Assessment and Future Directions of Nonlinear Model Predictive ...

Distributed MPC for Dynamic Supply Chain Management 609labeled τ 2 ,whereτ 2 is a constant representing the amount of delay in days tomove the goods. In the case of the supplier, the outgoing information flow (o S r )is converted through fabrication into materials, and this conversion process ismodeled as a simple delay. Since material flows downstream, we say that R isdownstream from M (likewise, M is downstream from S), while M is upstreamfrom R (likewise, S is upstream from M). The customer can be thought of as astage downstream (not shown) from R in our model.Each stage x ∈{S,M,R} in Figure 1 is characterized by 3 state variables,defined as follows. The stock level s x is the number of items currently availablein stage x for shipment to the downstream stage. The unfulfilled order of stocko x u is the number of items that stage x has yet to receive from the upstream stage.The backlog of stock b x is the number of committed items that stage x has yetto ship to the downstream stage. The exogenous inputs (assumed measureable)are the demand rate d x r , defined as the number of items per day ordered bythe downstream stage, and the acquisition rate a x r, defined as the number ofitems per day acquired from the upstream stage. The outputs are the order rateo x r , defined as the number of items per day ordered from the upstream stage,and the shipment rate lr x , defined as the number of items per day shipped tothe downstream stage. The order rate is the decision variable (control). By ournotation, all rate variables are denoted by an r subscript. The model, state andcontrol constraints for any stage x ∈{S,M,R} aresubject toṡ x (t) =a x r(t) − l x r (t)⎫⎪⎬ȯ x u (t) =ox r (t) − ax r (t) , t ≥ 0,ḃ x (t) =d x r (t) − lx r (t) ⎪ ⎭}(1)0 ≤ (s x (t),o x u(t),b x (t)) ≤ s max0 ≤ o x r (t) ≤ o r,max, t ≥ 0, (2)where lr x (t) =d x r (t − τ 1 )+b x (t)/t b . The dynamics of the supply chain in thepresent work arise either from rates of accumulation, or from one of two types ofmaterial flow delay (see [7], Chapter 11). Equation (1) describes the first-orderdynamics for stock, unfulfilled orders, and backlog, each arising from rates ofaccumulation. The constraints on the state and control in (2) reflect that stock,unfulfilled order and backlog are independently bounded from below by zero andfrom above by a common constant s max , and that the control (order rate) is nonnegativeand bounded by the positive constant o r,max . The objective of supplychain management is to minimize total costs, which includes avoiding backlog(keep near zero) and keeping unfulfilled orders and stock near desired (typicallylow) levels [7]. Specifically, the control objective for each stage is (s x (t),o x u (t)) →(s d ,o x ud (t)), where s d is a constant desired stock (common to every stage) ando x ud (t) =t llr x(t) is the desired unfulfilled order. The flow constant t l represents thelead time from the downstream stage. Note that if the demand rate converges toa steady value d x r (t) → d r , then backlog will converge to zero, the shipment rateconverges lr x(t) → d r, and the desired unfulfilled order becomes the constanto x ud = t ld r . For each stage x ∈ {S,M,R}, the acquisition rate a x r (t) andthe