are made only in the earliest period when the technology becomes available. Furthermore, in this case, there is an optimal solution where the capacity addition equals the availability bound (Uit ) in all periods except perhaps the last one in which capacity is added. Using these observations, we propose to perturb the LP relaxation solution by shifting technological capacity additions from later periods to earlier ones if procurable technology is available. The first step of the heuristic is to solve the LP relaxation of problem (P) using standard techniques. For large-scale problems, one can decompose the problem across time periods by relaxing constraint (3), and using subgradient methods. A Benders decomposition approach where capacity acquisition decisions are made in the master problem and allocation decisions are made in the subproblems can also be used. To construct a feasible solution to (P) from its LP relaxation solution, we retain the values of the allocation variables (Wijt ). For any technology, we consider only the time periods where the capacity was acquired in the LP relaxation solution and shift the capacity additions from later periods to an earlier period as long as the total installed capacity does not exceed the availability in that period. In this 12
way, we construct a solution where capacity additions are equal to the availability bounds in all periods except the last one where capacity is added. The final step is to round up the binary variables corresponding to periods where the capacity is acquired in the heuristic solution. A formal statement of this heuristic is presented next.
Temporal Capacity Shifting Heuristic LP 1. Solve the LP relaxation of (P). Let the (XitLP , YitLP , Wijt ) denote the optimal solution
to the LP relaxation. For each i ∈ I, let Ti := {t|YitLP > 0}, i.e., the set of time periods when capacity is added. Let Ti = {t1 , t2 , . . . , tpi }. H H LP 2. Denote the heuristic solution by (XitH , YitH , Wijt ). Set Wijt ← Wijt for all i, j, and t.
3. Repeat the following step for all i ∈ I: Do for h = 1, . . . , pi Set XitHh ← XitLP , YitHh ← 0, and k ← h + 1. h While XitHh < Uith and k ≤ pi do, Let δ = min{Uith − XitHh , XitLP }. k Set XitHh ← XitHh + δ and XitLP ← XitLP − δ. k k Set k ← k + 1. End While. If XitHh > 0, set YitHh ← 1. End Do. The time complexity of the above heuristic is O(T LP +|I|n2 ), where T LP is the complexity of solving the LP relaxation. The following two properties of the heuristic solution are obvious from the construction. 13
t
Lemma 3.1 For any technology type i ∈ I,
τ =1
n
Lemma 3.2 For any technology type i ∈ I,
t=1
XiτH ≥ XitH =
t
τ =1
n
t=1
XiτLP for all t = 1, . . . , n.
XitLP .
Using the above properties, the following result is easily obtained. Theorem 2 The capacity shifting heuristic produces a feasible solution to (P). We shall now establish a crucial property of the capacity shifting heuristic that is needed for the qualitative analysis in the next section. For the remainder of this section and the next one, we make the following assumption: Assumption 1 For any technology i ∈ I, Uit = Ui for all t = 1, . . . , n. According to this assumption, the available procurable capacity of any technology is constant in all periods. This assumption is made only for the ease of exposition of the analytical results presented next. In the Appendix, we prove that the main result of Section 4 is valid even without this assumption. n
Lemma 3.3 For any technology type i ∈ I,
t=1
YitH −
n
t=1
YitLP ≤ 1.
Proof. Under Assumption 1, XitH is either Ui or 0 for all time periods except at most one, for which XitH = where 0 < < Ui . Let m be the number of time periods for which XitH = Ui . Then, clearly
n
solution satisfies Thus,
n
t=1
t=1
XitH = mUi + and
n
LP t=1 Xit ≤ Ui
YitLP ≥ m +
Ui
n t=1
n
t=1
YitH ≤ m + 1. Note that the LP relaxation
YitLP . Using Lemma 3.2, mUi + ≤ Ui
n t=1
YitLP .
and the result follows.
2
Note that the above property will not be satisfied by a naive round-up strategy, since such a scheme could potentially lead to rounding up the binary variables in all periods. The capacity shifting heuristic can be improved by shifting only to periods that offer a cost benefit or by integrating it with other heuristic methods such as those proposed by Fong 14
& Srinivasan (1981) and Li & Tirupati (1994). Such improvements will only produce better quality solutions. However, in the next section, we show that the capacity shifting heuristic is asymptotically optimal even in its simple form.
4
Probabilistic Analysis
In this section, we carry out a probabilistic analysis of the error of the capacity shifting heuristic solution for the technology selection problem. We consider a fixed set of product families J , a fixed set of technology choices I, and a fixed set of yield rates µij . The probabilistic analysis will be carried out on instances of (P) consisting of increasingly more time periods where the problem parameters such as costs, product demands, and technological availability are drawn from the following probabilistic model. - The availability, Ui , of technology type i is drawn from distributions with bounded support, i.e., Ui ∈ [U i , U i ], with U i > 0 for all i ∈ I. - The demands of the various product families and the cost parameters for the various technologies are assumed to follow one of the following distributions: (D1) For each product family j, the demands, djt , are i.i.d. random variables with finite first and second moments. For a given technology i, the cost parameters αit and βit for each time period are either i.i.d. random variables with finite first and second moments or have a distribution (not necessarily i.i.d.) with a bounded support. (D2) For each product family j, the demands, djt , are independent random variables (not necessarily identically distributed) with bounded second moments. For a given technology i, the cost parameters αit and βit for each time period are random variables with a bounded support. 15
(D3) For each product family j, the demands, djt , are random variables with bounded support. For a given technology i, the cost parameters αit and βit for each time period are either i.i.d. random variables with finite first and second moments or have a distribution (not necessarily i.i.d.) with a bounded support. - The problem parameters are such that the random instance is feasible. This can be easily ensured by including an expensive artificial technology with infinite capacity that is capable of producing all product families. Distributions (D1)-(D3) permit the inclusion of problem instances with a wide variety of demand trends and cost structures. The main tool in our probabilistic analysis is the asymptotic convergence properties of extreme order statistics (Galambos 1987). For a sequence of random variables {x1 , x2 , . . . , xn }, consider the random variables xn = maxj=1,...,n {xj } and xn = minj=1,...,n {xj }. The asymptotic theory of extreme order statistics concerns with the study of the limiting distributions of the statistics xn and xn as n approaches infinity. We shall make use of the following results from asymptotic extreme value theory: Lemma 4.1 Suppose {xj }nj=1 is a sequence of non-negative i.i.d. random variables with finite second moment. Let zn = maxj=1,...,n {xj }, then limn→∞
zn √ n
= 0 with probability 1
(w.p. 1). Proof. Consider the sequence of random variables {yj }nj=1 with yj = kxj where k > 0 is an arbitrary constant. As {yj }nj=1 is also a sequence of non-negative i.i.d. random variables with finite second moment, E[yj2 ] < +∞. From integration theory (Galambos 1988, Chapter
2 2, Theorem 3), it is known that E[yj2 ] < +∞ if and only if ∞ n=1 P r(yj > n) < +∞ or, since √ yj is non-negative, ∞ n) < +∞. Since yj is identically distributed for all j, the n=1 P r(yj > √ n) < +∞. Let zn = maxj=1,...,n {yj }. From last condition is equivalent to ∞ n=1 P r(yn >
extreme value theory of ordered statistics (Galambos 1987, Corollary 4.3.1),
16
∞
n=1
P r(yn >
√
n) < +∞ implies P r(zn >
√
n infinitely often ) = 0, i.e., P r(lim supn→∞
zn = kzn , the latter statement is equivalent to P r(lim supn→∞ and k was arbitrary, it follows that P r(limn→∞
zn √ n
zn √ n
zn √ n
≤ 1) = 1. As
≤ k1 ) = 1. Since
zn √ n
≥0 2
= 0) = 1.
Lemma 4.2 Suppose {xj }nj=1 is a sequence of non-negative independent random variables with bounded first and second moments. Let zn = maxj=1,...,n {xj }. Then limn→∞
zn n
= 0
w.p. 1. Proof. For a non-negative random variable, we have from Markov’s inequality P r(xn > kBn ) ≤
E[x2n ] . (kBn )2
Since xn for all n have bounded first and second moments, there exist µ and
σ 2 such that µn ≤ µ and σn2 ≤ σ 2 for all n, where µn and σn2 are the first and second moments of xn . We thus have E[x2n ] ≤ σ 2 + µ2 for all n. Hence, Letting Bn = n, we have
∞
n=1
∞
n=1
P r(xn > kBn ) ≤
σ 2 +µ2 k2
∞
1 2. n=1 Bn
P r(xn > kBn ) < +∞ for all k > 0. It then follows from
Theorem 4.4.1 in Galambos (1987) that P r(lim supn→∞ znn = 0) = 1. Moreover, since for all n, we have limn→∞
zn n
zn n
≥0 2
= 0 w.p. 1.
Lemma 4.3 For any i ∈ I, under distribution (D1) and (D3), we have n
√
YitH = 0 w.p. 1 n
t=1
lim n→∞ and, under distribution (D2), we have
n
lim
n→∞
t=1
n
YitH
= 0 w.p. 1.
Proof. For a random instance of (P) with n time periods, the maximum demand of product family j is dmax = maxt=1,...,n {djt }. Clearly, in the optimal solution of the LP relaxation of j (P), the final capacity of any technology choice i will satisfy Xi0 + while, additionally,
n
t=1
XitLP =
n
t=1
n
t=1
XitLP ≤
Ui YitLP . From Lemma 3.3, we then have 17
j∈Ji
dmax j
n
YitH
≤
max j∈Ji (dj
− Xi0 )+
Ui
t=1
+ 1,
where (·)+ = max(·, 0). If the demands satisfy distribution (D1), then by Lemma 4.1, limn→∞
dmax j √ n
= 0 w.p. 1 for all j ∈ Ji . If the demands satisfy distribution (D2), then by
Lemma 4.2, limn→∞
dmax j n
= 0 w.p. 1 for all j ∈ Ji . If the demands satisfy distribution (D3),
then for each j ∈ Ji there exists a finite upper bound dj on the demand in all time periods. ≤ dj w.p. 1 and limn→∞ Then, dmax j
dmax j √ n
2
= 0 w.p. 1.
For an instance of (P) with n time periods, let znLP , znIP , and znH denote the optimal value of LP relaxation, the optimal value of the integer program, and the value of the heuristic solution, respectively. We now state the main results of this section. Theorem 3 For the specified probability model, limn→∞
H −z IP zn n n
= 0 w.p. 1.
Proof. We know that 0 ≤ znH − znIP ≤ znH − znLP . From the construction of the heuristic solution, we also have znH − znLP ≤
n
i∈I
t=1
αit XitH − αit XitLP + βit YitH − βit YitLP . Let
αimax = maxt=1,...,n {αit }, αimin = mint=1,...,n {αit }, βimax = maxt=1,...,n {βit }, and βimin = mint=1,...,n {βit }. Using Lemma 3.2 and Lemma 3.3, we have znH
−
znIP
≤
i∈I
≤
αimax
−
βimin
+
αimin
n
XitH
+
βimax
t=1
U i (αimax
n
YitH
−
βimin
t=1
−
αimin )
+
(βimax
−
n
βimin )
YitH
t=1
Since 0 ≤ αimin and 0 ≤ βimin ≤ βimax for all i ∈ I, we have: −
znIP
≤
βimax
+
(αimax U i
+
βimax )
n
YitH
.
t=1
i∈I
Dividing by n, we obtain:
βimax αimax βimax znH − znIP + √ ≤ + Ui √ n n n n i∈I
18
YitH
t=1
i∈I
znH
n
n
YH √ it . n
t=1
.
−1
Under distributions (D1) and (D3), the cost parameters are either i.i.d. or have bounded support. In either case, limn→∞
αmax i √ n
= 0 and limn→∞
βimax √ n
= 0 w.p. 1. Thus, the result
follows from Lemma 4.3. Similarly, we also have,
βimax znH − znIP ≤ + αimax U i + βimax n n i∈I
n
t=1
YitH
n
.
Under distribution (D2), the cost parameters have bounded support. Hence, (U i αimax + βimax ) < +∞ and, for all i, βimax < +∞ and are independent of n. Thus, the result follows 2
from Lemma 4.3.
To characterize the asymptotic properties of the relative error of the heuristic solution, we make the following assumptions: Assumption 2 For any t, δijt /µij ≥ 1 for all j ∈ Ji and i ∈ I. Assumption 3 For any t, djt ≥ 1 for at least one j ∈ J . The quantity δijt /µij can be interpreted as the unit production cost of family j from technology i in period t. Then, Assumption 2 states that the unit production cost of a product family in any period is at least 1. Similarly, Assumption 3 states that in each period there is unit demand of at least one of the product families. These assumptions are satisfied by appropriate scaling for positive lower bounds on the production costs and demands. Corollary 3.1 Under Assumptions 2 and 3 and the specified probability model, znH − znIP = 0 w.p. 1. n→∞ znIP lim
Proof. It suffices to show that znIP is Ω(n). Consider the dual of the LP relaxation of (P): znD = max
n
t=1
(
i∈I
Xi0 γit + ξit ) +
19
j∈J
djt ηjt
s.t.
λit −
n
γiτ ≤ αit
t = 1, . . . , n; i ∈ I
−Ui λit + ξit ≤ βit
t = 1, . . . , n; i ∈ I
γit + µij ηjt ≤ δijt
t = 1, . . . , n; i ∈ I; j ∈ Ji
τ =t
λit , γit , ξit ≤ 0 ηjt unrestricted
t = 1, . . . , n; i ∈ I j∈J
where λit , γit , and ηjt are the dual variables corresponding to constraints (2), (3) and (4), respectively. Here, ξit is the dual variable corresponding to the LP relaxation constraint Yit ≤ 1. Consider the following feasible solution to the above problem: λit = 0, ξit = 0, γit = 0, and ηjt = mini∈Ij {δijt /µij }. Then, under Assumptions 2 and 3, we have znD is Ω(n). Since the problem instances are assumed to be feasible, the result follows from Theorem 3. 2
The above result also establishes the asymptotic tightness of the LP relaxation and the Lagrangian relaxation as empirically observed by Liu & Sahinidis (1995) and by Chang & Gavish (1995). Let us denote the objective value of the Lagrangian dual solution of an instance of (P) with n periods by znLD . Corollary 3.2 Under Assumptions 2 and 3 and the specified probability model, lim n→∞
znIP − znLP = 0 w.p. 1 and znIP
lim n→∞
znIP − znLD = 0 w.p. 1. znIP
Proof. From the proof of Theorem 3, we have limn→∞ znLD ≤ znIP ≤ znH , the result follows from Corollary 3.1.
20
H −z LP zn n n
= 0 w.p. 1. Since znLP ≤ 2
5
Computational Results
Even though the main purpose of our paper is the theoretical analysis of the capacity shifting heuristic, in this section, we empirically investigate the performance of the heuristic on randomly generated instances as well as a large-scale problem with industrial data from the literature. The purpose of the computations is to investigate the convergence properties of the asymptotically optimal scheme and gain some insight on how model parameter choices affect solution characteristics. We also compare the proposed approach to a commonly used “myopic” heuristic.
Generation of test problems Test problem instances are generated from a production network with two product families (numbered 1 and 2) along with two dedicated technologies (numbered 1 and 2), one for each family. There is also a single flexible technology (numbered 3) capable of manufacturing both families. To guarantee feasibility, we include an artificial technology with infinite capacity that can produce both families at a very high cost. However, in the problem instances generated, this artificial technology was always redundant. No initial capacity is assumed for any technology type. The means and standard deviations used to generate problem parameters for various instances are shown in Table 1. Here, K denotes the ratio between fixed and variable cost of expansion. Unless otherwise specified, K = 2. The actual distributions used to generate the parameters are shown in Tables 2-4, where the distribution parameters are from Table 1. In these tables, N (µ, σ) is a normal distribution with mean µ and standard deviation σ, and U (a, b) is a uniform distribution with support [a, b]. Recall from the probability model of Section 4 that distribution (D1) was that of i.i.d. demand and i.i.d./bounded cost parameters, distribution (D2) was that of i.d demand and bounded cost parameters, and distribution (D3) was that of bounded demand and i.i.d./bounded cost parameters. The
21
independent (but not identical) demands in (D2) were generated by letting µdjt = 2tµdj /n for t ≤ n/2 and µdjt = µdj − 2(t − n/2)µdj /n for t > n/2, i.e., increasing mean for the first half of the planning horizon, and then decreasing mean for the remainder.
Effect of distribution Our first experiment is to investigate the convergence rate of the heuristic with respect to the various parameter distributions assumed in the probabilistic analysis of Section 4. For each distribution, the optimality gap of the heuristic solution for different planning horizon lengths is shown in Figure 1. Each data point in this and all subsequent figures is the average of ten sample runs. For bounded demand (distribution D3), convergence is achieved fastest. This can be explained from the fact that, since the demand is bounded, expansions in early periods are sufficient for the entire planning horizon.
Effect of demand variance Next, we investigate the sensitivity of the convergence rate to variance in demand. We assume distribution D1(a) and vary the standard deviation of the demand to 1, 3, and 5. The LP relaxation gap and the heuristic error are plotted in Figures 2 and 3. The higher the demand variability, the lower are the LP relaxation gaps and the heuristic errors. This can be explained by noting that, with a higher demand variance, more expansion set-ups are needed. Thus, the LP relaxation solution is closer to the IP solution.
Effect of expansion cost variance Our third experiment is to investigate the convergence behavior of the heuristic with respect to variability in expansion costs. We assume distribution D1(a) and vary the standard deviation of the cost parameters to 2, 4, and 6. The LP relaxation gap and the heuristic
22
error are plotted in Figures 4 and 5. Although the LP relaxation gap worsens with higher cost variability, the heuristic error still decreases.
Effect of fixed to variable cost ratio Next, we investigate the effect of the fixed to variable cost ratio on convergence rates. We assume the distribution D1(a) and vary the ratio K = 2, 3, 4. These ratios are similar to those considered by Rajagopalan (1994) and Li & Tirupati (1994). The LP relaxation gap and the heuristic error are plotted in Figures 6 and 7. As expected, higher fixed to variable cost ratio worsens the LP relaxation, thereby increasing the errors.
Flexible and dedicated technology mix To obtain some insight into the appropriate mix of flexible and dedicated technology in the heuristic solution, we examine the percentage of flexible capacity in the solution to a 10 period problem with parameters generated from distribution D1(a). In this model, the demands of the two product families are uncorrelated (independent). To investigate the effect of demand correlation on the use of flexible capacity (Li & Tirupati 1994), we generate instances with correlated demands by first generating the aggregate demand Dt ∼ N (µd1 + µd2 , σd ), then generating a uniform number pt ∼ U (0, 1), and then letting d1t = pt Dt and d2t = (1 − pt )Dt . Table 5 presents the percentage of total flexible capacity in the solution with respect to demand correlation and relative cost of dedicated and flexible capacity. From this table, we conclude that substantial investment in flexible capacity is economically justified even when it is significantly more expensive. Not surprisingly, the investment in flexible capacity decreases if it is relatively more expensive. Furthermore, investment in flexible capacity is more when demands across the product families are dependent. These observations are similar to those of Li & Tirupati (1994) and Kourpas (1997).
23
Comparison to myopic optimization A commonly used approach for constructing a heuristic solution to a multi-period optimization problem with n periods is the following p-period myopic or “limited look-ahead” scheme, with p
Nikolaos V. Sahinidis∗ Department of Chemical and Biomolecular Engineering University of Illinois at Urbana-Champaign June 20, 2000. Revised: July 1, 2002.
Abstract This paper addresses a multi-period investment problem for the selection, acquisition, and allocation of alternative technology choices to meet the demand of a number of product families over a long-range planning horizon. We first show that the problem is N P-hard. We then present a solution strategy based upon perturbing the linear programming relaxation solution of a multiperiod mixed-integer linear programming ∗
Corresponding author: [email protected]
1
formulation. With mild assumptions on the problem parameters, we carry a probabilistic analysis which proves that the proposed solution approach is asymptotically optimal almost surely.
1
Introduction
The evolution of manufacturing technology in recent years has made available a wide variety of technological options for the manufacturing of products. Increased competition along with a dynamic demand environment, characterized by short product life cycles and changing product mixes, have placed technology adoption decisions among the key strategic concerns for a firm. The type of technology to adopt depends, for example, on the flexibility afforded and the investment and operating cost structures. Typically, flexible technology that is able to manufacture a wide variety of products is more expensive to acquire and operate than dedicated technology. The technology selection decision then has to be made by evaluating the tradeoff between “economies of scale” of the investment decisions and the “economies of scope” afforded by the ability to meet the demand of a variety of products. In addition to the selection of the technology type, the manufacturing firm also has to decide when and how much of each technology type to acquire. These decisions are governed by the forecasted available technology and investment costs. The strategic technology adoption decisions have to be made in conjunction with operational level decisions of how the available capacities are to be allocated to meet the demands of the various products. It is often necessary to model strategic technology adoption problems over planning horizons with a very large number of time periods. For instance, integer programs that arise in the modeling of petroleum production involve quarterly discretization of planning horizons that span several decades (Haugland, Hallefjord & Asheim 1988, Nygreen, Christiansen, Haugen, Bjorkvoll & Kristiansen 1998, van den Heever & Grossmann 2000). The size of these
2
models, as well as others that arise in long-range capacity planning of electric utilities (Levin, Tishler & Zahavi 1983, Wang, Jaraiedi & Torreis 1996), is predominantly determined by the temporal rather than the spatial component of the problem. The purpose of the current paper is to present an optimization framework to aid technology adoption decisions problems with a large number of time periods. In particular, we develop a fast, linear programming-based, approximation scheme that is asymptotically optimal “with probability 1” as the number of time periods increases for a hard combinatorial optimization problem of selection, acquisition, and allocation of several types of technology choices for the production of multiple product families over a multi-period planning horizon. There exists a wide body of literature on optimization approaches to multifacility capacity expansion and evaluation of technology investments. For surveys of the early literature on capacity planning the reader is referred to Freidenfelds (1981) and Luss (1982). Fong & Srinivasan (1981) considered the problem of capacity expansion of multiple facilities to satisfy the demand in different regions to minimize combined capacity expansion and transportation costs. The authors developed an efficient heuristic for the resulting mixed-integer linear programming (MILP) based upon interchanging capacities of two facilities. Klincewicz, Luss & Yu (1988) addressed a large-scale capacity planning model for opening, expanding, and closing multiple facilities to satisfy demands of different regions. The authors developed a heuristic based upon network optimization techniques for this problem. Fine & Freund (1990) formulated the problem of acquisition and allocation of dedicated and flexible technology under demand uncertainty as a two-stage stochastic program. Linearity of the investment costs was a key assumption in their development, thus providing no ability to model economies of scale. Sahinidis & Grossmann (1992) considered the problem of multiperiod capacity expansion of chemical processing networks. Using a number of MILP reformulation techniques, the authors developed exact algorithms for this problem. Li & Tirupati (1994) considered the problem of selection, acquisition, and allocation of dedicated and flexible technology to 3
satisfy the demand of a number of product families. Under certain assumptions, the authors showed that the constraint matrix of the resulting formulation is totally unimodular, thus resulting into integral solutions for concave cost functions. This and other structural properties were exploited to develop two heuristics for the problem. Rajagopalan (1994) studied the problem of capacity acquisition for a number of alternative technologies to satisfy the demand of a single product family. Assuming non-decreasing demand, specific cost structures, and fixed charges to model economies of scale, the author used variable disaggregation to develop an MILP formulation whose linear programming relaxation has integral solutions. Recently, Rajagopalan, Singh & Morton (1998) presented a stochastic multiperiod capacity expansion and replacement model where technological capacity becomes available only at certain time periods. The authors exploited the structure of the optimal solution to develop a dynamic programming strategy. Because of the combinatorial complexity of most of the above-mentioned problems, the use of heuristic strategies is prevalent in the technology selection and capacity expansion literature (Fong & Srinivasan 1981, Klincewicz et al. 1988, Li & Tirupati 1994). However, the qualitative justification of these heuristics has been based upon empirical evidence. In a recent line of work, Liu & Sahinidis (1997) and Ahmed & Sahinidis (2000) proposed LP-based approximation schemes for a capacity expansion problem in the chemical process industries. Using probabilistic analysis tools, the schemes were theoretically proven to be asymptotically optimal in the number of time periods. The principal aim of probabilistic analysis of an algorithm for an optimization problem is to characterize the probable performance of the algorithm on a “typical” instance of the problem. At the onset of the analysis, a typical problem instance is described by assuming probability distributions on the problem parameters. Tools from probability theory are then used to draw inferences regarding the distributions of various measures of algorithmic performance. As pointed out by Coffman & Lueker (1991), complete knowledge of these distributions is usually out of reach and we must often settle for 4
weaker results such as asymptotic distributions. That is, probabilistic inferences about the algorithm’s performance can only be quantified when the problem size is extremely large. Beginning with the seminal works of Beardwood, Hamilton & Hammersley (1959) and Karp (1976), we have witnessed a wide body of literature on the probabilistic analysis of algorithms for classical combinatorial optimization problems such traveling salesman problems (Karp & Steele 1985), knapsack problems (Dyer & Frieze 1989), bin-packing problems (Coffman & Lueker 1991), vehicle routing problems (Bramel & Simchi-Levi 1997), and generalized assignment problems (Romeijn & Piersma 2000), only to name a few. Reviews of this topic appear in Karp, Lenstra, McDiarmad & Kan (1985) and Bramel & Simchi-Levi (1997). The remainder of the paper is organized as follows. In Section 2, we model a very general problem of selection, acquisition, and allocation of several types of technology choices for the production of multiple product families in a dynamic demand environment. The technology choices are classified according to their acquisition and allocation costs as well as the degree of flexibility afforded. We consider discrete time periods and fixed charge economies of scale to formulate the problem as a multiperiod MILP. The developed model is quite general in terms of the existing literature. We do not require non-decreasing demand patterns and allow several technology types that are capable of producing subsets of the product families. Furthermore, we allow for limited availability of procurable capacity of the various technology types along the planning horizon. We show that the problem under consideration is N Phard. This complexity characterization motivates the need for efficient heuristic methods for the problem. In Section 3, we present a polynomial time heuristic method that is guaranteed to produce feasible solutions for the problem under study. By means of probabilistic analysis tools, in Section 4, we prove that the relative error between the heuristic and the exact MILP solutions almost surely vanishes asymptotically as the problem size increases. Computational results are presented in Section 5 demonstrating that the heuristic produces good quality solutions for small- as well as large-scale problems. The computations also demonstrate 5
that, whereas the optimality gap of a myopic heuristic worsens with increasing number of time periods, the optimality gap of the proposed heuristic improves in accordance to our analytical results.
2
Model Formulation and Complexity
In this section, we present an MILP formulation for the problem of determining an optimal strategy for the selection, acquisition, and allocation of various technology types to satisfy the demands of a number of product families over a multiperiod planning horizon. The investment and allocation costs for the various technology types, as well as the demands of the various product families are assumed to be known over the planning horizon. The objective is to time and size technology acquisitions, and allocate the available capacity to various product families in a way that minimizes the total discounted investment and allocation cost for the entire planning horizon. A fixed-charge cost model captures the economies of scale in the investment costs. The following notation is used to describe the model.
Sets and indices: I set of technology choices; i
index for technology choices (i ∈ I);
J
set of product families;
j
index for product families (j ∈ J );
Ij
set of technology types capable of producing product family j;
Ji
set of product families that can be produced with technology i;
n
number of time periods in the planning horizon;
t
index for time periods (t = 1, . . . , n).
Parameters:
6
αit
variable cost of acquiring one unit of capacity of technology type i in period t;
βit
fixed cost associated with acquiring capacity of technology type i in period t;
δijt
cost associated with allocating a unit of capacity of technology type i to product family j in period t;
µij
yield rate of product family j per unit capacity of technology type i;
djt
demand of product family j in period t;
Uit
capacity of technology type i available for acquisition in time period t;
Xi0
initial capacity of technology type i.
Variables: Wijt amount of capacity of technology type i allocated to product family j in period t; Xit Yit
amount of capacity addition of technology type i in period t; binary variables indicating whether capacity of technology type i is acquired in period t (Yit = 1) or not (Yit = 0).
Without loss of generality, we assume that ∪i∈I Ji = J and ∪j∈J Ij = I, so that there is a technology available for each product family and I consists of only those technologies that can produce one or more of the product families in J . A technology i is dedicated if Ji is a singleton and flexible if |Ji | > 1. We also assume that all cost parameters are appropriately discounted. Following Rajagopalan (1994) and Li & Tirupati (1994), we ignore inventory fluctuations and do not permit disposal of excess capacity. With the above assumptions, the technology selection, acquisition, and allocation problem is formulated as follows.
Problem (P):
n
min
αit Xit + βit Yit +
t=1 i∈I
j∈Ji
Wijt ≤ Xi0 +
δijt Wijt
(1)
j∈Ji
Xit ≤ Uit Yit
s.t.
t
τ =1
7
t = 1, . . . , n; i ∈ I
(2)
Xiτ t = 1, . . . , n; i ∈ I
(3)
i∈Ij
µij Wijt = djt
Wijt , Xit ≥ 0 Yit ∈ {0, 1}
t = 1, . . . , n; j ∈ J
(4)
t = 1, . . . , n; i ∈ Ij ; j ∈ J
(5)
t = 1, . . . , n; i ∈ I
(6)
In the formulation above, the objective (1) minimizes the sum of investment and allocation costs over the planning horizon. The variable upper bound (2) ensures that the level of technology acquired in any period does not exceed the capacity available for acquisition in that period. Constraint (3) enforces the condition that, for any technology, the total capacity allocated to the product families does not exceed the installed capacity. Constraint (4) ensures that the allocated capacities will meet product demand. Replacing this constraint by a two-sided inequality or by any other system of linear equalities and inequalities in terms of the allocation variables (Wijt ) does not affect the analysis to follow. The non-negativity and binary restrictions of the variables are enforced through (5) and (6). The formulation presented above is fairly general with respect to the similar models in the existing literature. Unlike Rajagopalan (1994), we do not make assumptions on the structure of the cost components and demand with respect to time. Furthermore, we are allowing flexible technology types capable of producing several product families. An essential difference with the single flexible technology model of Li & Tirupati (1994) is that we are accommodating a wide variety of flexible technologies that are capable of producing different subsets of product families, and part of the decision process includes selecting among these various technology types. We also allow for limited availability of procurable capacity of the various technology choices along the planning horizon. Finally, the possibility of unequal yield rates (µij ) between any two technology types as well as the possibility of unavailability of a particular technology type in a particular period makes our model more general than that of Fong & Srinivasan (1981). These modeling generalizations introduce substantial complexity into the problem. Next, we prove that the problem under study is N P-hard. 8
Lee & Luss (1987) addressed the complexity of multi-facility capacity expansion problems with concave costs under the assumption that infinite capacities are available for expansion. They proved that, although the complexity is exponential in the number of facilities, the problem can be solved in polynomial effort with respect to the number of time periods when the set of facilities is fixed. Owing to the similarity between multi-facility capacity expansion problems and problem (P), it might appear that the analysis of Lee and Luss is applicable to (P). However, the presence of bounds (Uit ) on the procurable capacity makes problem (P) difficult even in the case of a single technology choice as discussed next. Consider first the n-period capacitated lot-sizing problem with zero holding costs (Bitran & Yanasse 1982): Problem (CLSP):
min
n
(pt xt + st yt )
t=1
s.t.
It−1 + xt = rt + It
t = 1, . . . , n
x t ≤ Kt y t
t = 1, . . . , n
It , x t ≥ 0
t = 1, . . . , n
yt ∈ {0, 1}
t = 1, . . . , n
An instance of (CLSP) is defined by the number of time periods n, the costs coefficients pt and st , the production capacities Kt , the demands rt and the initial inventory I0 . Given any such instance of the (CLSP) we can construct a single technology instance of (P) as follows. - The set of technology types is a singleton, i.e., I = {i}. - There are n time periods. - The set of product families is also a singleton J := {j}. - The yield rate is µij = 1. - The allocation costs are all zero, i.e., δijt = 0 for all t = 1, . . . , n. 9
- The investment cost components are αit = pt and βit = st for all t = 1, . . . , n. - The available levels of technology are given by Uit = Kt for all t = 1, . . . , n. - The initial available capacity is Xi0 = I0 . - The demand of the product family j is given by djt =
t
τ =1 rτ .
With the above data, problem (P) reduces to: Problem (P1):
min
n
(pt Xt + st Yt )
t=1
s.t.
Xt ≤ Yt Kt t = 1, . . . , n Wt ≤ I0 +
t
Xτ t = 1, . . . , n
τ =1 t
Wt =
τ =1
rτ t = 1, . . . , n
Xt , Wt ≥ 0 t = 1, . . . , n Yt ∈ {0, 1} t = 1, . . . , n
Note that the indices i and j have been dropped since there is a single technology choice and a single product. Lemma 2.1 Problems (P1) and (CLSP) are equivalent. 2
Proof. See Appendix I.
The (CLSP) with zero holding costs has been shown to be N P-hard for the following parameter structures with respect to time periods (Bitran & Yanasse 1982): (i) Constant st , non-increasing pt , non-increasing Kt , (ii) Constant st , non-decreasing pt , non-decreasing Kt , (iii) Non-decreasing st , zero pt , non-decreasing Kt , 10
(iv) Non-increasing st , zero pt , non-increasing Kt . Since for any given instance of (CLSP) we can construct an equivalent single technology instance of (P), we have the following result: Theorem 1 Problem (P) with one of the above combinations of investment cost structure and the procurement bound is N P-hard.
3
A Temporal Capacity Shifting Heuristic
The complexity characterization of problem (P) in the previous section motivates the need for efficient heuristic strategies. In this section, we develop a capacity shifting heuristic that provides a theoretically sound means for obtaining solutions to problems with long planning horizons. The motivation for the development of this heuristic and subsequent analysis comes from the empirical observations made by Chang & Gavish (1995) and Liu & Sahinidis (1995) on the effect of the length of planning horizon on the quality of relaxation gaps in MILP models of capacity expansion problems. While investigating lower bounding procedures for telecommunication network expansion problems, Chang & Gavish (1995) considered a capacity expansion problem similar to (P) as one of their subproblems (Chang & Gavish 1995, problem (SP2) in pg. 48). The authors conducted computational studies to study the effect of the length of the planning horizon on the quality of the relative gap of the Lagrangian relaxation of (SP2), and made the observation that the gap quality did not deteriorate as the number of planning periods grew. Liu & Sahinidis (1995) actually observed decreasing relative LP relaxation gaps with the increase in the length of the planning horizon in chemical process network expansion problems. These observations appear to be counter-intuitive since a larger number of time periods implies a larger number of binary variables in the
11
integer program and increased complexity per Corollary 1. In the next section, we provide a mathematical explanation of this phenomenon using the heuristic strategy developed next. The empirical evidence cited above suggests the possibility of construction of good quality solutions from the LP relaxation solution for instances of (P) with large planning horizons. Note that simply rounding up the values of the binary variables (Yit ) in the LP relaxation of (P) results in a feasible solution. However, such a naive rounding strategy might result in very poor solutions, possibly requiring technology adoption to be carried out in all periods. We propose to perturb the LP relaxation solution in a way that keeps the number of technology acquisitions small. Since fixed costs of technological capacity addition are typically high, we wish to acquire as much capacity as possible whenever the procurement decision is taken. It is easy to see that, if the investment costs are equal in all periods, there is an optimal solution to (P) where capacity acquisitions are made only in the earliest period when the technology becomes available. Furthermore, in this case, there is an optimal solution where the capacity addition equals the availability bound (Uit ) in all periods except perhaps the last one in which capacity is added. Using these observations, we propose to perturb the LP relaxation solution by shifting technological capacity additions from later periods to earlier ones if procurable technology is available. The first step of the heuristic is to solve the LP relaxation of problem (P) using standard techniques. For large-scale problems, one can decompose the problem across time periods by relaxing constraint (3), and using subgradient methods. A Benders decomposition approach where capacity acquisition decisions are made in the master problem and allocation decisions are made in the subproblems can also be used. To construct a feasible solution to (P) from its LP relaxation solution, we retain the values of the allocation variables (Wijt ). For any technology, we consider only the time periods where the capacity was acquired in the LP relaxation solution and shift the capacity additions from later periods to an earlier period as long as the total installed capacity does not exceed the availability in that period. In this 12
way, we construct a solution where capacity additions are equal to the availability bounds in all periods except the last one where capacity is added. The final step is to round up the binary variables corresponding to periods where the capacity is acquired in the heuristic solution. A formal statement of this heuristic is presented next.
Temporal Capacity Shifting Heuristic LP 1. Solve the LP relaxation of (P). Let the (XitLP , YitLP , Wijt ) denote the optimal solution
to the LP relaxation. For each i ∈ I, let Ti := {t|YitLP > 0}, i.e., the set of time periods when capacity is added. Let Ti = {t1 , t2 , . . . , tpi }. H H LP 2. Denote the heuristic solution by (XitH , YitH , Wijt ). Set Wijt ← Wijt for all i, j, and t.
3. Repeat the following step for all i ∈ I: Do for h = 1, . . . , pi Set XitHh ← XitLP , YitHh ← 0, and k ← h + 1. h While XitHh < Uith and k ≤ pi do, Let δ = min{Uith − XitHh , XitLP }. k Set XitHh ← XitHh + δ and XitLP ← XitLP − δ. k k Set k ← k + 1. End While. If XitHh > 0, set YitHh ← 1. End Do. The time complexity of the above heuristic is O(T LP +|I|n2 ), where T LP is the complexity of solving the LP relaxation. The following two properties of the heuristic solution are obvious from the construction. 13
t
Lemma 3.1 For any technology type i ∈ I,
τ =1
n
Lemma 3.2 For any technology type i ∈ I,
t=1
XiτH ≥ XitH =
t
τ =1
n
t=1
XiτLP for all t = 1, . . . , n.
XitLP .
Using the above properties, the following result is easily obtained. Theorem 2 The capacity shifting heuristic produces a feasible solution to (P). We shall now establish a crucial property of the capacity shifting heuristic that is needed for the qualitative analysis in the next section. For the remainder of this section and the next one, we make the following assumption: Assumption 1 For any technology i ∈ I, Uit = Ui for all t = 1, . . . , n. According to this assumption, the available procurable capacity of any technology is constant in all periods. This assumption is made only for the ease of exposition of the analytical results presented next. In the Appendix, we prove that the main result of Section 4 is valid even without this assumption. n
Lemma 3.3 For any technology type i ∈ I,
t=1
YitH −
n
t=1
YitLP ≤ 1.
Proof. Under Assumption 1, XitH is either Ui or 0 for all time periods except at most one, for which XitH = where 0 < < Ui . Let m be the number of time periods for which XitH = Ui . Then, clearly
n
solution satisfies Thus,
n
t=1
t=1
XitH = mUi + and
n
LP t=1 Xit ≤ Ui
YitLP ≥ m +
Ui
n t=1
n
t=1
YitH ≤ m + 1. Note that the LP relaxation
YitLP . Using Lemma 3.2, mUi + ≤ Ui
n t=1
YitLP .
and the result follows.
2
Note that the above property will not be satisfied by a naive round-up strategy, since such a scheme could potentially lead to rounding up the binary variables in all periods. The capacity shifting heuristic can be improved by shifting only to periods that offer a cost benefit or by integrating it with other heuristic methods such as those proposed by Fong 14
& Srinivasan (1981) and Li & Tirupati (1994). Such improvements will only produce better quality solutions. However, in the next section, we show that the capacity shifting heuristic is asymptotically optimal even in its simple form.
4
Probabilistic Analysis
In this section, we carry out a probabilistic analysis of the error of the capacity shifting heuristic solution for the technology selection problem. We consider a fixed set of product families J , a fixed set of technology choices I, and a fixed set of yield rates µij . The probabilistic analysis will be carried out on instances of (P) consisting of increasingly more time periods where the problem parameters such as costs, product demands, and technological availability are drawn from the following probabilistic model. - The availability, Ui , of technology type i is drawn from distributions with bounded support, i.e., Ui ∈ [U i , U i ], with U i > 0 for all i ∈ I. - The demands of the various product families and the cost parameters for the various technologies are assumed to follow one of the following distributions: (D1) For each product family j, the demands, djt , are i.i.d. random variables with finite first and second moments. For a given technology i, the cost parameters αit and βit for each time period are either i.i.d. random variables with finite first and second moments or have a distribution (not necessarily i.i.d.) with a bounded support. (D2) For each product family j, the demands, djt , are independent random variables (not necessarily identically distributed) with bounded second moments. For a given technology i, the cost parameters αit and βit for each time period are random variables with a bounded support. 15
(D3) For each product family j, the demands, djt , are random variables with bounded support. For a given technology i, the cost parameters αit and βit for each time period are either i.i.d. random variables with finite first and second moments or have a distribution (not necessarily i.i.d.) with a bounded support. - The problem parameters are such that the random instance is feasible. This can be easily ensured by including an expensive artificial technology with infinite capacity that is capable of producing all product families. Distributions (D1)-(D3) permit the inclusion of problem instances with a wide variety of demand trends and cost structures. The main tool in our probabilistic analysis is the asymptotic convergence properties of extreme order statistics (Galambos 1987). For a sequence of random variables {x1 , x2 , . . . , xn }, consider the random variables xn = maxj=1,...,n {xj } and xn = minj=1,...,n {xj }. The asymptotic theory of extreme order statistics concerns with the study of the limiting distributions of the statistics xn and xn as n approaches infinity. We shall make use of the following results from asymptotic extreme value theory: Lemma 4.1 Suppose {xj }nj=1 is a sequence of non-negative i.i.d. random variables with finite second moment. Let zn = maxj=1,...,n {xj }, then limn→∞
zn √ n
= 0 with probability 1
(w.p. 1). Proof. Consider the sequence of random variables {yj }nj=1 with yj = kxj where k > 0 is an arbitrary constant. As {yj }nj=1 is also a sequence of non-negative i.i.d. random variables with finite second moment, E[yj2 ] < +∞. From integration theory (Galambos 1988, Chapter
2 2, Theorem 3), it is known that E[yj2 ] < +∞ if and only if ∞ n=1 P r(yj > n) < +∞ or, since √ yj is non-negative, ∞ n) < +∞. Since yj is identically distributed for all j, the n=1 P r(yj > √ n) < +∞. Let zn = maxj=1,...,n {yj }. From last condition is equivalent to ∞ n=1 P r(yn >
extreme value theory of ordered statistics (Galambos 1987, Corollary 4.3.1),
16
∞
n=1
P r(yn >
√
n) < +∞ implies P r(zn >
√
n infinitely often ) = 0, i.e., P r(lim supn→∞
zn = kzn , the latter statement is equivalent to P r(lim supn→∞ and k was arbitrary, it follows that P r(limn→∞
zn √ n
zn √ n
zn √ n
≤ 1) = 1. As
≤ k1 ) = 1. Since
zn √ n
≥0 2
= 0) = 1.
Lemma 4.2 Suppose {xj }nj=1 is a sequence of non-negative independent random variables with bounded first and second moments. Let zn = maxj=1,...,n {xj }. Then limn→∞
zn n
= 0
w.p. 1. Proof. For a non-negative random variable, we have from Markov’s inequality P r(xn > kBn ) ≤
E[x2n ] . (kBn )2
Since xn for all n have bounded first and second moments, there exist µ and
σ 2 such that µn ≤ µ and σn2 ≤ σ 2 for all n, where µn and σn2 are the first and second moments of xn . We thus have E[x2n ] ≤ σ 2 + µ2 for all n. Hence, Letting Bn = n, we have
∞
n=1
∞
n=1
P r(xn > kBn ) ≤
σ 2 +µ2 k2
∞
1 2. n=1 Bn
P r(xn > kBn ) < +∞ for all k > 0. It then follows from
Theorem 4.4.1 in Galambos (1987) that P r(lim supn→∞ znn = 0) = 1. Moreover, since for all n, we have limn→∞
zn n
zn n
≥0 2
= 0 w.p. 1.
Lemma 4.3 For any i ∈ I, under distribution (D1) and (D3), we have n
√
YitH = 0 w.p. 1 n
t=1
lim n→∞ and, under distribution (D2), we have
n
lim
n→∞
t=1
n
YitH
= 0 w.p. 1.
Proof. For a random instance of (P) with n time periods, the maximum demand of product family j is dmax = maxt=1,...,n {djt }. Clearly, in the optimal solution of the LP relaxation of j (P), the final capacity of any technology choice i will satisfy Xi0 + while, additionally,
n
t=1
XitLP =
n
t=1
n
t=1
XitLP ≤
Ui YitLP . From Lemma 3.3, we then have 17
j∈Ji
dmax j
n
YitH
≤
max j∈Ji (dj
− Xi0 )+
Ui
t=1
+ 1,
where (·)+ = max(·, 0). If the demands satisfy distribution (D1), then by Lemma 4.1, limn→∞
dmax j √ n
= 0 w.p. 1 for all j ∈ Ji . If the demands satisfy distribution (D2), then by
Lemma 4.2, limn→∞
dmax j n
= 0 w.p. 1 for all j ∈ Ji . If the demands satisfy distribution (D3),
then for each j ∈ Ji there exists a finite upper bound dj on the demand in all time periods. ≤ dj w.p. 1 and limn→∞ Then, dmax j
dmax j √ n
2
= 0 w.p. 1.
For an instance of (P) with n time periods, let znLP , znIP , and znH denote the optimal value of LP relaxation, the optimal value of the integer program, and the value of the heuristic solution, respectively. We now state the main results of this section. Theorem 3 For the specified probability model, limn→∞
H −z IP zn n n
= 0 w.p. 1.
Proof. We know that 0 ≤ znH − znIP ≤ znH − znLP . From the construction of the heuristic solution, we also have znH − znLP ≤
n
i∈I
t=1
αit XitH − αit XitLP + βit YitH − βit YitLP . Let
αimax = maxt=1,...,n {αit }, αimin = mint=1,...,n {αit }, βimax = maxt=1,...,n {βit }, and βimin = mint=1,...,n {βit }. Using Lemma 3.2 and Lemma 3.3, we have znH
−
znIP
≤
i∈I
≤
αimax
−
βimin
+
αimin
n
XitH
+
βimax
t=1
U i (αimax
n
YitH
−
βimin
t=1
−
αimin )
+
(βimax
−
n
βimin )
YitH
t=1
Since 0 ≤ αimin and 0 ≤ βimin ≤ βimax for all i ∈ I, we have: −
znIP
≤
βimax
+
(αimax U i
+
βimax )
n
YitH
.
t=1
i∈I
Dividing by n, we obtain:
βimax αimax βimax znH − znIP + √ ≤ + Ui √ n n n n i∈I
18
YitH
t=1
i∈I
znH
n
n
YH √ it . n
t=1
.
−1
Under distributions (D1) and (D3), the cost parameters are either i.i.d. or have bounded support. In either case, limn→∞
αmax i √ n
= 0 and limn→∞
βimax √ n
= 0 w.p. 1. Thus, the result
follows from Lemma 4.3. Similarly, we also have,
βimax znH − znIP ≤ + αimax U i + βimax n n i∈I
n
t=1
YitH
n
.
Under distribution (D2), the cost parameters have bounded support. Hence, (U i αimax + βimax ) < +∞ and, for all i, βimax < +∞ and are independent of n. Thus, the result follows 2
from Lemma 4.3.
To characterize the asymptotic properties of the relative error of the heuristic solution, we make the following assumptions: Assumption 2 For any t, δijt /µij ≥ 1 for all j ∈ Ji and i ∈ I. Assumption 3 For any t, djt ≥ 1 for at least one j ∈ J . The quantity δijt /µij can be interpreted as the unit production cost of family j from technology i in period t. Then, Assumption 2 states that the unit production cost of a product family in any period is at least 1. Similarly, Assumption 3 states that in each period there is unit demand of at least one of the product families. These assumptions are satisfied by appropriate scaling for positive lower bounds on the production costs and demands. Corollary 3.1 Under Assumptions 2 and 3 and the specified probability model, znH − znIP = 0 w.p. 1. n→∞ znIP lim
Proof. It suffices to show that znIP is Ω(n). Consider the dual of the LP relaxation of (P): znD = max
n
t=1
(
i∈I
Xi0 γit + ξit ) +
19
j∈J
djt ηjt
s.t.
λit −
n
γiτ ≤ αit
t = 1, . . . , n; i ∈ I
−Ui λit + ξit ≤ βit
t = 1, . . . , n; i ∈ I
γit + µij ηjt ≤ δijt
t = 1, . . . , n; i ∈ I; j ∈ Ji
τ =t
λit , γit , ξit ≤ 0 ηjt unrestricted
t = 1, . . . , n; i ∈ I j∈J
where λit , γit , and ηjt are the dual variables corresponding to constraints (2), (3) and (4), respectively. Here, ξit is the dual variable corresponding to the LP relaxation constraint Yit ≤ 1. Consider the following feasible solution to the above problem: λit = 0, ξit = 0, γit = 0, and ηjt = mini∈Ij {δijt /µij }. Then, under Assumptions 2 and 3, we have znD is Ω(n). Since the problem instances are assumed to be feasible, the result follows from Theorem 3. 2
The above result also establishes the asymptotic tightness of the LP relaxation and the Lagrangian relaxation as empirically observed by Liu & Sahinidis (1995) and by Chang & Gavish (1995). Let us denote the objective value of the Lagrangian dual solution of an instance of (P) with n periods by znLD . Corollary 3.2 Under Assumptions 2 and 3 and the specified probability model, lim n→∞
znIP − znLP = 0 w.p. 1 and znIP
lim n→∞
znIP − znLD = 0 w.p. 1. znIP
Proof. From the proof of Theorem 3, we have limn→∞ znLD ≤ znIP ≤ znH , the result follows from Corollary 3.1.
20
H −z LP zn n n
= 0 w.p. 1. Since znLP ≤ 2
5
Computational Results
Even though the main purpose of our paper is the theoretical analysis of the capacity shifting heuristic, in this section, we empirically investigate the performance of the heuristic on randomly generated instances as well as a large-scale problem with industrial data from the literature. The purpose of the computations is to investigate the convergence properties of the asymptotically optimal scheme and gain some insight on how model parameter choices affect solution characteristics. We also compare the proposed approach to a commonly used “myopic” heuristic.
Generation of test problems Test problem instances are generated from a production network with two product families (numbered 1 and 2) along with two dedicated technologies (numbered 1 and 2), one for each family. There is also a single flexible technology (numbered 3) capable of manufacturing both families. To guarantee feasibility, we include an artificial technology with infinite capacity that can produce both families at a very high cost. However, in the problem instances generated, this artificial technology was always redundant. No initial capacity is assumed for any technology type. The means and standard deviations used to generate problem parameters for various instances are shown in Table 1. Here, K denotes the ratio between fixed and variable cost of expansion. Unless otherwise specified, K = 2. The actual distributions used to generate the parameters are shown in Tables 2-4, where the distribution parameters are from Table 1. In these tables, N (µ, σ) is a normal distribution with mean µ and standard deviation σ, and U (a, b) is a uniform distribution with support [a, b]. Recall from the probability model of Section 4 that distribution (D1) was that of i.i.d. demand and i.i.d./bounded cost parameters, distribution (D2) was that of i.d demand and bounded cost parameters, and distribution (D3) was that of bounded demand and i.i.d./bounded cost parameters. The
21
independent (but not identical) demands in (D2) were generated by letting µdjt = 2tµdj /n for t ≤ n/2 and µdjt = µdj − 2(t − n/2)µdj /n for t > n/2, i.e., increasing mean for the first half of the planning horizon, and then decreasing mean for the remainder.
Effect of distribution Our first experiment is to investigate the convergence rate of the heuristic with respect to the various parameter distributions assumed in the probabilistic analysis of Section 4. For each distribution, the optimality gap of the heuristic solution for different planning horizon lengths is shown in Figure 1. Each data point in this and all subsequent figures is the average of ten sample runs. For bounded demand (distribution D3), convergence is achieved fastest. This can be explained from the fact that, since the demand is bounded, expansions in early periods are sufficient for the entire planning horizon.
Effect of demand variance Next, we investigate the sensitivity of the convergence rate to variance in demand. We assume distribution D1(a) and vary the standard deviation of the demand to 1, 3, and 5. The LP relaxation gap and the heuristic error are plotted in Figures 2 and 3. The higher the demand variability, the lower are the LP relaxation gaps and the heuristic errors. This can be explained by noting that, with a higher demand variance, more expansion set-ups are needed. Thus, the LP relaxation solution is closer to the IP solution.
Effect of expansion cost variance Our third experiment is to investigate the convergence behavior of the heuristic with respect to variability in expansion costs. We assume distribution D1(a) and vary the standard deviation of the cost parameters to 2, 4, and 6. The LP relaxation gap and the heuristic
22
error are plotted in Figures 4 and 5. Although the LP relaxation gap worsens with higher cost variability, the heuristic error still decreases.
Effect of fixed to variable cost ratio Next, we investigate the effect of the fixed to variable cost ratio on convergence rates. We assume the distribution D1(a) and vary the ratio K = 2, 3, 4. These ratios are similar to those considered by Rajagopalan (1994) and Li & Tirupati (1994). The LP relaxation gap and the heuristic error are plotted in Figures 6 and 7. As expected, higher fixed to variable cost ratio worsens the LP relaxation, thereby increasing the errors.
Flexible and dedicated technology mix To obtain some insight into the appropriate mix of flexible and dedicated technology in the heuristic solution, we examine the percentage of flexible capacity in the solution to a 10 period problem with parameters generated from distribution D1(a). In this model, the demands of the two product families are uncorrelated (independent). To investigate the effect of demand correlation on the use of flexible capacity (Li & Tirupati 1994), we generate instances with correlated demands by first generating the aggregate demand Dt ∼ N (µd1 + µd2 , σd ), then generating a uniform number pt ∼ U (0, 1), and then letting d1t = pt Dt and d2t = (1 − pt )Dt . Table 5 presents the percentage of total flexible capacity in the solution with respect to demand correlation and relative cost of dedicated and flexible capacity. From this table, we conclude that substantial investment in flexible capacity is economically justified even when it is significantly more expensive. Not surprisingly, the investment in flexible capacity decreases if it is relatively more expensive. Furthermore, investment in flexible capacity is more when demands across the product families are dependent. These observations are similar to those of Li & Tirupati (1994) and Kourpas (1997).
23
Comparison to myopic optimization A commonly used approach for constructing a heuristic solution to a multi-period optimization problem with n periods is the following p-period myopic or “limited look-ahead” scheme, with p
