A Combined Inventory and Lateral Re-Supply Model for Repairable Items—Part I: Modeling
4.3 A Model with Repair and Lateral Re-supply
4.3.1 Extensions to the FZC Model
The FZC model really consist of the Traveling-Salesmen sub-problem and the Delivery-Allocation sub-problem. The former is manifested in terms of the VRP, while the latter is manifested in a newsvendor problem. The VRP is concerned with routing a fleet of vehicles to the depots. Because of the perishable nature of a newspaper, the newsvendor problem consists of seeking a balanced inventory that minimizes the sum of over- and under-stocking costs. First, a possible extension can be made on the newsvendor problem. The other extension is related to VRP, especially the delivery part.
To start, we should define the extra decision variables and parameters needed:
Here, we add two components to the original FZC vehicle-routing decision-variable xhij. They are related to the quantity repaired at depot i and the inventory at the de- pot. These repaired and inventoried items are available for both local consumption as well as re-supplies to other depots. In other words, both the repaired and inven- toried items can be delivered from depot i to depot j—denoted by xhij ( u) and xhij ( d) respectively. Here the two new decision variables, xhij ( u) and xhij ( d), lie within the upper bound and lower bound as mandated respectively by the repair capability [Ui ( u), Li ( u)] and storage capacity [Ui ( d), Li ( d)] respectively at depot i.
Correspondingly, x123( u) = 1 suggests vehicle 1 picks up the repaired items at depot 2 and delivers it to depot 3, where it is needed. x202( u) = 1 suggests that ve- hicle 2 picks up the repaired item at the main depot (node 0) and delivers to depot 2.
(Should both x202( u) and x123( u) pick up the unitary value, then vehicle 1 picked up a repaired item at location 0 and drove to depot 2 and then deliver it to depot 3—as illustrated in Fig. 4.2.) In Table 4.2, the bounds L2( u) and U2( u) for x123( u) reflect the repair capability at depot 2, which ranges between [0–9]—as illustrated also in Fig. 4.2. In future extensions, a probability density function can ideally be used in lieu of a uniform distribution within a range. Naturally, this would introduce another random variable into the model, aside from failure probabilities.
On the other hand, x123( d) denotes the amount of inventoried items vehicle 1 picks up at depot 2 and delivers to depot 3. x202( d) suggests that vehicle 2 picks up
x if L u x u U u L d x d U d
ijh i ijh
i i ijh
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eerwise
Table 4.2 Inventory/Allocation for the example problem Depot0123 Max. inventory (Pi)10455 Min. inventory0000 Initial inventory (βi)2121 Demand: uniform pdf0.10.250.40.2 Shortage cost ( Ci)$ 5$ 20$ 25$ 15 Surplus cost ( ci)$ 10$ 5$ 5$ 5 Inventory-cost function18 + 0.5z10 + 0.5z20 + 0.5z3048.75 − 3.75z01 − 3.75z21 − 3.75z3151 − 4z02 − 4z12 − 4z3239 − 2z03 − 2z13 − 2z23 Failure range[0–9][0–10][0–9] Repair capability range[0–15][0–6][0–9][0–9]
stored items at main depot 0 and delivers to depot 2. Obviously, x123( d) is deter- mined by the remaining inventory of depot 2. Similarly, x202( d) is governed by the remaining inventory at depot 0. As long as parts are available, they can be delivered.
The amount of storage at depot i is determined by the manager on the basis of the newsvendor inventory-cost function, as shown in Table 4.2. Notice this cost func- tion is computed based on Eq. (4.2) below. The unit cost of storage and repair will decide—along with transportation cost—the relative merits of re-supplying from the repair shop vis-à-vis the stored inventory.
We assume that every depot can supply the other depot and itself, as shown by the delivery decision variable zij. Thus, z01 is the shipment from depot 0 to depot 1.
This amount is the sum of the repaired and inventoried items delivered from depot 0 to depot 1 by any vehicle. To see the significance of these new variables, let us define zij explicitly in terms of the allocation of available products from depot i to depot j (via the fleet of vehicles H):
Setting aside transportation cost, we suggest that a re-supply at j can be satisfied equally well by repaired or inventoried items available at depot i. Perhaps it is more desirable to consider the unit cost of repair and storage in the above equation for zij. Barring nonavailability, the cheaper way to re-supply is preferred. Let wi be the unit repair-cost at depot i to supplement the delivery unit cost ci′.
We highlight the decision variable, zii, to show that each depot can re-supply itself.
Thus z11 is the amount that depot 1 re-supplies itself, and z22 is the amount that de- pot 2 re-supplies itself. Notice that z11 is the sum of x11( u) and x11( d), or that local demand can be satisfied by both repaired and inventoried items:
Let us now examine the “placing-order” assignment-variable yijh. Thus y101 equals unity if vehicle 1 ships supply from depot 0 to depot 1; and zero otherwise. In other words, the delivery variable z01 takes on a nonzero value if y101 = 1, in accordance with this generalization. (More will be said about this in the formal model formula- tion in Sect. 4.3.2).
We refer to yi (and its allied variable yij) as a “bookkeeping” binary variable. It is equal to unity if depot i can supply other depots; we say the depot i is sufficient, in the sense it is fully endowed with an inventory of spare parts. It is equal to zero otherwise. In the latter case, it accepts supplies but cannot deliver to others. Thus yi suggests that either it can send an item to the other depots or it accepts items from
zij =
∑
h H∈ x uijh( )+x dijh( )zij =
∑
h H∈ x uijh( ) if w ci < ′i zij =∑
h H∈ x dijh( ) if ci′ ≤wizii=x (u) x (d)ii + ii
other depots, but not both-if we assume the triangle inequality. Remember we are covering shortage with both repaired and inventoried items, it would make little sense to supply (say) three items to the outside, while accepting one repaired item from outside. In sum, we use this variable to mark either one of these two cases: If y1 is equal to one, then it means that depot 1 will be sufficient enough to re-supply itself and can supply the other depots. If y1 is zero, it means that depot 1 has used up all of its available resources and has nothing to offer.