Abstract
In this paper, we study the periodic-review stochastic Joint-replenishment Problem (JRP), with backorders-lost sales mixtures, controllable lead times, and investment to reduce the major ordering cost. The purpose is to determine a strict cyclic replenishment policy, the length of lead times, and the major ordering cost that minimize the total system cost. We first present an effective heuristic algorithm to approach the problem. However, results illustrate how computationally expensive the algorithm would be for a practical application. Hence, we then propose an efficient and more practically applicable solution procedure. In particular, approximating part of the cost function with its second-order Taylor series expansion, we obtain an expression that resembles the deterministic cost structure. Therefore, the problem can be approached exploiting a standard algorithm suitable for the deterministic JRP. Numerical tests compare the performances of the algorithms developed and show that the approximated approach is actually promising for a practical application.
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Appendix
Appendix
1.1 Proof of Proposition 1
The limit properties are relatively easy to prove. We only demonstrate the convexity properties.
Point (i):
For (T, A, k, z) fixed and L n ∈[L m,n , L m−1,n ], for n=1, 2, …, N, we have:
which is true for m=1, 2, …, M n and for each n. Hence, for (T, A, k, z) fixed, C(T, A, k, z, L) is concave in L, with L n ∈[L m,n , L m−1,n ] for each n.
Point (ii):
For (T, k, z, L) fixed, letting ξ≡τ/δ, we have ∂2 C(T, A, k, z, L)/∂A 2=ξ/A 2>0, which proves the convexity of C(T, A, k, z, L) in A for (T, k, z, L) fixed.
Point (iii):
Let us keep (A, k, L) fixed. We have 
, where C
n
(T
n
, z
n
, L
n
) is given by (2) and T
n
≡k
n
T. Annadurai and Uthayakumar (2010) proved that a function structurally identical to C
n
(T
n
, z
n
, L
n
) is convex in (T
n
, z
n
). Since convexity is invariant under affine maps, we can affirm that C
n
(T, k
n
, z
n
, L
n
), which is given by (2) replacing T
n
with k
n
T, is convex in (T, z
n
). This is true for each n. Moreover, noting that ∂2(τI(A)+A/T)/∂T
2=2A/T
3>0, the convexity of C(T, A, k, z, L) in (T, z) is readily deduced.
To prove that C(T, A, k, z, L), with 
and (k, L) fixed, is convex in (T, z), it is simply needed to observe that 
and that 
. Reminding that C
n
(T, k
n
, z
n
, L
n
) (for each n) is convex in (T, z
n
), it thus follows that C(T, A, k, z, L), with 
and (k, L) fixed, is convex in (T, z).
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Braglia, M., Castellano, D. & Frosolini, M. Joint-replenishment problem under stochastic demands with backorders-lost sales mixtures, controllable lead times, and investment to reduce the major ordering cost. J Oper Res Soc 67, 1108–1120 (2016). https://doi.org/10.1057/jors.2016.13
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DOI: https://doi.org/10.1057/jors.2016.13