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...(Thomas and Griffin, 1996; Fumero and Vercellis, 1999; Brown al., 2001; Lee and Whang, 2001; Bloomquist et al., 2002; and Gupta et al., 2002). This issue becomes especially critical after companies merge since system redundancy (in terms of a low capacity utilization) and inefficiency (in terms of a high distribution cost) are inevitably introduced into the combined supply/distribution network. A representative example of this is the recent merger of Nabisco and Kraft Food Inc. who had to expend a large amount of effort to create a new joint supply network (Harps, 2003).
In this study, we are interested in the following integrated Production, Inventory, and Distribution Routing Problem (PIDRP). We are given a single product and a set of plants, each with its own production capacity, inventory capacity, raw material supply contract, inventory holding cost, and production cost. Associated with each plant is a heterogeneous fleet of transporters, each with its own operation cost, required loading/unloading time, loading capacity, available time, and traveling speed. We are also given a set of customer demand centers (DCs) located over a wide geographical region, each with its own demand per time period in the planning horizon, its own inventory capacity, holding cost, and safety stock requirement. The problem is to determine the operation schedules to coordinate the production, inventory, and transportation routing operations so that the customer demand, transporter travel time and capacity constraints, plant production, and inventory constraints are all satisfied, while the resulting operation cost (i.e., the sum of production, inventory, and transportation cost) over a given planning horizon is minimized.
Our research on this problem was motivated by the post-merger demand-supply coordination problem encountered by a leading chemical company in North America. After acquiring a large manufacturing facility from one of its competitors, the company experienced operational difficulties including: (i) there being no guideline as how much should be produced at an individual facility since the production costs, material costs, supply contract, logistics partner contract, production costs, and inventory capacities are different at each facility; and (ii) a large number of DCs (most of them being ocean terminals) to be supplied. The DCs have individual inventory capacities, geographical locations, and demand patterns and consequently different safety stock requirements over a year. Due to the finite nature of production capacity, plants often use any excess capacity to build up inventories. Similarly, the DCs build up inventories whenever arrivals exceed consumption over a period. Very often products are directly moved from the delivery vessel to a waiting ship for export to a foreign market. Any surplus stock is held in the customer's inventory and can be used at a later time often saving a delivery from the producer. It is unclear which DC should be supplied by which facility or group of facilities, and how much inventory DC should hold. The ships used to transport the chemicals are owned by the individual facilities. These vessels have different capacities, operation costs, travel speeds, loading/unloading times, and ability to access particular DCs in different seasons (the shipping lanes may freeze over in winter). Whenever the company's own fleet capacity is insufficient, it charters vessels from a third-party company whose rates depend on the tonnage and travel distance. These rates are generally higher than in-house shipping costs and thus failing to optimally solve the integrated planning/scheduling problem can result in significant additional expenditure.
Motivated by this application, we consider in this study a hypothetical PIDRP that contains multiple production plants, all of which are able to produce the same product, and many customers, each with a specific demand for the product in each time period over a T-period planning horizon. Each plant has a limited production capacity and owns a fleet of transporters each with a different capacity, speed, and availability. The customer demands are assumed to be deterministically known over the periods in T. For the in-house transporters, the operational costs are proportional to the travel time rather than the shipping quantity. However, for chartered transporters, the industry practice is that costs are based on both the quantity shipped and the distance traveled. We will assume that there is no fixed cost associated with transporter usage and that each transporter can make multiple trips during each time period (a trip is defined as a sequence of DC locations a transporter visits beginning and ending at the same plant.) Different transporters (from either the same plant or different plants) can deliver to an individual DC in a period. It is also assumed that the plant inventory cost is based on an average inventory level, a DC's inventory cost is based on the ending inventory in each period, and the loading and unloading times are quantity-independent. Our aim is to develop an integrated operations plan, which minimizes production, inventory, and transportation cost, subject to all the constraints involved.
We use the following notation in our mathematical analyses.
Model parameters
I = set of plants;
J = set of DCs;
T = set of time periods in the planning horizon;
V(i) = set of transporters owned by plant i;
N(v) = maximum number of trips made by transporter v during each period;
[a.sub.i](t) = production cost at plant i during time t;
[h.sub.i](t) = inventory holding cost at plant i during time t;
[bar.h.sub.j](t) = inventory holding cost at D[C.sub.j] during time t;
[c.sub.v] = variable shipping cost for transporter v (per hour);
[c.sub.i,j.sup.c] = shipping rate (per unit load) for chartered transporters to go from location i to location j;
[C.sub.v] = maximum loading capacity for transporter v;
[T.sub.v](t) = available operation time (travel and loading/unloading) for transporter v during period t;
[t.sub.j,k.sup.v] = traveling time from location j to location k by transporter v (includes loading time if location j is a plant, unloading time if location k is a DC);
[D.sub.j](t) = demand of D[C.sub.j] in time period t that must be satisfied by either the inventory at D[C.sub.j], or by the shipment that arrives during t, or by both (i.e., no backlogs);
[p.sub.i.sup.max] = maximum production capacity of plant i;
[s.sub.i.sup.max], [s.sub.i.sup.min] = maximum ending inventory capacity and safety stock (i.e., the minimum inventory) requirement, respectively, at plant i;
[z.sub.i.sup.max], [z.sub.i.sup.min] = maximum ending inventory capacity and safety stock (i.e., the minimum inventory) requirement, respectively, at D[C.sub.i].
Variables
[p.sub.i](t) = production quantity by plant i during time t;
[s.sub.i](t) = ending inventory of plant i at time t;
[z.sub.j](t) = ending inventory of D[C.sub.j] at time t;
[x.sub.i,j,k.sup.v,n](t) = equal to one, if transporter v of plant i visits D[C.sub.k] immediately after visiting D[C.sub.j] during its nth trip in time period t, [for all]i [member of] I, v [member of] V(i), n [member of] N(v), j [member of] {i} [union] J, k [member of] {i} [union] J, j [not equal to] k, t [member of] T;
[g.sub.i,j,k.sup.v,n](t) = quantity carried by transporter v of plant i while it is traveling from D[C.sub.j] to D[C.sub.k] during its nth trip in period t, [for all]i [member of] I, v [member of] V(i), n [member of] N(v), j [member of] {i} [union] J, k [member of] {i} [union] J, j [not equal to] k, t [member of] T;
[q.sub.i,j.sup.v,n](t) = quantity delivered by transporter v, v [member of] V(i), to D[C.sub.j] from plant i during its nth trip in period t;
[Q.sub.i,j](t) = quantity shipped from plant i to D[C.sub.j] by chartered transporters in period t.
Our integrated optimization problem is to minimize the sum of the plants' fleet transportation costs, chartered transporter shipping costs, production costs plus inventory costs, and DC inventory costs. The following mixed-integer program gives a formal definition of PIDRP with a single product (which we denote as problem (P) in the remainder of this paper):
P: min [summation over (t[member of]T)] [summation over (i[member of]I)] [summation over (v[member of]V(i))] [summation over (n[member of]N(v))] [summation over (j,k[member of]{i}[union]J,j[not equal to]k)] [c.sub.v][t.sub.j,k.sup.v][x.sub.i,j,k.sup.v,n](t) + [summation over (t[member of]T)][summation over (i[member of]I)] [summation over (j[member of]J)] [c.sub.i,j.sup.c][Q.sub.i,j](t) + [summation over (t[member of]T)][summation over (i[member of]I)][a.sub.i](t)[p.sub.i](t) + [summation over (t[member of]T)][summation over (i[member of]I)][h.sub.i](t)([p.sub.i](t)/2 + [s.sub.i](t)) + [summation over (t[member of]T)][summation over (j[member of]J)][bar.h.sub.j](t)[z.sub.j](t), (1)
subject to
plant inventory balance constraints:
[s.sub.i](t) = [s.sub.i](t - 1) + [p.sub.i](t) - [summation over (j[member of]J)][summation over (v[member of]V(i))] [summation over (n[member of]N(v))] [q.sub.i,j.sup.v,n](t) - [summation over (j[member of]J)] [Q.sub.i,j](t) [for all]i [member of] I, t [member of] T, (2)
customer inventory balance constraints:
[z.sub.j](t) = [z.sub.j](t - 1) + [summation over (i[member of]I)] [summation over (v[member of]V(i))] [summation over (n[member of]N(v))] [q.sub.i,j.sup.v,n](t) + [summation over (i[member of]I)] [Q.sub.i,j](t) - [D.sub.j](t) [for all]j [member of] J, t [member of] T, (3)
storage capacity and safety stock requirement constraints:
[s.sub.i.sup.min] [less than or equal to] [s.sub.i](t) [less than or equal to] [s.sub.i.sup.max] [for all]i [member of] I, t [member of] T, (4)
[z.sub.j.sup.min] [less than or equal to] [z.sub.j](t) [less than or equal to] [z.sub.j.sup.max] [for all]i [member of] J, t [member of] T, (5)
production capacity constraints:
[less than or equal to] [p.sub.i](t) [less than or equal to] [p.sub.i.sup.max] [for all]i [member of] I, t [member of] T (6)
trip integrity constraints:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[for all]i [member of] I, v [member of] V(i), n [member of] N(v), k [member of] {i} [union] J, t [member of] T, (7)
[summation over (j[member of]J)] [x.sub.i,i,j.sup.v,n](t) [less than or equal to] 1 [for all]i [member of] I, v [member of] V(i), n [member of] N(v), t [member of] T, (8)
transporter capacity constraints:
[g.sub.i,j,k.sup.v,n](t) [less than or equal to] [C.sub.v][x.sub.i,j,k.sup.v,n](t) [for all]i [member of] I, j [member of] {i} [union] J, k [member of] {i} [union] J, j [not equal to] k, v [member of] V(i), n [member of] N(v), t [member of] T, (9)
commodity flow conservation constraints:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[for all]i [member of] I, v [member of] V(i), k [member of] J, n [member of] N(v), t [member of] T, (10a)
[summation over (j[member of]J)] [g.sub.i,j,i.sup.v,n](t) - [summation over (l[member of]J)] [g.sub.i,i,l.sup.v,n](t) = -[summation over (j[member of]J)] [q.sub.i,j.sup.v,n](t) [for all]i [member of] I, v [member of] V(i), n [member of] N(v), t [member of] T, (10b)
transportation duration constraints:
[summation over (n[member of]N(v))] [summation over (j[member of]{i}[union]J)][summation over (k[member of]{i}[union]J)] [t.sub.j,k.sup.v][x.sub.i,j,k.sup.v,n](t) [less than or equal to] [T.sub.v](t)
[for all]i [member of] I, v [member of] V(i), t [member of] T, (11)
non-negativily and integer...
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