JOINT OPTIMISATION OF FACILITY LOCATION AND TWO-ECHELON INVENTORY CONTROL WITH RESPONSE TIME REQUIREMENT AND LATERAL TRANSSHIPMENT

Abstract

Lateral Transshipment (LT) (stock movement between facilities on the same echelon), has been used as an option for reducing the occurrences of stockout and excess stock in many multi-echelon environments. Several LT models have been formulated for many supply chain systems. However, the incorporation of LT into a system which jointly optimises facility location and two-echelon inventory decisions with Response Time Requirement (RTR) has not been considered. Therefore, this study was designed to incorporate LT into a two-echelon system which jointly minimises expected cost emanating from facility location and inventory decisions subject to RTR. The customer arrival at facilities was modelled as a single server queue with Poisson arrivals and exponential service rate. The balance equation of this queue along with the distribution of the number of orders in replenishment (Nvw) was used to derive service center steady state expected level for on-hand inventory (Ivw), backorder (Bvw), and LT (Tvw). The derived steady state expected levels were used to formulate the two echelon LT model. This model was decomposed using Lagrange relaxation. Relaxation of the assignment variable’s integrality was used to further reduce the model. The reduced model was checked for convexity using second order conditions. Karush-Kuhn Tucker (KKT) conditions were used to investigate global optimality, which was also examined for the case of stochastic occurrences. Multiple computational experiments were performed on three data sets using general algebraic modelling system for the values: duvw(max) = 100, 150; ρ = 0.5, 0.9 and τ = 0.2, 0.3, 0.5, where, duvw(max) , ρ and τ are customer distance, utilisation rate and RTR, respectively. The expected number of customers in queue at a service center was: E[Nvw] = P u∈U λuYuvw λ0 ρ S0+1 1−ρ + P u∈U λuYuvwαw. The derived steady state expected levels were: Ivw = PSvw−1 s=0 (Svw − s)P{Nvw = s}, Bvw = P u∈U λuYuvw λ0 ρ S0+1 1−ρ + P u∈U λuYuvwαw + P u∈U λuYuvw λw P|w|Svw−1 s=0 Fw(s) − |w|Svw and Tvw = PSvw−1 s=0 Fvw(s) − Svw − P u∈U λuYuvw λw P|w|Svw−1 s=0 Fw(s) − |w|Svw ii The two-echelon LT model formulated was: min X w∈W X v∈V

fvwXvw + hvwIvw + pvwBvw + qvwTvw + X u∈U λuYuvwduvw! + h0S0 Subject to P v∈V Yuvw = 1 Yuvw ≤ auvwXvw Svw ≤ Cvw S0 ≤ C0 h ρ S0+1 λ0(1−ρ) + αw − τ i ≤ P|w|Svw−1 s=0 [1−Fw(s)] λw Xvw, Yuvw ∈ {0, 1}. The Lagrange dual problem was: max θ,π≥0 min X,Y,S X w∈W X v∈V ( fvwXvw + (hvw + qvw) S Xvw−1 s=0 Fvw(s) − qvwSvw + (pvw − qvw + θvw) P u∈U λuYuvw λw + (pvw − qvw) P u∈U λuYuvw λw ( |w| X Svw−1 s=0 Fw(s) − |w|Svw) +X u∈U λuYuvw (pvw + θvw)ρ S0+1 λ0(1 − ρ) + X u∈U (((pvw + θvw)αw + duvw − θvwτ )λu − πu) Yuvw) + X u∈U πu Subject to Yuvw ≤ auvwXvw Svw ≤ Cvw S0 ≤ C0 Xvw, Yuvw ∈ {0, 1} iii The reduced model obtained was: min 0≤Yuvw (hvw + qvw) S Xvw−1 s=0 Fvw(s) − qvwSvw + (pvw − qvw + θvw) P u∈U λuYuvw λw + (pvw − qvw) P u∈U λuYuvw λw ( |w| X Svw−1 s=0 Fw(s) − |w|Svw) +X u∈U λuYuvw (pvw + θvw)ρ S0+1 λ0(1 − ρ) + X u∈U (((pvw + θvw)αw + duvw − θvwτ )λu − πu) Yuvw where λu, λw, λ0, Yuvw, Lw,(Svw, S0),(Cvw, C0), Xvw, auvw, τ, fvw, hvw, pvw, qvw and duvw are, customer demand, pool demand, plant demand, assignment variable, lead time, base stock levels, capacity, location variable, distance variable, facility, holding, backorder, LT and transportation costs, while, θvw, πu are Lagrange multipliers and Fvw, Fw are facility and pool distribution functions, respectively. The reduced model was convex and satisfied KKT conditions, establishing the existence of global minimum for the two echelon LT model. The stochastic case was also shown to be convex. The computational experiment showed that expected cost remained stable with increasing RTR, and that the model resulted to lower cost when compared with the model without LT. The two-echelon joint location-inventory model with response time requirement and lateral transshipment obtained lower expected cost than the model without lateral trans shipment. Stability of expected cost with varying response time requirement was also established.