In this paper, we assess the benefits of risk pooling in the service parts logistics systems. We formulate a special case of the Network Design Inventory Problem ...
Proceedings of the 2007 Industrial Engineering Research Conference G. Bayraksan, W. Lin, Y. Son, and R. Wysk, eds.
Benefits of Inventory Pooling for Service Parts Logistics Systems Ilyas Mohamed Iyoob and Erhan Kutanoglu Graduate Program in Operations Research and Industrial Engineering The University of Texas at Austin Austin, TX 78705, USA Abstract In this paper, we assess the benefits of risk pooling in the service parts logistics systems. We formulate a special case of the Network Design Inventory Problem (NDIP) and define conditions for which complete centralization and complete decentralization are optimal. Computational results show that the range of cost parameter values within which partial decentralization is optimal is very small. As a result, in most cases the optimal solution is either close to complete decentralization or close to complete centralization. Finally, we develop an algorithm that evaluates a small set of solutions that form an efficient frontier over the rest of the feasible solutions.
Keywords Risk pooling, inventory pooling, service parts logistics, inventory.
1. Introduction Shrinking profit margins in the high tech industry has led to an increased focus on revenues from after market service parts. This involves improving service parts logistics by identifying locations for service facilities and allocating customers to them such that the overall cost is minimized. Owing to the low demand – high cost nature of service parts, we would also like to simultaneously allocate inventory at the facilities such that inventory cost is minimized as well. However, decisions should be made such that contractual service agreements are satisfied as well. The combined Network Design and Inventory Problem (NDIP) with time based service level constraints is an ongoing research project. In this paper, we formulate a special case of the NDIP to isolate the effects of risk pooling which we hope will give us valuable insight to the overall problem. The general benefit of risk pooling has been extensively studied in the literature. Schwarz [3] analyzes the benefit of risk pooling in terms of the lead time values for complete centralization versus complete decentralization. His main conclusion was that the degree of benefit of risk pooling is highly dependent on the pipeline inventory holding cost when inventory is completely centralized. As the pipeline inventory holding cost decreases, the benefit of risk pooling increases. Xu and Evers [6] prove that partial centralization can never be better than complete centralization based on demand correlation alone. However, transportation cost and lead times may favor partial centralization. Das and Tyagi [1] develop an optimization model to analyze the optimal degree of centralization while balancing transportation and inventory costs. In production-inventory systems, Kim and Benjaafar [2] show that the benefit of pooling is sensitive to the utilization of the production system and the variability in demand and production times. Counter to intuition, the benefit of pooling is minimal when utilization of the production system is high and/or the variability in demand and production times is high. Wong et al. [4] use Markov chains to obtain computational results on the benefits of pooling repairable service parts. Stocking decisions are then made by minimizing the inventory holding cost, downtime cost, and transshipment costs (Wong et al. [5]). In this paper, we provide results similar to those of Das and Tyagi [1], but in the context of service parts, in which demand is very low, inventory is replenished through a base stock policy, and a time-based service level must be satisfied.
Iyoob and Kutanoglu
2. Model Formulation Consider a network of one facility surrounded by n customers such that each customer has the same demand rate per unit time d and transportation cost per unit of demand c (see Figure 1). We assume that demand from each customer is Poisson and the facility is using one-for-one replenishment (or base stock) policy. If the facility is out of stock for an attempted customer demand, the demand is satisfied from a higher-echelon facility outside the system, which we assume has infinite stock level. Hence, the demand is “lost” for fill rate calculations at the facility. Customer demands can be satisfied either from stock at the centralized “shared” facility or from a “dedicated” facility at the customer’s site. Hence each customer site is a potential location for a dedicated facility. When an onsite dedicated facility is used for a customer, its demand is not assigned to the shared facility and is satisfied from the dedicated stock; hence it is “decentralized.” However, each dedicated facility needs to stock exactly one unit of the service part to guarantee practically 100% part availability for very low customer demand rate. The goal is to minimize the total transportation cost and inventory cost subject to a time based service level constraint, which requires that a certain percentage of total demand is satisfied within a service time window. We assume that all the customers are within a distance from the shared facility such that they are within the time window provided that the part is available at the shared facility. A dedicated facility is always within the time window of the respective customer. Without loss of generality, we assume that the replenishment lead time for the shared and dedicated facilities is 1 unit of time.
Figure 1: System of interest Suppose N is the number of dedicated facilities to be opened and S is the (base) stock level at the shared facility. The fill rate at the shared facility is given by b (S , l ) = 1 -
lS
S
å
lk
, where
lS
S
å
lk
is the Erlang loss (or S ! k = 0 k! S ! k = 0 k! block) formula, for stock level S, and mean demand rate during lead time λ, computed using the lost-sales fill rate formula. The service level constraint states that 100 a % of the total demand should be satisfied within the time window. Given that individual demand rate is very low (say d £ 0.01 ) ensures that the single unit stocked at dedicated facilities is sufficient for practically a 100% fill rate ( d £ 0.01 is consistent with demand values in the service parts systems). Since all customers are within the time window, we can write the service level constraint as Nd + β (S , (n - N )d )(n - N )d (1) ³a nd where l = (n - N )d is the mean demand during lead time at the shared facility when N of the customers are served by dedicated facilities.
/
/
Therefore, assuming that the shared facility has no capacity limit, we now formulate the optimization model as follows:
Minimize z(N , S ) = (n - N )cd + (N + S )h = (h - cd )N + hS + ncd Subject to N + β(S , (n - N )d )(n - N ) ³ an N Î {0,1,..., n}
(2) (3) (4)
Iyoob and Kutanoglu
S Î {0,1,...}
(5)
h æ h ö Dividing z (N , S ) by the constant cd , the proportional objective function is ç -1÷ N + S + nh . Therefore, the cd è cd ø value of ratio h / cd plays a significant role in controlling the decision of centralization versus decentralization. Lower h / cd values favor decentralization, whereas centralization becomes increasingly favorable as this ratio increases. Another important observation is that the service level function for a fixed stock level is concave with respect to the number of dedicated facilities. In order to prove this, let us define f (S , N ) = N + b (S , (n - N )d )(n - N ) as the left hand side of the constraint for given stock level S at the shared facility and number of dedicated facilities, N. Theorem 1. For a given value of S, f is a monotonically non-decreasing function in N, i.e. f (S , N + 1) - f (S , N ) ³ 0 . Proof: We have f (S , N ) = N + b (S , (n - N )d )(n - N ) and f (S , N + 1) = (N + 1) + b (S , (n - (N + 1))d )(n - (N + 1)) . Therefore, f (S , N + 1) - f (S , N ) = 1 + b (S , (n - N - 1)d )(n - N - 1) - b (S , (n - N )d )(n - N ) . = 1 + (n - N )(b (S , (n - N - 1)d ) - b (S , (n - N )d )) - b (S , (n - N - 1)d ) But for a given stock level S , fill rate increases as mean demand decreases. Hence, (b (S , (n - N -1)d ) - b (S , (n - N )d )) ³ 0 . Since N £ n , (n - N )(b (S , (n - N -1)d ) - b (S , (n - N )d )) ³ 0. Adding ‘1’ to both sides, 1 + (n - N )(b (S , (n - N -1)d ) - b (S , (n - N )d )) ³ 1 . Since the fill rate function has a probabilistic value (between 0 and 1), 1 + (n - N )(b (S , (n - N - 1)d ) - b (S , (n - N )d )) - b (S , (n - N - 1)d ) ³ 0 . Hence, f (S , N + 1) - f (S , N ) ³ 0 . ■
The implication of Theorem 1 is that for each stock level at the shared facility, we can now calculate the number of dedicated facilities required to satisfy the target service level (Corollary 1.1). Let S be the stock level required at the shared facility to satisfy the demand of all the customers while maintaining the target service level: S = min {S : b (S , nd ) ³ a }. The associated solution represents stocking S at the shared facility without any SÎ{0,1,...}
dedicated facilities, hence “complete centralization” with total cost of h S + ncd. In comparison, suppose S (< S ) units are stocked at the shared facility, let Nˆ S be the smallest number of dedicated facilities that need to be opened in order to maintain the target service level, and let lˆ S be the corresponding mean demand at the shared facility.
ë
û
Corollary 1.1. For a given value of S, Nˆ S = n - lˆS / d where lˆS = max{l : l (1 - b (S , l )) £ nd (1 - a )}. Proof:
l £ nd
(
)
By definition, l = (n - N )d . Therefore, for a given value of S , lˆS = n - Nˆ S d . Rearranging the terms, the value of Nˆ S is
ê lˆ ú Nˆ S = n - ê S ú êë d úû since Nˆ S is defined as the smallest number of dedicated facilities that need to be opened. Now, in order for equation (3) to be satisfied, we require
( (
) )(
Iyoob and Kutanoglu
)
Nˆ S + β S , n - Nˆ S d n - Nˆ S ³ an . Multiplying across by -d and adding nd to both sides, nd - Nˆ S d - β S , n - Nˆ S d n - Nˆ S d £ nd - and .
( ( ) )( Substituting lˆ = (n - Nˆ )d , lˆ (1 - β (S , lˆ )) £ nd (1 - a ). S
S
)
S
S
Again, since Nˆ S is defined as the smallest number of dedicated facilities that need to be opened, lˆ S should be as large as possible. However, l £ nd is an upper bound on lˆ "S Î 0,1,..., S . Therefore,
{
S
lˆS = max{l : l (1 - b (S , l )) £ nd (1 - a )}. ■
}
l £ nd
For given system parameters (a , n, d ), and cost parameters (h, c ), we now find conditions favoring centralization and decentralization. Theorem 2. Sufficient optimality conditions.
(
)
h £ 1. cd ìï éanù - Nˆ k üï h £ min í S * = 0, N * = éanù is optimal if 1 < ý. cd kÎ{1, 2,...,S }ïî éanù - Nˆ k - k ïþ
(a) Complete decentralization S * = 0, N * = n is optimal if (b)
(
)
(
)
(c) Complete centralization S * = S , N * = 0 is optimal if
ìï Nˆ k h ³ max í cd kÎ{0,1,...,S -1}ïî Nˆ k - S - k
(
üï ý. ïþ
)
Proof: (a) Complete decentralization: The coefficient of N in the objective function z (N , S ) is negative since h - cd £ 0 . optimal solution is to set S = 0 and N = n . ìï éanù - Nˆ k üï h (b) If 1 < £ min í ý , then for every k Î 1,..., S we have cd kÎ{1, 2,...,S }ïî éanù - Nˆ k - k ïþ h éanù - Nˆ k . £ cd éanù - Nˆ k - k
{
(
)
Multiplying both sides by cd éanù - Nˆ k - k ,
(
Therefore, the
}
) (
)
h éanù - Nˆ k - k £ cd éanù - Nˆ k .
Rearranging the terms,
(h - cd )éanù £ (h - cd )Nˆ k + hk .
This can be rewritten as,
(h - cd )éanù + h(0) £ (h - cd )Nˆ k + hk .
The left hand side of the last inequality is z (éanù,0) . Since this holds for all k under the condition, the optimal solution is to set S = 0 and N = éanù. (c) Complete centralization: ìï Nˆ k h If ³ max í cd kÎ{0,1,...,S -1}ïî Nˆ k - S - k
(
üï ý , then for every k Î 0,..., S -1 we have ïþ Nˆ k h . ³ cd Nˆ k - S + k
)
{
}
Iyoob and Kutanoglu
(
)
Multiplying both sides by cd Nˆ k - S + k ,
(
)
h Nˆ k - S + k ³ cdNˆ k . Rearranging the terms,
(h - cd )Nˆ k + hk ³ hS .
This can be rewritten as,
(h - cd )Nˆ k + hk ³ (h - cd )(0) + hS .
( )
The right hand side of the last inequality is z 0, S . Since this holds for all k , the optimal solution is to set
S = S and N = 0 . ■ We can now extend the results from theorems 1 and 2 to derive an algorithm for solving the risk pooling problem. The algorithm takes advantage of the fact there are only S + 1 feasible solutions worth comparing because they form an efficient frontier over the rest of the feasible solutions (from Theorem 1). Algorithm Step 0: Initialize k = 0. h Step 1: If £ 1 then set S * = 0, N * = n and stop. Otherwise, go to step 2. cd Step 2: Calculate S = min {S : b (S , nd ) ³ a }, and Nˆ k = n - lˆk / d for each k Î 0,..., S , where
ë
SÎ{0,1,...}
{
û
}
ìï éanù - Nˆ k üï h * * £ min í ý , then set S = 0, N = éanù , and stop. cd kÎ{1, 2,... ,S }ïî éanù - Nˆ k - k ïþ
lˆS = max{l : l (1 - b (S , l )) £ nd (1 - a )}. If l £ nd
Otherwise, go to step 3.
ìï Nˆ k h ³ max í cd kÎ{0,1,...,S -1}ïî Nˆ k - S - k
üï * * ý , then set S = S , N = 0 , and stop. Otherwise go to step 4. ïþ * ˆ Step 4: Among all k Î 0,..., S , if k , N k * is the pair that yields the smallest objective function value, then set S * = k * , N * = Nˆ * and stop. Step 3: If
{
(
} (
)
)
k
3. Computational Results Consider a network of n = 100 customers each with demand d = 0.01 . For a service level of a = 95% , S = 4 units need to be stocked at the shared facility under the complete centralization policy. Suppose the holding cost is h h = 0.105 and transportation cost is c = 10 , then the ratio = 1.05 . Table 1 shows the calculations required for cd the solution algorithm.
S
lˆS
0 1 2 3 4
0.05 0.25 0.55 0.93 1.00
Table 1: Calculations for solution algorithm Nˆ éanù - Nˆ
Nˆ S 95 75 45 7 0
S
éanù - Nˆ S - S N/A 1.0526 1.0416 1.0352 1.0439
(
S
Nˆ S - S - S 1.0439 1.0416 1.0465 1.1666 N/A
)
(
z Nˆ S , S
)
10.475 10.480 10.435 10.350 10.420
h h = 1.05 is greater than 1, complete decentralization is not optimal. Since = 1.05 is also greater cd cd than 1.0352 (minimum of fourth column in Table 1), we move on to step 3 of the algorithm. The maximum value of Given that
Iyoob and Kutanoglu
h = 1.05 . Therefore, in step 4 of the algorithm we identify that cd z(3,7) = 10.350 is the smallest value in column six of Table 1 and hence the optimal solution. Figure 2 is a graphical representation of the solution. the fifth column in Table 1 (1.1666) is greater than
(h /cd ) = 1.05
(h /cd ) = 1.05 10.5
10.6
10.48
10.55
10.46
10.5
Obj
Obj
10.44 10.42 10.4
10.45 10.4
10.38
10.35
10.36
10.3
10.34 0
1
2
S
3
4
0
5
20
(a) S * = 3
40
N
60
80
100
(b) N * = 7
(
)
Figure 2: Plots of z Nˆ S , S with respect to S and N The fact that S + 1 = 5 solutions form an efficient frontier over all other feasible solutions can also be seen in Figure 2(b). We can also use this example to illustrate Theorems 1 and 2. Figure 3 shows the non-decreasing nature of the service level for given S with respect to N. In Figure 4, we can see that the objective function curve changes based on the transportation cost and inventory cost parameters.
120 S=0
80
S=1
60
S=2
40
S=3 S=4
20
10
90
80
70
60
50
40
30
20
10
0
0
f (S,N )
100
N
Figure 3: Illustration of Theorem 1
Iyoob and Kutanoglu (h /cd ) = 1.03
(h /cd ) = 1.30
10.42
13
10.4
12.5
10.38
12
Obj
10.36
Obj
Obj
(h /cd ) = 0.90 10.6 10.4 10.2 10 9.8 9.6 9.4 9.2 9 8.8
10.34 10.32
11
10.3
10.5
10.28 10.26 0
1
2
S
3
4
5
11.5
10 0
1
(a) h = 0.09
2
S
3
4
5
0
1
(b) h = 0.103
2
S
3
4
5
(c) h = 0.13
Figure 4: Illustration of Theorem 2 Figures 4(a), 4(b), and 4(c) correspond to (a), (b), and (c) in Theorem 2. Note that as the h/cd ratio increases, the optimal solution turns from complete decentralization (Figure 4(a)) to partial decentralization (Figures 4(b) and 2(a)) to complete centralization (Figure 4(c)). We now test the likelihood that a problem would yield a partially decentralized optimal solution. We know that this æ ìï an - Nˆ k üï ìï üï ö Nˆ k ÷ . Define occurs only when the h/cd ratio falls in the interval ç min í é ù , max í ý ç kÎ{1, 2,..., S }ï éan ù - Nˆ - k ï kÎ{0,1,..., S -1}ï Nˆ - S - k ýï ÷ k î þ î k þø è ˆ ˆ Nk ïì éanù - N k ïü ïì ïü LB = min í ý and UB = kÎ{0max íˆ ý , then the size of the UB - LB gap would give us ˆ } , 1 ,... , S 1 kÎ{1, 2,... ,S }ï an - N ïî N k - S - k ïþ k -kï îé ù þ an idea of the likelihood of partially decentralized optimal solutions.
(
(
)
)
Consider the experimental design presented in Table 2, with 10 different values of n, α, and d, each represented as a factor in the design. For each of the 1000 combinations, we calculate the LB and UB values. The average gap is 0.033 and the maximum gap is 0.994. Therefore, we can conclude that in most cases, the optimal solution will either be complete centralization or decentralization (with N ³ an dedicated facilities). Table 2: Input system parameters n α d 100 0.75 0.001 200 0.8 0.002 300 0.85 0.003 400 0.9 0.004 500 0.94 0.005 600 0.95 0.006 700 0.96 0.007 800 0.97 0.008 900 0.98 0.009 1000 0.99 0.01 Furthermore, each combination in Table 2 yielded an optimal solution with S * = 0 , S * = S - 1 , or S * = S , regardless of the cost parameters h and c . In other words, suppose S * = k where k Î 1,2,..., S - 2 . This could
(
)
(
) ( )
(
{
) (
}
)
potentially occur only when k satisfies z Nˆ k , k < z(an,0), z Nˆ k , k < z 0, S , and z Nˆ k , k < z Nˆ S -1 , S - 1 . This
ìï üï Nˆ k - Nˆ S -1 Nˆ k h < min í , ý . This interval does not exist for an - Nˆ k - k cd ïî Nˆ k - Nˆ S -1 - S - k - 1 Nˆ k - S - k ïþ any of the combinations in Table 2, thus resulting in either S * = 0 , S * = S - 1, or S * = S . can be rewritten as
an - Nˆ k
0.01 ).
References 1. 2. 3. 4. 5. 6.
Das C. and Tyagi R., 1997. “Role of inventory and transportation costs in determining the optimal degree of centralization,” Transportation Research E: Logistics and Transportation Review, 33, 171-179. Kim, J. and Benjaafar, S., 2002. “On the benefits of inventory-pooling in production-inventory systems,” Manufacturing & Service Operations Management, 4(1), 12-16. Schwarz, L. B., 1989. “A model for assessing the value of warehouse risk-pooling: Risk-pooling over outside-supplier leadtimes,” Management Science, 35(7), 828-842. Wong, H., Cattrysse, D., and Oudheusden D. V., 2005. “Inventory pooling of repairable spare parts with non-zero lateral transshipment time and delayed lateral transshipments,” European Journal of Operations Research, 165, 207-218. Wong, H., Cattrysse, D., and Oudheusden D. V., 2005. “Stocking decisions for repairable spare parts pooling in a multi-hub system,” International Journal of Production Economics, 93-94, 309-317. Xu, K. and Evers, P. T., 2003. “Managing single echelon inventories through demand aggregation and the feasibility of a correlation matrix,” Computers & Operations Research, 30, 297-308.
