Sep 1, 2008 - ature in facility location and multi-echelon inventory models. ... effect (Eppen 1979) since safety stock can be reduced by pooling demand from ...
A Two-Echelon Inventory-Location Problem with Service Considerations Ho-Yin Mak Zuo-Jun Max Shen September 1, 2008
Abstract We study the problem of designing a two-echelon spare parts inventory system consisting of a central plant and a number of service centers each serving a set of customers with stochastic demand. Capacity at the plant and service centers are limited and based on the storage space available. The manufacturing process is modeled as queuing system at the plant. The goal is to optimize the base-stock levels at both echelons, the location of service centers and the allocation of customers to centers simultaneously, subject to service constraints. A mixed integer nonlinear programming model (MINLP) is formulated to minimize the total expected cost of the system. The problem is NP-hard and a Lagrangian heuristic is proposed. We present computational results and discuss the trade-off between cost and service.
1
Introduction
Logistics is a vital aspect in business management. Logistics (transportation, inventory and administrative) costs amounted to 8.7% of the US GDP in 2002 (MacroSys Research and Technology, 2005). This proportion decreased significantly from the early 1980s (16.2% in 1981), suggesting the improvement in efficiency of logistics management over the past two decades. Traditionally, companies have managed distribution and storage decisions separately, in part due to the complexity of combining them. Furthermore, the tactical and operational decisions of stocking and distribution are considered separately from strategic decisions of facility location and network design. Shen et al. (2003), Daskin et al. (2002) and Candas and Kutanoglu (2007) have shown that ignoring the effect of inventory in facility location decisions can lead to suboptimal supply chain designs. However, it remains a challenging task to integrate facility
1
location with inventory management and distribution decisions, especially with considerations of customer service levels. In this paper, we are concerned with the problem of designing a two-echelon service parts supply chain under the constraint of maintaining customer service. There is a central manufacturing plant with limited production and storage capacity. The firm locates service centers (SCs) to satisfy demand from spatially dispersed customers. The service centers replenish inventory from the plant. The problem is to determine the number and location of DCs, the assignment of customers to SCs and the inventory levels at SCs and the plant, subject to response time requirement for demand. Base stock (S − 1, S) inventory policies are suitable for products with relative low demand and high inventory holding costs. Moinzadeh and Lee (1986) provide an analytical relationship to determine whether base stock policies are optimal given certain problem parameters. Their results suggest that a base stock policy is optimal when demand rates are low and setup costs are low relative to holding cost rates. One example where this class of policies is used is in spare parts inventory systems where a failed part is replaced by a new one from inventory. If the failure rate is low, a base stock policy is desirable. In this paper, we assume that the plant and SCs manage inventory using a base stock policy. Such systems exist in a wide range of sectors, including military, airlines and computer manufacturing. We may treat our problem as designing a spare parts inventory system, although the model is applicable in a variety of production-inventory systems as well. One might think that when demand rates are low, it is always a better option to ship directly from the plant to the customer instead of keeping inventory at SCs. However in many applications (especially in spare parts systems), the customers are sensitive to service times. It is then desirable to store inventory at SCs that are close to the customers located far from the plant. Caglar et al. (2004) state that such a system is suitable for spare parts systems where SCs equipped with spare parts inventory and technicians are located close to customers. Kutanoglu (2008) and Cohen et al (1990) discusses the example of IBM’s multi-echelon service parts system, where items are sent to customers from stocking locations close to them when needed. In this paper, we consider this class of inventory systems where customers wish to be served within a time limit and thus SCs must be located. The remainder of this paper is organized as follows: In Section 2, we review the related literature in facility location and multi-echelon inventory models. Then we present the formulation of the model, its properties and the solution approach in Sections 3 and 4 respectively. In view of making strategic location decisions under uncertainty, a scenario-based model is discussed in
2
Section 5. Finally, we present results of computational experiments in Section 6 and conclude in Section 7.
2
Literature Review
Location analysis has been studied extensively in the operations research, economics and geography literature. Daskin (1995), Drezner (1995) and Drezner and Hamacher (2002) provide an excellent summary of the theory. Location-allocation models are concerned with choosing the optimal subset of candidate locations to open facilities at, and assigning demand to the open facilities. Many traditional location models are deterministic, assuming that all problem parameters are known and constant. On the other hand, stochastic location models assume that some of the problem parameters are uncertain and attempt to determine the best decision under uncertainty. Snyder (2005) provides a review of stochastic location models. Traditionally, inventory management has been treated separately from facility location in logistics research. Nahmias (2005) and Zipkin (2000) are examples of texts that cover inventory management in great detail. The inventory literature is extensive, and we only review models for multi-echelon inventory systems related to our problem. Throughout this paper, an echelon refers to a level of facilities holding inventory. One of the earliest and most influential papers in the area of multi-echelon inventory management is by Sherbrooke (1968). He considers a two-echelon spare parts inventory system for repairable parts. All facilities operate under (S − 1, S) policies. Demand is satisfied with inventory at the lower echelon and the failed item are repaired at the upper echelon where there is ample repair capacity. Sherbrooke develops the METRIC model to approximate the expected backorders at each facility for minimization. Generalizing Sherbrooke’s model, Muskstadt (1973) develops the MOD-METRIC model which considers items under hierarchical parts structures. Another classical paper is written by Graves (1985) who proposes a two-parameter approximation for a problem structure identical to Sherbrooke’s. He also derives a computational costly exact method for determining the expected inventory and backorder levels. He shows numerically that the two-parameter approximation is more accurate than METRIC under the problem structure. For a textbook treatment of multi-echelon inventory theory including the above models, see, for example, Sherbrooke (2004) and Axs¨ater (2006). There have been research on two-echelon inventory systems that consider time-based service requirements. In a recent paper, Kutanoglu (2008) study a system with the possibility of emergency lateral transhippments between the local facilities (i.e. SCs). There are service level constraints based on the portion of demand fulfilled within certain time windows. His evalu-
3
ation model points out important insights including how time-based service requirements are more relevant in service parts systems than fill rates, and how emergency lateral transshipment improves response time performances. Caglar (2000) and Caglar et al. (2004) study a two-echelon spare parts inventory system using (S − 1, S) replenishment policies at both levels. There is a constraint on the expected response time, i.e. the time between demand arrival and fulfillment. They develop algorithms to optimize stocking levels at both levels. While the papers mentioned above provide important insights on the tactical and operational aspects of two-echelon inventory systems, we aim to apply some of their results to a model that covers the strategic design phase. Specifically, we would like to determine how many SCs to have, where the optimal locations are and the allocation of customers to SCs. After the location and allocation decisions are determined, the system operates in a similar fashion as those studied in the papers mentioned above. We shall see later in this paper that many of the results in these papers are critical to our model formulation. Shen et al. (2003) studies a facility location problem with the facilities managing inventory with an continuous (r, Q) policy. They extend the fixed charge location problem to include inventory costs using an EOQ-based approximation. The model captures the economies of scale effect (Eppen 1979) since safety stock can be reduced by pooling demand from more retailers to a single distribution center. Because of the inventory cost term, the objective function is nonlinear with two square root terms. Under the assumption that the demand at each retailer has the same mean-variance ratio, the two square root terms can be combined into one and the problem can be solved using a column generation algorithm. Their computational results show that if location and inventory decisions are optimized separately, the number of facilities opened will be more than optimal. Daskin et al. (2002) improve the computational time using a Lagrangian relaxation technique. Shu et al. (2005) relax the assumption of constant meanvariance ratio and solve the problem based on submodular function minimization. Shen and Qi (2006) further incorporate operational routing costs into the model. The technique of continuous approximation (Daganzo 2005) is used to estimate routing cost. Shen and Daskin (2005) evaluate the trade-offs between cost-minimization and servicemaximization. They consider service levels based on the distance between distribution centers and retailers. It is shown that model with service constraints is structurally identical to the Shen et al. (2003) model and can be solved using the same algorithms. The author further introduce a genetic algorithm that can generate the cost-service trade-off curve efficiently. Extending the Shen et al. model, Ozsen et al. (2006a, 2006b) consider the effect of storage
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capacity at facilities under single and multiple sourcing. Unlike the traditional approach in which capacity is defined on the maximum demand that can be assigned to a facility (e.g. Geoffrion and McBride 1978, Holmberg et al. 1999), the authors impose an upper bound on the amount of space that can be occupied by inventory at a facility. The new definition is more flexible because the inventory level held at a facility can be adjusted even for the same amount of demand assigned. In another extension, Shen (2006a) relaxes the assumption that all demand must be served and formulates a profit-maximizing model with demand-choice and pricing flexibility. The results show that demand-choice flexibility can significantly improve the supply chain profit because the firm has the freedom to serve more profitable customers with higher priority. For more information regarding these models, see Shen (2006b) for a more detailed survey. While the models point out the importance of the integrated planning approach, only single-echelon problems have been studied, i.e. inventory is held at one level of facilities only. Multi-echelon inventory-location models have received relatively little attention. Nozick and Turnquist (2001) consider the location of distribution centers (lower echelon) in a two-echelon system with inventory held at DCs and a plant. The system structure is the same as the one considered in the classical two-echelon inventory systems literature, e.g. Sherbrooke (1968) and Graves (1985). The DCs and plant operate under (S − 1, S) policies, and the plant has ample production capacity. They utilizes a linear approximation for safety stock costs as a function of the number of DCs. The resulting location model is structurally identical to the uncapacitated fixed charge problem, with inventory costs included in the fixed location cost as a constant. More recently, Candas and Kutanoglu (2007) consider a multi-commodity two-echelon inventorylocation problem where stocking levels and fill rates can be optimized to achieve a system-wide time-based service level. They formulate a nonlinear integer programming model and propose a linearization-based technique to solve small to medium sized instances. Moreover, they compare the integrated approach of simultaneously considering inventory and location decisions in the same model to the decoupled approach that solves for the location decisions first (ignoring inventory costs) and then optimizing inventory levels given the facility locations. Their results show that the integrated approach can yield solutions that achieve the same service level at a lower cost. Despite the important contributions of the papers mentioned above, all of them treat replenishment lead time as deterministic with the exception of the paper by Nozick and Turnquist which considers the replenishment process as an M/G/∞ queue. In the literature, there has only been a small number of inventory-location models that allow replenishment lead times to be stochastic.
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Eskigun et al. (2005) and Eskigun et al. (2007) study supply chain network design models that factor in the effect of modal choice on the overall service time and customer satisfaction. Their models allow different modes to have different lead times. Moreover, the dwell time (the waiting part of lead time) at a node depends on the amount of demand processed according to an approximate queuing formula. However, their resulting models are deterministic and use the average lead time of different modes as input. Their models also do not consider inventory effects. Sourirajan et al. (2007) study a single-product inventory-location model where the replenishment lead time is stochastic and depends on the amount of demand processed by a facility. The mean lead time is modeled by an approximate queuing formula. The aim for considering stochastic lead time in their model is to determine the required safety stock at each facility. Moreover, Benjaafar et al. (2006) study the problem of locating facilities and managing inventory in a single-echelon system with the replenishment process modeled by an M/M/1 queue which captures the congestion effect. In this paper, we build on the idea of Benjaafar et al. and consider a more general two-echelon system. Our problem also includes the consideration of service time and distance constraints. To the best of our knowledge, there has been no published work in joint inventory-location problems that consider both stochastic replenishment lead time and service time requirements.
3
Model Formulation
3.1
Basic Formulation
We consider a single product supply chain which consists of a single plant (upper echelon), a set of service centers (lower echelon) and a set of customers. The plant manufactures the item and holds inventory to meet demands from SCs. The SCs hold inventory to fulfill customer orders. Each customer places orders at an assigned SC following a Poisson process, consistent with the classical exponential failure model in reliability theory. We assume that orders by different customers are independent. This assumption is valid when customers are geographically dispersed. The plant and SCs have fixed amount of storage space to hold inventory and operate with continuous review (S − 1, S) replenishment policies. Since the items are typically expensive for many spare parts applications, it is also possible to interpret the capacity restriction as budget constraints. Precisely, when a customer places an order, the SC sends one unit of the product from its inventory (if there is stock) to the customer and immediately places an order with the plant.
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When the plant receives an order, it sends one unit of the product from its inventory (if there is stock) to the SC, and immediately releases an order of one unit to its production line. If a service center or the plant is out of stock, the demand is backordered until they are filled. Backorders are handled in a first-come, first-served (FCFS) manner. Finished goods from the production line are used to fill backorders or stored as inventory at the plant immediately after they leave the production line. We assume that the production line has c parallel servers, each of which can process one unit at a time with an exponential service rate. Since each SC faces a superposition of Poisson demand from customers and places replenishment orders in a one-toone manner, the plant faces Poisson demand. This property is exploited by many two-echelon inventory models with base stock policies (e.g. Graves 1985, Caglar et al. 2004). It implies that the production line behaves as a queue with Markovian arrivals. The problem is to determine simultaneously the optimal number and location of SCs, the assignment of customers to the opened SCs and the inventory stocking levels at SCs and the plant. The costs considered in the model include fixed location costs of SCs, transportation costs from the plant to the SCs and from the SCs to the customers, and inventory holding costs at the plant and the SCs. We begin by introducing the following notation: Sets I = Set of customers J = Set of candidate service center locations
Cost Parameters fj = Fixed cost of opening a SC at location j hj = Holding cost per unit of inventory per unit time at location j p = Backorder cost per unit of inventory per unit time dij = Shipping cost per unit from location j to customer i
Demand and Other Parameters λi = Demand rate (Poisson) at customer i P λ = Total demand of all customers (= i∈I λi ) µ = Order processing or manufacturing rate at the plant ρ = Utilization rate of the plant (= λ/µ) τ = Average response time requirement αj = Deterministic transportation lead time from the plant to location j dmax = Maximum distance allowed between customer and the SC assigned
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aij = 1 if costumer i is within dmax distance from candidate location j, 0 otherwise Cj = Storage space available at candidate SC location j, in number of units of the product
Decision Variables Xj = 1 if a SC is located at j, 0 otherwise Yij = 1 if demand at customer i is assigned to SC at j, 0 otherwise Sj = Base stock level maintained at SC at j S0 = Base stock level maintained at the plant
Service Variables I¯j = Steady state expected inventory level at SC j B¯j = Steady state expected backorder level at SC j ¯j = Steady state expected response time at SC j W I¯0 = Steady state expected inventory level at the plant B¯0 = Steady state expected backorder level at the plant Nj (t) = Number of replenishment orders made by DC j that has not yet arrived by time t
Using the above notation, the model can be formulated as follows: ! min
X
fj Xj + hj I¯j + pB¯j +
j∈J
X
dij λi Yij
+ h0 I¯0
(1)
i∈I
Subject to: X
Yij = 1, for each i ∈ I
(2)
j∈J
Yij ≤ aij Xj , for each i ∈ I, j ∈ J
(3)
Sj ≤ Cj Xj , for each j ∈ {0} ∪ J
(4)
¯j ≤ τ , for each j ∈ J W
(5)
Xj ∈ {0, 1}, for each j ∈ J
(6)
Yij ∈ {0, 1}, for each i ∈ I, j ∈ J
(7)
Sj ≥ 0, integer, for each j ∈ {0} ∪ J
(8)
The objective (1) is to minimize the sum of the (annualized) fixed location costs, shipment costs, inventory holding costs at the plant and the SCs and backorder costs at SCs. Backorders at the plant do not incur a monetary cost as they are considered “internal” to the system. Constraints (2) require that all demand must be assigned to SCs. Constraints 3 state that demand assignments cannot be made to a candidate location unless a SC is opened and that the
8
resulting distance is shorter than dmax . Constraints (4) require that the base stock level at a SC cannot exceed the storage capacity. The service time constraints (5) state that the expected response time (time between order arrival and shipment made) cannot exceed the required level. Finally, (6 - 8) are nonnegativity and integrality constraints on the decision variables. In this model, the service requirements are that (i) the distance between any customer and the assigned SC is no longer than a specified limit dmax , which is captured by constraint (3); and (ii) the average time between receiving an order and sending out the item at any SC must not exceed the service guarantee (5). Most models in the spare parts inventory management literature consider shortage costs instead of service requirements, often for tractability reasons. The models that consider service requirements have such constraints either on service level (Cohen et al., 1989) or response time (Caglar, 2001). The former type of constraints limit the chance of stocking out and essentially ignore the differences between stock-outs with short response times (e.g. an hour) and those with long response times (e.g. 2 days). In our twoechelon system, stock-outs at SCs may have a short response time if the plant has the item in stock and ships immediately; or a long lead time if the plant also has a stock-out. Since we are mainly concerned with designing spare parts inventory systems, a response time requirement is more appropriate. The service distance requirement (3) can also be interpreted as a travel time requirement from SCs to customers, if we assume that the travel speed is constant. In principle we may combine this with (5) to obtain a single time-based service requirement. However, this would make the resulting optimization model highly nonlinear. In many practical settings, we are more concerned with how much time it takes for the item to be shipped out than the time it takes for the shipped out item to arrive. For this reason, we argue that our service time constraint is appropriate. In addition, we may require dmax to be small enough such that the (pure) transportation lead time from SCs to customers is short. Consider each SC as a queuing system (Kruse, 1981) where the entities passing through are orders. Then the number of entities in queue represent the backorder level and the time in system for an entity represents the service time. It is possible to apply Little’s law to show that the expected response time at a SC is given by : ¯ ¯ j = P Bj W i∈I λi Yij
(9)
We may replace (5) by the following: B¯j ≤ τ
X i∈I
9
λi Yij
(10)
Before proposing the solution algorithm, we would like to express the inventory and backorder levels in our formulation in terms of the decision variables, i.e. (X, Y, S).
3.2
Inventory Level at the Plant
Since the demand for each customer follows a Poisson process, the demand seen by each SC is the superposition of independent Poisson processes and therefore is Poisson. Following the same idea and that each SC operates under a (S − 1, S) policy, the demand faced by the plant also follows a Poisson process. With a single production line with exponential service rate, the plant operates as a queuing system. Let N0 denote the steady state number of orders in the queuing system (in line and in service). Then Caglar (2001) show that the steady state expected inventory and backorder levels are given by: I¯0 B¯0
= S0 − E[N0 ] + B¯0 = E[N0 ] −
SX 0 −1
(11)
[1 − F0 (s)]
(12)
s=0
where F0 (s) =
s X
P (N0 = m)
m=0
By substituting steady state probabilities into the above formulas, we can easily obtain the expected plant inventory and backorder levels of different manufacturing queue structures. For example, the steady state probabilities for the M/M/c queue is as follows:
P (N0 = n) =
Pc−1 m=0
(cρ)n m!
+
n
λ n!µn P (N0
(cρ)c c!(1−ρ)
−1
= 0)
λn cn−c c!µn P (N0
if n = 0 if 1 ≤ n ≤ c
= 0)
otherwise
It is also possible to allow a batch ordering policy at the plant.
1
Suppose the plant sends
a job request to the production line after receiving Q > 0 orders. Then the interarrival time distribution at the manufacturing queue is Erlang with parameters (Q, λ). The steady state of the EQ /M/1 queue is given by the following (Jackson and Nickols 1956):
P (N0 = n) =
ρ Q −(n=1)Q z0−Q )z0
1−
ρ Q (1
−
if n = 0 otherwise
where z0 is the only root outside the unit circle of: ρz Q+1 − (1 + ρ)z Q + 1 = 0 1
We would like to thank an anonymous referee for pointing out this fact.
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For the simplicity of notation we will only consider the M/M/1 case. Buzacott and Shanthikumar (1993) show that the average inventory and backorder levels at the plant are given by: ρ 1 − ρS0 1−ρ
I¯0
=
S0 −
B¯0
=
ρS0 +1 1−ρ
(13) (14)
We may apply Little’s law to obtain the expected response time at the plant: B¯0 ρS0 +1 W¯0 = = λ λ(1 − ρ)
3.3
(15)
Inventory Level at SCs
Similar to the case for the plant, the steady state expected inventory and backorder levels at each SC as follows: I¯j
= Sj − E[Nj ] + B¯j
(16)
Sj −1
B¯j
= E[Nj ] −
X
[1 − Fj (s)]
(17)
s=0
where Fj (s) =
s X
P (Nj = m)
m=0
Therefore, the steady state distribution of the outstanding orders Nj , defined to be the number of items that SC j has ordered but not yet arrived, is required to determine the inventory and backorder levels. At a point of time t, the number of outstanding orders from SC j is the sum of: (a) the plant backorders due to SC j at time t − αj (this quantity will not reach j before t) and (b) the number of new orders made during the interval (t − αj , t). If the plant processes orders in a FCFS manner, it is possible to disaggregate plant backorders randomly (Graves 1985). The idea is that the probability that a backorder belongs to a particular SC is proportional to the demand rate (i.e. reorder rate) faced by the SC. The steady state expected value of the backorder due to SC j is given by (λj /λ)B¯0 . The expected number of new orders made in a time interval of length αj is λj αj . Therefore the steady state expected value of Nj is given by:
E[Nj ] =
λj ¯ λj ρS0 +1 B0 + λj αj = + λj αj λ λ 1−ρ
(18)
From equations (13) and (14), we can express the steady state expected inventory and backorder levels at SC j using the steady state distribution of Nj . However, the exact method
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presented by Graves (1985) is computationally costly and difficult to be embedded into optimization problems. Therefore, various approximation schemes are proposed in the literature. The earliest of such approximations is the METRIC method (Sherbrooke 1968). It applies Palm’s theorem and approximates the outstanding order distribution using a Poisson distribution with matching mean. Another influential approximation scheme in the literature is the two-parameter approximation proposed by Graves (1985). The idea is that the distribution of outstanding orders is approximated by a negative binomial distribution with matching mean and variance (i.e. two parameters), as opposed to the Poisson distribution (i.e. one parameter) in METRIC. Graves uses computational tests to compare the two approximation schemes with the exact method and concludes that the two-parameter model is more accurate. It is possible to adapt these two approximation models for our problem. We construct a METRIC-like method by approximating the distribution of Nj with a Poisson distribution with mean given in (18). Similarly, it is possible to construct a two-parameter model by approximating the distribution of Nj with matching mean and variance. The METRIC-like approximation makes the expected SC inventory levels a convex function of assigned demand (Lemma 1). This property can be exploited to design efficient solution algorithms. On the other hand, the twoparameter approximation does not ensure convexity. Moreover, fitting the negative binomial distribution requires rounding since the parameter r has to be integer-valued. This makes the resulting expected inventory and backorder level expressions indifferentiable with respect to assigned demand and increases the difficulty of optimization. In integrated inventory-location problems, we do not attempt to optimize the tactical and operational decisions (inventory management in this case) in the strategic phase and keep them fixed forever. Instead, we include an estimate of the costs of the later phases as a function of the strategic decisions to support better design of the supply chain. For this purpose, approximations for the inventory costs suffice. Strategic decisions are always subject to uncertainty in the future environment (e.g. demand rate). In this case, pursuing further precision in the future cost estimates may destroy model tractability, and does not improve model accuracy since the available data is not accurate to begin with (Daganzo 2005). Therefore, we proceed with the reasonably accurate and much more tractable METRIC-like approximation instead of the twoparameter scheme. Using the METRIC-like method, we approximate Nj in (13) and (14) with a Poisson random variable. Therefore we replace Fj (s) by: Fj (s) =
s ¯ X ¯ j )m e−λj Lj (λj L m! m=0
In the above, λj is the demand assigned to SC j (=
12
P
i∈I
(19) ¯ j is the expected replenλi Yij ). L
ishment lead time which consists of expected response time of the plant and the delivery lead time: S0 +1 ¯ j = W¯0 + αj = ρ L + αj λ(1 − ρ)
(20)
With the manipulations described above, it is possible to reformulate the problem as follows:
min
X
Sj −1
f X − pSj + (hj + p) j j ρ S0 1−ρ +h0 S0 − 1−ρ j∈J
X s=0
X ρS0 +1 Fj (s) + p + αj + dij λi Yij λ(1 − ρ) i∈I
(21)
Subject to: X
Yij = 1, for each i ∈ I
(22)
j∈J
Yij ≤ aij Xj , for each i ∈ I, j ∈ J
(23)
Sj ≤ Cj Xj , for each j ∈ {0} ∪ J S0 +1 X Sj −1 X ρ [1 − Fj (s)] , for each j ∈ J + αj − τ λi Yij ≤ λ(1 − ρ) s=0
(24) (25)
i∈I
Xj ∈ {0, 1}, for each j ∈ J
(26)
Yij ∈ {0, 1}, for each i ∈ I, j ∈ J
(27)
Sj ≥ 0, integer, for each j ∈ {0} ∪ J
(28)
In the next section, we present the solution approach based on Lagrangian relaxation.
4
Solution Approach
4.1
Obtaining a Lower Bound
Lagrangian relaxation has been used to solve many facility location models with great success. Examples include the classical P-median and fixed charge location models (Daskin 1995) and location-inventory models (Daskin et al. 2002, Ozsen et al. 2006). Fisher (1985) provides an excellent guide of this technique for practical purposes. It can be easily seen that our problem includes the uncapacitated fixed charge location problem as a special case (and thus is NP-hard). Therefore we believe that Lagrangian relaxation is a promising technique to solve our problem. We begin by proving the following model property that provides an upper bound on the maximum plant base stock level in many instances. Property 1. If p = 0 and τ > αmax = maxj∈J {αj }, i.e. there is no backorder cost and the service time requirement is longer than the pure transportation lead time between the plant and
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any candidate SC site, then an upper bound to the plant base stock level S0 exists. This is given by:
S0max
ρS+1 = min S ≥ 0 : + αmax ≤ τ λ(1 − ρ)
(29)
Proof. The definition (29) and the condition τ > αmax imply that for any values of Yij and Sj , LHS of (25) will be nonpositive. Since RHS of the constraint is nonnegative, it always holds at S0max . S0 is only constrained by the service constraints and nonnegativity, therefore S0max is feasible for all values of the other decision variables. Since the objective function (21) is strictly increasing in S0 , any solution with S0 > S0max will be suboptimal.
The above property states that if there is no backorder cost and all the deterministic transportation lead times between the SCs and the plant are less than the response time requirement, there will be a plant base stock level that ensures all service constraints will be satisfied with all SCs holding no inventory. The optimal plant base-stock level will not exceed this level. The property, in addition to the existence of a capacity constraint and the assumption that overall system demand is low, imply that the range of stock levels that we need to consider in order to ensure the service level is small, and bounded above by the capacity. Similar properties have likewise been exploited by Candas and Kutanoglu (2007) in developing solution algorithms. For our problem, it is possible to solve a small number of continuous problems by fixing the plant base-stock level to each of the possible values. The best (in terms of cost) of these continuous solutions will be the optimal solution for the original problem. When the plant base stock level is fixed, all terms that depend only on S0 become constants. For instance, the plant holding cost will be a constant and can be ignored when solving the restricted problem for the current value of S0 . We choose to relax the assignment constraints (22) and the service constraints (25) in the restricted problem. Using πi and θj to denote the corresponding dual multipliers for constraints (22) and (25) respectively, the following Lagrangian Dual problem is obtained:
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max min
θ≥0,π X,Y,S
X
Sj −1
fj Xj + (hj + p + θj )
X
Fj (s) − (θj + p)Sj +
s=0
j∈J
(θj + p)ρS0 +1 X λi Yij λ(1 − ρ) i∈I
! +
X
[(dij + pαj + θj αj − θτ ) λi − πi ] Yij
i∈I
+
X
πi
(30)
i∈I
Subject to: Yij ≤ aij Xj , for each i ∈ I, j ∈ J
(31)
Sj ≤ Cj Xj , for each j ∈ J
(32)
Xj ∈ {0, 1}, for each j ∈ J
(33)
Yij ∈ {0, 1}, for each i ∈ I, j ∈ J
(34)
Sj ≥ 0, integer, for each j ∈ J
(35)
For given values of dual multipliers, the Lagrangian problem decomposes by candidate SC locations into subproblems in the following form (for each j ∈ J):
Sj −1
min
Yij ,Sj
(hj + p + θj )
X
Fj (s) − (p + θj )Sj +
s=0
+
X
[(dij + pαj + θj αj − θτ ) λi − πi ] Yij
i∈I
θj ρS0 +1 X λi Yij λ(1 − ρ)
(36)
i∈I
Subject to: 0 ≤ Sj ≤ Cj
(37)
Yij ≤ aij , for each i ∈ I
(38)
Yij ∈ {0, 1}, for each i ∈ I
(39)
Sj integer
(40)
To solve the above subproblem, we use the fact that the SCs are constrained in storage capacity. This implies that the possible values of Sj in the optimal solution lies between 0 and Cj . In practice, the possible range is small even in the absence of physical storage limit since the management may want to limit the amount of inventory held due to the high item cost. Therefore a promising approach is to solve the subproblem by fixing Sj to each of the possible values. Moreover, we relax the integrality of Yij , allowing them to take on any value between 0 and 1. The continuous relaxation provides a lower bound for the Lagrangian relaxation approach. As shown in Section 6, the optimality gap of the Lagrangian algorithm is small in general, suggesting that the continuous relaxation of the subproblem is tight. We will now show that the continuous relaxation of the subproblem can be solved efficiently.
15
With the value of Sj fixed, the continuous subproblem can be expressed in the following form:
! X
min
0≤Yij ≤aij
Aij Yij + G
i∈I
X
λi Yij
(41)
i∈I Sj −1
where
G(x) = (hj + p + θj )
X
Fj (s, x) , Fj (s, x) =
s=0
Aij = (dij + pαj + θj αj − θj τ ) λi − πi +
and
s ¯ X e−xLj (xL¯j )m m! m=0
(θj + p)λi ρS0 +1 λ(1 − ρ)
(42) (43)
The solvability of the subproblem depends on properties of the function G(x) to be discussed below. Lemma 1. The function G(x) is decreasing and convex in x when x ≥ 0.
Proof. Note that G(x) represents the holding and penalty costs on the steady state expected inventory level defined in (16 - 17), with demand rate x assigned to the SC with base-stock level Sj . Taking the partial derivative of Fj (s) with respect to the assigned demand x (dropping subscript j for simplicity): ∂ F (s, x) ∂x
¯
= −Le−xL +
s i X ¯m h L ¯ ¯ m e−xL¯ mxm−1 e−xL − Lx m! m=1
Ls+1 s −xL¯ x e ≤0 s! S−1 X ∂ (h + p + θ) F (s, x) ≤ 0 ∂x s=0
= − ∂ G(x) ∂x
=
(44)
Therefore G(x) is decreasing in x. Taking second derivative of F (s): ¯ 2 e−xL¯ 2 L t=0 ∂ F (s) = − L¯ s+1 xs−1 e−xL¯ + L¯ s+2 xs e−xL¯ otherwise ∂x2 (s−1)!
2
∂ G(x) ∂x2
S−1 X
s!
2
∂ F (s) ∂x2
=
(h + p + θ)
=
¯ S+1 xS−1 e−xL¯ L (h + p + θ) ≥0 (S − 1)!
s=0
(45)
Therefore G(x) is convex. Corollary 1. The continuous subproblem (41) is a convex optimization problem. Corollary (1) suggests that it is possible to solve the continuous subproblem using standard convex optimization solvers. Alternatively, we propose an efficient procedure that exploits the
16
problem properties. The algorithm is based on the following theorems, similar to the results proposed by Ozsen et al. (2006a). To begin, we divide customers within the coverage distance (i.e. aij = 1) into two subsets Ij+ and Ij− , with Aij > 0 and Aij ≤ 0 respectively. Suppose there are m customers in the subset Ij+ which are sorted in the following order: 0≤
A1j A2j Amj ≤ ≤ ... λ1 λ2 λm
(46)
Theorem 1. There exists an optimal solution Yj∗ to the continuous subproblem (41) with the following properties: 1. Yij = 1 for all i ∈ Ij− . 2. At most one of the assignment variables Yij∗ takes on a (strictly) fractional value. ∗ 3. If Ykj > 0 for some 1 ≤ k ≤ m, then Yij∗ = 1 for all 1 ≤ i ≤ k − 1.
4. Yij = 0 for all i where aij = 0. Proof. The first property follows from Lemma 1 which states that G(x) is decreasing. Suppose that in an optimal solution, Yij < 1 for some i ∈ Ij− . Then by adding a positive quantity ≤ 1 − Yij , the objective function value decreases, since the linear term does not increase (Aij ≤ 0) and the nonlinear term decreases with the argument of G(.) increasing. Therefore there is a contradiction. To prove the second property, let Yj0 be an optimal solution vector to the continuous sub0 problem with Ykj and Ylj0 taking on strictly fractional values, k, l ∈ Ij+ and k < l. Let the
objective value of this solution be Z 0 . Then define a new solution Yj00 as follows: Yij0 if i 6= l, k 00 0 Yij = Ykj + if i = l Y 0 − λk if i = l lj λl
(47)
n o 0 λl 0 Let = min 1 − Ykj , λk Ylj which implies that Yj00 is feasible. Denote the objective value with Yj00 by Z 00 . Then the difference between the two solutions is given by:
00
Z −Z
0
! ! X X λk λk 0 0 = Akj − Alj +G λi Yij + λk − λl −G λi Yij λl λl i∈I i∈I Alj = Akj − λk λl Akj λk ≤ Akj − λk = 0
17
(48)
The inequality (48) holds because k is ranked before l in Ij+ . The above implies that Yj00 is n o 0 λl 0 optimal. Recall that ≤ min 1 − Ykj , λk Ylj . For each of the possible values: 0 00 1. If = 1 − Ykj , Ykj = 1, 0 < Ylj00 < 1
2. If =
λl 0 λk Ylj ,
00 Ylj00 = 0, 0 < Ykj 0
Ij−
= i ∈ I : Aij ≤ 0
Step 2: Sort customers in the subset Ij+ such that: A1 j A2 j Am j ≤ ≤ ... λ1 λ2 λm where m = |Ij+ | Step 3: Fix Yij = 1 for all i ∈ Ij− . Step 4: Compute the following: Aˆ0j
:=
X
Aij
i∈Ij−
ˆ 0j D
:=
X
λi
i∈Ij−
Define: Rj (i, x) :=
Sj −1 s+1 X L ¯ Aij ¯ xs e−xL − (hj + p + θj ) λi s! s=0
(55)
Let k = 1 and go to Step 5. Step 5: Compute the following: Aˆkj
:= Aˆk−1,j + Akj
ˆ kj D
ˆ k−1,j + λk := D
Step 5a: If Rj (k, Dk−1 ) < 0 and Rj (k, Dk ) > 0, a root to equation (49) exists in the interval Dk−1,j < x < Dkj . This root corresponds to an optimal solution with 0 < Ykj < 1, Yij = 1 for P i = 1..k − 1 and i λi Yij = x. Apply a bracketed root-finding method (e.g. Brent’s method) to find the root to the equation Rj (k, x) = 0, given by x∗ . Then the algorithm terminates and the optimal solution is given by:
Yij∗ =
1
if i ∈ Ij− ∪ {1..k − 1}
x −Di−1,j λi
i=k
0
otherwise
∗
19
(56)
Step 5b: If Rj (k, Dk−1 ) and Rj (k, Dk ) have the same sign, compute: Pˆkj := Aˆkj + G(Dˆkj ) Increment k by 1. If k ≤ m, go to the beginning of Step 5. If k > m, go to Step 6. Step 6: If the set Ij+ is looped through once without finding a fractional solution, the optimal cost is given by Pˆk∗ j = mink=1..m Pˆkj . The optimal solution is given by: 1 if i ∈ I − ∪ {1..k ∗ } j Yij∗ = 0 otherwise
(57)
Corollary 2. As the algorithm proceeds, if for any given i∗ ∈ Ij+ and Sj we find that the LHS of (49) is greater than the RHS, there will be no solution to (49) for any i ≥ i∗ . Proof. The LHS of (49) is increasing in i. The RHS is a decreasing function of x, i.e. decreasing as demand is added. Therefore if the LHS is greater than the RHS, there cannot be a solution for i ≥ i∗ .
The above corollary allows us to save computational time as it specifies conditions under which the existence of a root to (49) does not need be checked. The most costly step computationally in the algorithm is solving for the root of equation (49). The algorithm checks the necessary condition for a root to exist within the interval in question. In practice, a root seldom exists. Therefore in the vast majority of iterations, the root-finding procedure is not called. The next most costly step in the algorithm is sorting the set Ij+ , which has complexity |Ij+ |log|Ij+ |. However, the set needs to be sorted only once in order to solve the subproblems for all possible values of Sj .
4.2
Obtaining an Upper Bound
Section 4.1 outlines the method to find a lower bound to the problem given a set of Lagrangian multipliers. In each iteration of the Lagrangian procedure, we make use of the lower bound solution to find a upper bound (feasible) solution to the problem. Recall that the service constraints (25) and demand assignment constraints (22) are relaxed. Therefore a lower bound solution represents a system where demands may not be exactly fulfilled and waiting time requirements may not be satisfied. In this section, we outline a heuristic procedure to construct a feasible solution using the lower bound solution.
20
We begin by opening the SCs that are opened in the lower bound solution and set the plant base stock level equal to the value obtained in the lower bound solution. Then, we partition the set of customers into three groups and determine the assignment as follows: 1. The ones that are assigned to exactly one SC in the lower bound solution: Follow the assignment in the lower bound solution. 2. The ones that are assigned to more than one SC in the lower bound solution: Among the SCs to which the customer is assigned in the lower bound solution, consider the one that is closest to the customer. If assigning the customer to this SC does not increase the required base stock level to satisfy service constraints, assignment is made. Otherwise, choose the one among the SCs to which the customer is assigned in the lower bound solution that increases the total cost the least, based on assignments made so far. If none of such assignments is feasible (i.e. adding demand makes the required base stock level exceed the capacity), pick the open SC (which the customer may not be assigned to in the lower bound solution) that increases the cost the least. If none of such assignments is feasible, open the next closest unopened SC and assign demand to it. 3. The ones that are unassigned in the lower bound solution: Among all open SCs, if the nearest one to the customer can accommodate the demand without increasing the required base stock level, make the assignment. Otherwise, pick the open SC that increases the cost the least. If none of such assignments is feasible, open the next closest unopened SC and assign demand to it. The above assignment procedure is performed in order. All customers in the first group are processed first, then the second group and finally the third group. Within each group, customers are processed in decreasing order of demand rate. When all customers are assigned, the SCs that are unused are closed. For each feasible assignment, the optimal base stock levels at SCs and thus inventory-related costs can be computed easily by applying the following result: Property 2. For any feasible demand allocation (X, Y ) and plant base stock level (S0 ) (i.e. the response time constraint for SC j can be satisfied with some base stock level at j less than or equal to capacity Cj ), the optimal base stock level at an SC is given by Sj∗ = min Cj , max{Sjmin , SjN V } , where: Sjmin SjN V
S0 +1 X Sj −1 X ρ = min Sj ≥ 0 : + αj − τ λi Yij ≤ [1 − Fj (s)] λ(1 − ρ) s=0 i∈I p = min Sj ≥ 0 : F (Sj ) > p + hj
21
(58)
(59)
Proof. Given a demand allocation, the terms in the objective value that depend on Sj are: Sj −1
(hj + p)
X
Fj (s) − pSj
(60)
s=0
Let the part of the objective function that depends on Sj be denoted by: Sj −1
H(Sj ) = (hj + p)
X
Fj (s) − pSj
s=0
Let ∆(Sj ) be the change in objective value if the base stock level at j is increased from Sj to Sj + 1. Then: ∆(Sj ) = H(Sj + 1) − H(Sj ) = (hj + p)Fj (Sj ) − p
(61)
When ∆(Sj ) is negative, increasing Sj by one will result in a cost decrease. Furthermore, Fj (Sj ) is monotonically increasing in Sj and lies between 0 and 1 by definition. Therefore, the unconstrained optimal Sj can be found in the following manner: Starting from Sj , increment Sj by one as long as ∆(Sj ) ≤ 0. The optimal Sj in the absence of capacity and response time constraints is given by SjN V , which is the newsvendor quantity under a discrete demand distribution. However, the unconstrained optimal level may not be feasible under the service and capacity constraints. The value Sjmin is the minimum level required to satisfy the response time constraint. If SjN V < Sjmin , then Sjmin (the smallest feasible value) is the optimal value since the objective value is monotonically increasing in Sj for Sj > SjN V . On the other hand, if Sjmin > Cj , the optimal value is Cj (the largest feasible value) since the objective value is monotonically decreasing in the range Sj < SjN V . This implies that Sj∗ as defined above is the constrained optimal base stock level.
4.3
Detecting Infeasibility
Many classical facility location problems such as the fixed charge and P-median problems are always feasible. In our problem, it is not trivial to determine infeasibility for many instances due to the service constraints. Based on Property 3, we propose a method of detecting infeasibility. Once the problem is judged to be infeasible, the algorithm will be terminated. In each iteration of the Lagrangian relaxation procedure, a lower bound to the objective value is provided. If the instance is feasible, a relatively loose upper bound (U Bf eas ) is available from the following observation:
22
Property 3. If an instance is feasible, then the following provides an upper bound to the objective value: UB =
X
fj +
j∈J
X
λi dˆmax + h0 C0 +
i∈I
X j∈J
X ρ hj Cj + p + λαj 1−ρ
(62)
j∈J
where dˆmax corresponds to the shipping cost of a distance equal to dmax . Proof. If the instance is feasible, the first 4 terms in (62) are clearly upper bounds to the fixed costs, the transportation costs, the holding costs at the plant and the holding costs at SCs respectively. The last term is an upper bound to the backorder cost because for each j ∈ J: ρ + λαj ≥ 1−ρ
P
Sj −1 X X λi Yij ρS0 +1 + αj λi Yij = E[Nj ] ≥ E[Nj ] − [1 − Fj (s)] = B¯j λ 1−ρ s=0
i∈I
i∈I
If the lower bound determined in any iteration is larger than the value U Bf eas , we know that the instance is infeasible.
4.4
Summary of Lagrangian Relaxation Algorithm
We summarize the algorithm discussed previously as follows: Algorithm 2. Step 0: Iteration count n = 1. Initialize dual multipliers (θ1 , π 1 ). They can be set at any feasible value (e.g. all 0). Define a loose upper bound U Bf eas according to Property 3. Step 1: Find lower bound using the procedure described in Section 4.1 by solving subproblem (41) for each j and Sj = 0...Cj using Algorithm 1. Let the resulting lower bound value be LB n and the lower bound solution be (Xn , Yn , Sn ). If LB n > U Bf eas , terminate the algorithm and conclude that instance is infeasible. Step 2: Find upper bound with the procedure described in Section 4.2, using lower bound solution (Xn , Yn , Sn ) as input. If this upper bound is better than the tightest one (i.e. lowest ¯ Y¯ , S). ¯ objective value) so far, let its cost be U B and the corresponding (feasible) solution be (X, If
U B−LB n UB
< where > 0 is a pre-defined tolerance level, terminate the algorithm and the best
¯ Y¯ , S). ¯ Otherwise, continue to Step 3. solution found is (X, Step 3: Update the Lagrangian multipliers using the standard subgradient procedure from the lower bound solution (Xn , Yn , Sn ). Let the new multipliers be (θn+1 , π n+1 ) and increment n by 1. If n is larger than a pre-defined iteration limit, terminate the algorithm. Otherwise go to Step 1.
23
5
Scenario-Based Model
Facility location is a long-term decision. One challenge that often arises in strategic supply chain design is the unavailability of accurate demand and cost forecasts at the time of making the design decisions such as facility location. Since such decisions are difficult and very costly to reverse, it is important that they are robust against the uncertain environment. Snyder (2006) surveys the various location models that aim at producing robust or reliable solutions. Scenario-based modeling is a very popular technique for this class of models. Under this approach, uncertainty is characterized by a finite set of discrete scenarios. There are two possible interpretations of the set of scenarios. The dynamic interpretation is that while the business environment changes over time, the scenarios and the associated probabilities represent the long-run proportion of time that the environment is in each state. On the other hand, the static interpretation is that the set represents a range of possible outcomes, one of which will occur and remain fixed afterwards. In practice, companies often perform similar scenario-based analysis for considering the performances of projects under different possible future outcomes. In location modeling, it is possible to apply stochastic optimization or robust optimization techniques, depending on the defined objective function, to generate one or multiple “good” solutions that perform relatively well under most or all possible scenarios. One example is the paper by Snyder et al. (2007), who extend the inventory-location model of Shen et al. to incorporate uncertainty in the business environment (i.e. demand and shipping costs). In this section, we assume that the demand rates (λi ) are uncertain and formulate a twostage stochastic programming model to generate robust solutions. It is trivial to extend the model and allow other parameters such as shipping, holding and penalty costs to be uncertain as well. The first-stage decision is to determine the subset of candidate sites to locate SCs (i.e. the Xj variables) given a finite set of possible scenarios of demand rates. Then in the second stage, the company decides the customer-SC allocations and base stock levels after observing which one of the scenarios realizes. Define K as the set of scenarios and qk as the probability of the scenario k ∈ K happening. Moreover, we add a subscript k to any parameter (e.g. λik ) and decision variable (e.g. Sjk that may take on different values under different scenarios. Then we may formulate the scenariobased problem as follows:
24
min
" X X ρS0k +1 + p fj Xj + qk + αj + dij λik Yijk λk (1 − ρk ) i∈I j∈J j∈J k∈K Sjk −1 X ρ k 1 − ρSk 0k +(hj + p) Fjk (s) − pSjk + h0 S0k − 1 − ρ k s=0
X
X
Subject to: X
Yijk = 1, for each i ∈ I, k ∈ K
j∈J
Yijk ≤ Xj , for each i ∈ I, j ∈ J, k ∈ K Sjk ≤ Cj Xj , for each j ∈ {0} ∪ J, k ∈ K Xj ∈ {0, 1}, for each j ∈ J Yijk ∈ {0, 1}, for each i ∈ I, j ∈ J, k ∈ K Sjk ≥ 0, integer, for each j ∈ {0} ∪ J, k ∈ K Note that the objective value in the above formulation is to minimize the expected cost over all scenarios. However, optimizing the average outcome can generate solutions that perform badly under certain scenarios. It is possible to produce more robust solutions by using alternative objectives that reflect risk-averseness. Examples of such objectives include minimax regret (Serra and Marianov 1998), expected failure cost (Snyder and Daskin 2005) and conditional value-at-risk or CV aR (Chen et al. 2006). Moreover, it is often desirable to consider both an average-case objective and risk-averse objective simultaneously. Using multi-objective optimization techniques, it is possible to generate a efficient set of Pareto-optimal solutions with respect to the chosen objectives. The manager may then evaluate the trade-offs between the objectives and select one solution among the efficient set. This technique is demonstrated by Shen and Daskin (2005) with a model that captures the trade-off between cost and degree of customer coverage. In this paper, we apply a similar technique and consider two objectives: the expected cost and the worst-case “service time” across all scenarios. The “service time” for a customer under a scenario considered in this section is the sum of the expected response time at the assigned SC plus the deterministic transportation lead time from the SC to the customer. We attempt to minimize the longest service time among all customers across all scenarios. By considering the worst-case service time over all possible scenarios, this objective reflects the desire for the supply chain to be robust. Note that the service time objective replaces the service time and distance constraints (23 and 25) that are included in the single-scenario model. Population search techniques such as genetic algorithms (GAs) are very effective in multiobjective optimization (see, for example, Abraham et al. 2005 and Coello Coello et al. 2007).
25
By generating a large population and attempting to improve through genetic operations such as crossover and mutation, GAs can generate a set of solutions with desired properties (e.g. low cost, high service level) through an evolutionary process. We apply a GA similar to the one used by Shen and Daskin (2005) with the two objectives stated above. Details of the algorithm can be found in the appendix. The GA generates a set of non-dominated solutions that approximate the efficient frontier. Details of the computational results can be found in the next section. Finally, it should be noted that alternative objectives, such as the minimizing the worst cost or maximum regret over all scenarios, can be easily incorporated using slight variations of the algorithm. Before ending this section, we note that the model discussed in this section can be applied to multi-commodity problems. Service parts logistics systems often handle multiple nonsubstitutable parts or commodities. By interpreting K as the set of commodities, the demand rates (λik ) as well as the assignment and inventory variables (Yijk , Sjk ) become commodityspecific. Then the same formulation is equivalent to locating common SCs to store and distribute multiple commodities. For each commodity, the customer-SC assignment and the base stock levels at the plant and SCs are optimized. The same GA can be applied to produce high-quality solutions.
6 6.1
Computational Results Experiment Setup
In this section, we describe the computational experiments conducted to test the performance of our solution procedure. Three data sets from Daskin (1995) are used in the following experiments. The first data set contains 49 nodes representing the 48 continental state capitals of the US and Washington DC. The second set has 88 nodes, containing the same 49 cities in the first set and the 50 most populated cities (according to the 1990 census) in the US. The third data set contains 150 nodes which represent the 150 most populated cities in the US. The cost coefficients are adjusted in the following manner: The fixed costs fj are unchanged from the Daskin (1995) dataset. The transportation costs (dij ) are obtained by dividing the distance between two nodes from Daskin (1995) by 10. Demand rates (λi ) are set to 10−6 times the population figure given in Daskin (1995). The transportation lead time (αj ) is set to be equal to 1/10 of the distance between the corresponding demand node and Springfield, IL in the 49-node dataset and Chicago in the 88-node and 150-node datasets. Finally, the per-unit holding and penalty costs (h0 , hj , p) are set to 50 and 150 respectively. The algorithm is coded in C++ and run on a PC with Intel Core 2 Duo CPU (2.13GHz), 2GB of RAM and Windows Vista.
26
6.2
Algorithmic Performance
In the first experiment, we are concerned with the performance of the algorithm, i.e. speed and optimality gap. We run the algorithm with the three data sets described previously for different values of plant and SC capacities (C0 and Cj = 10 or 5), utilization rate (ρ = 0.9 or 0.5), response time requirement (τ = 5.5 or 1.5) and distance requirement (dmax = 2000 or 500 miles). The results are summarized in Table 1. For all instances tested, the algorithm converges to a feasible solution within approximately 1 to 2% to optimality in reasonable amount of time. In the majority of the instances, the optimality gap is less than 1%. For the 49-node data set, the solution time range from a few seconds up to approximately 3 minutes. All the 88-node instances are solved within 2 minutes. The solution time for the 150-node instances range from a minute up to 35 minutes. The solution times recorded are reasonable for the respective problem sizes. As observed from our experimental results, the solution time is longer for instances where C0 and Cj are larger, i.e. the plant and SCs have larger capacities. In our solution algorithm, a subproblem is solved for each feasible base stock level. As the number of feasible base stock levels increases with capacity, the computational effort required is larger. Recall that our model and algorithm is suitable for systems where demand is low and items are expensive (e.g. spare parts systems), i.e. the practical range of base stock levels under consideration is small. The computational time is also longer for instances with looser assignment distance requirements (dmax ). This observation may appear to be counter-intuitive because it suggests that a problem that is less tightly constrained is also harder to solve. However, this is a property of our proposed algorithm. When solving for a lower bound in each iteration, the algorithm solves |J| subproblems with complexity |Ij+ |log|Ij+ | each (for the case where equation (49) does not have a root). Ij+ is the subset customers which lie within dmax from the candidate SC location j in question and with parameter Aij > 0. As dmax increases, the number of customers that lie within dmax from each candidate facility location increases as well. Therefore the size of Ij+ as well as the computational effort also tend to increase. Similarly in solving for an upper bound, for each particular customer i, the algorithm considers all open facilities that lie within dmax from i. An increase in dmax would increase the number of SCs to consider and therefore increase computational effort. A tighter response time requirement (τ ) tends to increase computational time slightly. Comparing pairs of instances where the values of all other parameter are the same, the instances with smaller τ values (i.e. tighter time requirements) are consistently harder to solve. However,
27
the differences are not significant. The utilization parameter (ρ) does not appear to have a significant impact on the solution time. If we compare each pair of instances with different utilization rates and all other parameters equal, we can observe that the low-utilization instance requires a shorter solution time in the vast majority of the cases. However, the difference is usually small (a few seconds). Finally, we observe that all instances can be solved to (provably) within about 2% from optimality. Recall that our algorithm computes lower bounds by solving subproblems with the integrality of assignment variables (Yij ) relaxed. The small gap suggests that the relaxation is tight since the upper bounds hold for the problem with integrality constraints. The implication of this is that even if we relaxed the integrality of the assignment variables in the original problem, the resulting cost improvement would be small.
6.3
Effects of Response Time Requirement and Penalty Cost
In the third experiment, we wish to test the impact of the response time service requirement and the backorder penalty cost and their relationship. In traditional inventory problems with service considerations, it can be shown that there is a one-to-one correspondence between service level and penalty cost. In reality, it is often difficult to quantify “penalty costs” of having a stock-out or not fulfilling an order within a specified time. We would like to see whether a service constraint can act as a perfect substitute for a penalty cost in our problem. The 49-node dataset is used with utilization (ρ) set to 0.95, distance requirement dmax set to 2000 miles, and storage capacity (C0 and Cj ) set to 10 for both the plant and SCs. We ran two set of instances. In the first set, the unit penalty cost (p) is 0, and the response time requirement (τ ) is varied between 0.01 and 3 time units. In the second set, the response time requirement is fixed at 3 (a requirement loose enough that no inventory needs to be carried) and the unit penalty cost is varied between 0 and 600. For each instance, we record the costs and expected response time achieved (i.e. the longest expected response time among all open SCs). Then we plotted the total costs against the expected response time achieved in Figure 1. The penalty costs are excluded from the total costs for a fair comparison since the unit penalty costs are different across different instances. Figure 1 shows how high the total costs of location, transportation and inventory holding need to be (vertical axis) in order to maintain a expected response time at a certain level (horizontal axis), by using a penalty cost or a tight service requirement. The figure shows that the curve of using penalty costs lies above the one of using response time requirement. This means that setting a tight response time requirement can ensure a certain response time in a
28
Minimum Cost to Achieve Service Level 9500 9400
Minimum Cost
9300 9200 9100
Constraint Penalty
9000 8900 8800 8700 8600 8500 0.3
0.5
0.7
0.9
Service Level (Response Time)
Figure 1: Optimal Costs of Using Penalty Cost and Service Constraint to Achieve a Certain Average Response Time
more cost-effective manner than by using penalty costs. In our solution algorithm, the response time constraints (5) are relaxed by adding penalty terms to the objective function. We notice that this is structurally equivalent to increasing the penalty cost p by the dual multiplier θj . As the Lagrangian relaxation algorithm converges, the optimal dual multiplier values would almost ensure that the constraints are satisfied. Therefore we would expect the following two instances: the first one with 0 penalty cost and a tight response time constraint, and the second one with very loose response time constraint and unit penalty cost equal to the optimal θj values found in solving the first one, to give optimal solutions that are almost the same. This means that penalty costs and response time requirement are to some degree, equivalent in our model. However, this is not true in the practical sense. We do not know the optimal dual multipliers for the instance with tight response time constraints before solving it. Even if the values are known, they are generally not equal across candidate SC locations. However, there is no reason to define different unit penalty costs for different candidate SC locations, before even deciding which SCs will be opened and which customers they will serve. Instead, it is typical to set the penalty cost equal across all candidate SC locations. Our computational experiment shows that using a uniform (across candidate SC locations) penalty cost parameter will require higher costs to achieve a certain response time level than using a tight response time requirement. This inefficiency arises from the requirement that penalty costs are equal across candidate locations.
29
6.4
Trade-off Between Service and Cost
In spare parts inventory systems, service levels are important. However, focusing on cost minimization in supply chain design often results in poor service. As pointed out by Shen and Daskin (2005), cost and service can act as two opposing performance metrics in supply chain design. It is important to evaluate the trade-off between cost and service level. In this computational test, we solve the problem at different levels of service requirement. Then we plot the trade-off curve of expected cost against service time requirement (τ ). On the same graph we also show the optimal number of facilities against service requirements. The results are shown in Figure 2. With a loose service time requirement (τ large), it is optimal to locate 3 SCs. As the response time requirement tightens (τ decreases), expected cost increases as the optimal number of SCs remains unchanged. The increase in cost is due to holding extra units and adjusting the locations of SCs (relatively minor) to decrease response time. When the response time is decreased to a certain point (approximately 0.5 time units), it is optimal to open one more SC. The expected cost experiences a large jump at this point because of the extra fixed cost of opening a facility. From this point onwards, the cost continues to increase due to increasing holding costs and adjustments of SC locations until the service requirement becomes infeasible. We may also observe that it costs more to improve service when the response time requirement is tighter, a similar pattern as found by Shen and Daskin (2005). From this test, we point out the importance of finding the appropriate service-cost tradeoff. The marginal cost often increases to improve service rises with a tighter response time requirement. The manager should observe the trade-off curve and choose a solution base on the organizational needs.
6.5
Comparisons with One-Echelon Systems
It is often convenient to manage inventory at a single level. Benjaafar et al. consider the service parts inventory-location problem where inventory can only be held at the lower echelon (the SCs). In this experiment, we would like to study the benefits from the possibility of holding inventory at the plant also. Note that our formulation models a single-echelon system when the plant storage capacity S0 is set to 0. In the experiment, we allow the response time requirement τ to vary between 0.3 and 1.3. For each level of τ , we solve the problem with plant storage capacity C0 equals 0 (one-echelon) and 10 (two-echelon). The storage capacity at SCs, Cj is set to 10. We plot the expected cost against the response time requirement in Figure 3. From the graph, we observe that when the response time requirement is loose (high τ ), it
30
Trade-off Between Cost and Response Time Requirement 8800
5
Optimal Cost
8400 8200
4
8000 7800 7600
3
7400
Optimal Number of SCs
8600
Cost Number of SCs
7200 7000
2 0.3
0.8
1.3
1.8
2.3
Response Time Requirement
Figure 2: Trade-off Curve of Optimal Cost vs Response Time Requirement
is optimal to operate a single-echelon system and not to stock at the plant. The two-echelon solutions have plant base stock levels equal to 0. However when the response time requirement becomes tight, it is beneficial to keep inventory at the plant and manage a two-echelon system. Keeping inventory at the plant allows fewer units to be kept at SCs. Although the total inventory in the system may increase, the increased inventory at the plant are shared among all SCs. This sharing enhances availability and therefore the same response time requirements can be met with lower costs. For some instances (e.g. when τ = 0.4, the expected cost can increase by 7% if keeping stock at the upper echelon is not allowed.
6.6
Results for Scenario-Based Problem
In this experiment, we use the genetic algorithm described in the Appendix to solve the scenariobased problem discussed in Section 5. For the 49-node data set, 30 scenarios are generated by perturbing the demand rates with a random variable. Specifically, the demand rate for customer i under scenario k is given by λik = λi (0.5 + ik ), where λi is the demand rate for customer i used in the previous tests and ik is a [0, 1] random variable drawn independently for each (i, k) pair. We assign equal probability to all scenarios. The two objective functions used are the expected cost over all scenarios and the worst-case service time, defined as the longest expected response plus transportation lead time among all customers in all scenarios. While the first objective optimizes the average case performance, the second objective provides a robustness measure and ensures that the service level in the worst
31
(a) Expected Cost 9600 9400 9200
Cost
9000 One-Echelon Two-Echelon
8800 8600 8400 8200 8000 0.2
0.4
0.6
0.8
1
1.2
1.4
Response Time Requirement
(b) Base Stock Level 50 45
Base Stock Level
40 35
25
One-Echelon, Lower and Total Two-Echelon, Total
20
Two-Echelon, Lower
30
15 10 5 0 0.2
0.4
0.6
0.8
1
1.2
1.4
Response Time Requirement
Figure 3: Costs of One-Echelon and Two-Echelon Systems
32
(a) 25 Generations
2.5
(b) 50 Generations
2 1.8
2
1.6
Service Time
Service Time
1.4 1.5
1
1.2 1 0.8 0.6
0.5
0.4 0.2
0
0 0
5000
10000
15000
20000
25000
30000
35000
40000
0
5000
10000
Expected Cost
20000
25000
30000
35000
40000
30000
35000
40000
Expected Cost
(c) 100 Generations
(d) 500 Generations
2
2
1.8
1.8
1.6
1.6
1.4
1.4
Service Time
Service Time
15000
1.2 1 0.8
1.2 1 0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
5000
10000
15000
20000
25000
30000
35000
40000
Expected Cost
0
5000
10000
15000
20000
25000
Expected Cost
Figure 4: Trade-off Curve of Expected Cost vs Service Time for Scenario-Based Problem
scenario is not too bad. Figure 4 shows the non-dominated solutions (i.e. not worse than any other solution in both objectives) after 25, 50, 100 and 500 solutions. It can be observed that the efficient frontier moves towards the bottom-left corner (i.e. improving) as the number of generations increases. Similar to the results in Section 6.4, we observe that the slope of the efficient frontier is steeper when the expected cost is low. This suggests that starting from a solution with low average-case cost, the marginal cost of improving the worst-case service time is small. For instance, improving the worst-case service time by a factor of 3 (1.8 to 0.6) only increases the average-case cost by about 20% (8000 to 10000). Furthermore, we observe that the improvement of the efficient frontier appears to diminish with the number of generations. The relevant region (the bottom-left region) of the 100generation frontier approximates the 500-generation frontier closely. Therefore it may be sufficient to run a small number of generations to obtain fairly good results. The running time for 100 generations is less than 10 minutes.
33
7
Conclusions and Future Research
In this paper, we formulate a model for designing of a service parts network to ensure service quality. Fixed costs of locating service centers, shipping costs and inventory holding costs at both the plant and the service centers are considered. The model is formulated as an nonlinear integer programming problem. The formulation can be viewed as an extension of the uncapacitated fixed-charge location problem. The inventory-related costs and service performance metrics are approximated using a similar idea as in Sherbrooke (1968). As a result, we are able to decompose the problem into a series of convex optimization problems using Lagrangian relaxation. Then the algorithm is tested using datasets with 49, 88 and 150 nodes. Computational times are relatively short for such a complex integer nonlinear programming problem, reflecting the efficiency of the proposed solution approach. In addition, we compare the difference of using response time requirement and penalty cost in our model. We demonstrate that by using a response time requirement, it is less costly to achieve certain service levels than by using imaginary penalty costs. Finally, we highlight the importance of selecting the right trade-off between service and cost. The marginal cost for improving service is typically higher when the service requirement is tight than when it is loose. Therefore by selecting a solution carefully, it is possible to achieve significant gains in service without incurring large cost increases. Our computational results also demonstrate the possible cost savings by implementing a two-echelon system as opposed to a single-echelon one when response time requirements are tight. By storing inventory at the plant that is shared by all SCs, it is possible to achieve the same response time requirement at a lower cost. Finally, we formulate a scenario-based model and propose a genetic algorithm to solve it. Using two contrasting objectives of expected cost and worst-case service time, our model captures the trade-off between average-case performance and robustness in view of future uncertainty. The same model and solution approach can also be applied to solving multiple-commodity problems which are common in service parts systems. We plan to further this work in several areas. Firstly, we would like to relax the assumption of the (S − 1, S) inventory policy to allow batch ordering by SCs. This extension will make the model applicable to a larger variety of supply chain design environments. Secondly, we would like to consider the case where parts of the network may be disrupted. In many supply chain and military settings, disruptions may occur and certain facilities (the plant or service centers) or arcs may break down. We would like to address important questions such as how to design robust systems that are able to operate under unanticipated breakdowns of nodes or arcs on the network.
34
Another important issue is the nature of the service time requirement. The model we present in this paper considers only the mean response time. However, it is desired that service be not just quick on average but also uniform. In classical inventory models, service levels are usually defined using probabilistic constraints (e.g. type I service levels in continuous review problems). As an extension, we would like to re-define the service requirements using either chance constraints on the response time distribution or constraints on the service time variance. Finally, we have limited our analysis in this paper to single sourcing, i.e. demand at a customer is assigned to one SC only. Multiple sourcing has been studied with capacitated facility location problems (e.g. Geoffrion and McBride 1978, Guinard and Spielberg 1979, and Nauss 1978) using continuous allocation variables. If the demand for a certain node is split among multiple facilities, customer requests are assumed to be randomly assigned to each of the facilities with the specified probability. However, we argue that this traditional approach for modeling multiple-sourcing is not directly applicable to joint inventory-location problems. In this more complex class of problems, sourcing and inventory management must be considered in an integrated manner. We cannot assume that the demand at a node is split among facilities randomly without considering the availability of inventory at the facilities. Multiple sourcing is a strategy for sharing inventory and improving service level. We do not believe that simply having continuous assignment variables can adequately model the mechanics. We believe that multiple-sourcing is not just an interesting extension to this problem, but also a promising area of future research in the general area of integrated inventory-location models.
References [1] Abraham, A., L. Jain and R. Goldberh, Eds. (2005). Evolutionary Multiobjective Optimization. Springer, New York. [2] Axs¨ ater, S. (2006). Inventory Control, Second Edition. Springer, New York. [3] Benjaafar, S., Y. Li and D. Xu. (2006). “Demand Allocation in Systems with Multiple Inventory Locations and Multiple Demand Sources”. Working Paper, University of Minnesota, Minneapolis. [4] Buzacott, J.A. and J.G. Shanthikumar. (1993). Stochastic Models of Manufacturing Systems. Prentice Hall, New Jersey. [5] Caglar, D. (2001). A Multi-Echelon Spare Parts Inventory System with Emergency Lateral Shipments subject to a Response Time Constraint. Ph.D. Thesis, Northwestern University, Evanston, IL.
35
Table 1: Test on Algorithmic Performance # Nodes
C0 , Cj
ρ
τ
dmax
Lower Bound
Upper Bound
Gap %
CPU Time (s)
Iterations
49
5
0.9
5.5
2000
9841.8
9904.5
0.63%
34.03
565
49
5
0.5
5.5
2000
9482.6
9493.3
0.11%
31.39
549
49
5
0.9
1.5
2000
9841.8
9881.6
0.40%
36.19
599
49
5
0.5
1.5
2000
9482.6
9529.4
0.49%
36.07
618
49
5
0.9
5.5
500
10587.6
10804.1
2.00%
8.51
537
49
5
0.5
5.5
500
10308.2
10392.0
0.81%
6.87
482
49
5
0.9
1.5
500
10587.7
10733.1
1.35%
9.28
601
49
5
0.5
1.5
500
10308.2
10451.8
1.37%
7.35
522
49
10
0.9
5.5
2000
9297.5
9317.7
0.22%
196.58
521
49
10
0.5
5.5
2000
9148.0
9181.5
0.36%
183.79
507
49
10
0.9
1.5
2000
9297.5
9309.8
0.13%
198.46
551
49
10
0.5
1.5
2000
9144.7
9152.8
0.09%
52.98
154
49
10
0.9
5.5
500
10375.8
10611.1
2.22%
36.25
460
49
10
0.5
5.5
500
10232.6
10398.8
1.60%
32.68
461
49
10
0.9
1.5
500
10375.8
10601.3
2.13%
38.68
493
49
10
0.5
1.5
500
10232.6
10356.0
1.19%
36.72
506
88
5
0.9
5.5
2000
15368.9
15501.9
0.86%
23.03
513
88
5
0.5
5.5
2000
14898.7
14959.0
0.40%
20.71
526
88
5
0.9
1.5
2000
15368.9
15434.3
0.42%
29.10
615
88
5
0.5
1.5
2000
14898.7
15038.9
0.93%
25.07
596
88
5
0.9
5.5
500
15368.9
15501.9
0.86%
23.40
513
88
5
0.5
5.5
500
14898.7
14959.0
0.40%
21.23
526
88
5
0.9
1.5
500
15368.9
15434.3
0.42%
26.87
615
88
5
0.5
1.5
500
14898.7
15038.9
0.93%
23.32
596
88
10
0.9
5.5
2000
14107.6
14185.6
0.55%
117.68
486
88
10
0.5
5.5
2000
13879.4
13912.7
0.24%
113.94
497
88
10
0.9
1.5
2000
14107.6
14185.6
0.55%
136.02
577
88
10
0.5
1.5
2000
13879.4
13923.0
0.31%
119.11
537
88
10
0.9
5.5
500
14107.6
14185.6
0.55%
116.00
486
88
10
0.5
5.5
500
13879.4
13912.7
0.24%
114.00
497
88
10
0.9
1.5
500
14107.6
14185.6
0.55%
136.19
577
88
10
0.5
1.5
500
13879.4
13923.0
0.31%
119.11
537
150
5
0.9
5.5
2000
17528.3
17557.2
0.16%
298.73
557
150
5
0.5
5.5
2000
17417.1
17443.7
0.15%
288.51
552
150
5
0.9
1.5
2000
17528.3
17550.8
0.13%
340.25
637
150
5
0.5
1.5
2000
17386.2
17440.9
0.31%
1996.70
3776
150
5
0.9
5.5
500
17891.6
17909.7
0.10%
52.33
546
150
5
0.5
5.5
500
17769.0
17797.2
0.16%
107.94
1206
150
5
0.9
1.5
500
17891.6
17913.3
0.12%
61.45
627
150
5
0.5
1.5
500
17781.3
17803.3
0.12%
54.27
594
150
10
0.9
5.5
2000
17088.4
17116.5
0.16%
1757.90
557
150
10
0.5
5.5
2000
17039.1
17066.0
0.16%
1717.65
544
150
10
0.9
1.5
2000
17088.4
17113.5
0.15%
2047.83
641
150
10
0.5
1.5
2000
17039.1
17066.0
0.16%
1962.70
641
150
10
0.9
5.5
500
17494.7
17543.7
0.28%
1288.19
2484
150
10
0.5
5.5
500
17420.2
17488.1
0.39%
1593.67
3063
150
10
0.9
1.5
500
17526.8
17547.6
0.12%
314.91
557
150
10
0.5
1.5
500
17471.7
17491.4
0.11%
251.71
491
36
[6] Caglar, D., C.L. Li and D. Simchi-Levi. (2004). Two-Echelon Spare Parts Inventory System Subject to a Service Constraint. IIE Transactions, 36, 655-666. [7] Candas, M.F. and E. Kutanoglu. (2007). Benefits of Considering Inventory in Service Parts Logistics Network Design Problems with Time-based Service Constraints. IIE Transactions, 39, 159-176. [8] Chen, G., M.S. Daskin, Z.J. Shen and S. Uryasev. (2006). The α-Reliable Mean-Excess Regret Model for Stochastic Facility Location Modeling. Naval Research Logistics, 53, 617626. [9] Cohen, M.A., P.A. Kleindorfer and H.L. Lee. (1989). Near-Optimal Service Constrained Stocking Policies for Spare Parts. Operations Research. 37, 104-117. [10] Coello Coello, C.A., G.B. Lamont and D.A. Van Veldhuizen. (2007). Evolutionary Algorithms for Solving Multi-Objective Problems, Second Edition. Springer, New York. [11] Daskin, M. (1995). Network and Discrete Location: Models, Algorithms, and Applications. John Wiley & Sons, New York. [12] Daskin, M., C. Coullard and Z.J. Shen (2002). An Inventory-Location Model: Formulation, Solution Algorithm and Computational Results. Annals of Operations Research, 110, 83-106. [13] Drezner, Z., Eds. (1995). Facility Location: A Survey of Applications and Methods. Springer, New York. [14] Drezner, Z. and H.W. Hamacher, Eds. (2002) Facility Location: Applications and Theory. Springer, New York. [15] Eskigun, E., R. Uzsoy, P.V. Preckel, G. Beaujon, S. Krishnan and J.D. Tew. (2005). Outbound Supply Chain Network Design with Mode Selection, Lead Times and Capacitated Vehicle Distribution Centers. European Journal of Operations Research, 165, 182-206. [16] Eskigun, E., R. Uzsoy, P.V. Preckel, G. Beaujon, S. Krishnan and J.D. Tew. (2007). Outbound Supply Chain Network Design with Mode Selection and Lead Time Considerations. Naval Research Logistics, 54, 282-300. [17] Fisher, M. (1985). An Applications Oriented Guide to Lagrangian Relaxation. Interfaces, 15(2), 2-21. [18] Geoffrion, M. and McBride, R. (1978). Lagrangian Relaxation Applied to Capacitated Facility Location Problem. AIIE Transactions 10, 40-47. [19] Graves, S.C. (1985). A Multi-Echelon Inventory Model for a Repairable Item with One-forone Replenishment. Management Science, 40, 597-602. [20] Gross, D. and C.M. Harris (1998). Fundamentals of Queuing Theory (3rd ed.). John Wiley & Sons, New York.
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[38] Shen, Z.J., C. Coullard, and M.S. Daskin (2003). A Joint Location-Inventory Model. Transportation Science, 37, 40-55. [39] Shen, Z.J. and M.S. Daskin (2005). Trade-offs Between Customer Service and Cost in Integrated Supply Chain Design. Manufacturing and Service Operations Management, 7(3), 188-207. [40] Shen, Z.J. and L. Qi. (2006). Incorporating Inventory and Routing Costs in Strategic Location Models. European Journal of Operational Research, 179, 372-389. [41] Sherbrooke, C.C. (1968). METRIC: A Multi-Echelon Technique for Recoverable Item Control. Operations Research, 34, 311-319. [42] Sherbrooke, C.C. (2004). Optimal Inventory Modeling of Systems. Kluwer Academic Publishers, Norwell, MA. [43] Sourirajan, K., L. Ozsen and R. Uzsoy. (2007). A Single-Product Network Design Mode with Lead Time and Safety Stock Considerations. IIE Transactions, 39(5), 411-424. [44] Snyder, L. (2006). Facility Location Under Uncertainty: a Review. IIE Transactions, 38, 537-554. [45] Snyder, L. and M.S. Daskin (2005). Reliability Models for Facility Location: The Expected Failure Cost Case. Transportation Science, 39(3), 400-416. [46] Zipkin, P.H. (2000). Foundations of Inventory Management. McGraw-Hill, Boston, Massachusetts.
39
Appendix: Genetic Algorithm for Scenario-Based Problem The scenario-based problem formulated in Section 5 can be solved using a genetic algorithm similar to the one used by Shen and Daskin (2005). We outline our algorithm as follows: Algorithm 3. Step 0 (Initialization): Generate an initial population (counter g = 0) of a given size (we used 500). Each individual solution in the population consists of a binary vector (the X vector) with dimension |J| representing the set of SCs located. For each solution t, we generate a random cutoff level from a uniform [0, 1] distribution. Then for each j ∈ J, another [0, 1] random number is drawn. Xj set to 1 (an SC is located) if the new random number is below the cutoff value and 0 (not located) otherwise. Go to Step 1 after generating the entire population. Step 1 (Elite Preservation): For each solution in generation g, the two objective values: the expected cost and the worst-case service time are evaluated using a procedure similar to the one described in Section 4.2. Furthermore, a score Vmg is assigned to each solution m. We use a simple scoring rule, where Vmg is equal to a weighted sum of the two objective values: the expected cost and the service time. We note that it is possible to apply more complex scoring rules such as the one used by Shen and Daskin (2005) that compare each solution with the entire population and therefore are more computationally costly. In our problem, the simple scoring rule works satisfactorily. All non-dominated solutions in generation g are copied to generation g + 1. If more than half of the population in generation g is non-dominated, the population size of generation g + 1, Pg , is set to the minimum of 2 times the number of non-dominated solutions in g and 2000 (expansion). Increment g by 1 and go to Step 2. Step 2 (Importation): Generate 20% of the remaining solutions (after Step 1) for generation g by importation. Each of these solutions are generated using the random procedure described in Step 0. Then go to Step 3. Step 3 (Crossover and Mutation): The remaining solutions are generated by crossover and mutation. To generate a new solution, two solutions from generation g − 1 are selected randomly as parents. The probability of being chosen decreases with the score Vm : 2 maxn Vng − Vmg g g m0 ∈Pg (2 maxn Vn − Vm0 )
P (t Chosen as parent) = P
After selecting two distinct parents (selection procedure is started over if both parents are the same), a random crossover point (an integer between 1 and |J| is generated. The genes (Xj variables) to the left of the crossover point are copied from the first parent and those to the right of the crossover point are copied from the second parent. Finally, a child solution is mutated with a given probability (0.2 is used in our computational tests). If a solution is to be mutated, a gene (Xj variable) is identified randomly and switched
40
(from 1 to 0 and vice versa). After generating all solutions, go to Step 1.
41
