Problem Statement

A Combined Inventory and Lateral Re-Supply Model for Repairable Items—Part I: Modeling

4.1 Problem Statement

73

A Combined Inventory and Lateral Re-Supply

Keeping the fact that the problem laid out in this paper is the extension of Fed- ergruen & Zipkin (FZ) model (1984); the comparison of the two models in short is presented in Table 4.1 in order to make the contribution clear.

For the sake of completeness, the details of the FZ model data are shown in Fig. 4.1. The objective of the FZ model is to minimize the traveling cost and inven- tory cost function occurring at each demand sites. The traveling cost is related to the distance travelled between nodes. The inventory cost function is the newsvendor inventory cost function with shortage and surplus cost. The traveling cost matrix is symmetric and the “distance” is measured in terms of time. For illustration, the inventory cost function is calculated based on uniform distribution. Note that the inventory cost function is nonincreasing function. There is no inventory cost func- tion associated with the supply depot (Node-0) because its main role is to store enough supplies to satisfy the demand occurred at each node. The main depot only supplies all nodes.

Let us refer to the example problem as shown in Fig. 4.2. While making lateral re-supply decisions between depots 1, 2, and 3, we formulate a delivery-allocation plan using the vehicle fleet of two based at the main depot 0. Lateral re-supplies

Table 4.1  The comparison of two models

Topics covered Federgruen & Zipkin (FZ)

model Extended model (research

model)

Lateral resupply Not allowed Each base can resupply the

others

Delivery type Only delivery Both delivery and pick-up

Local storage capability Yes Yes

Local repair capability No Yes

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from all depots are then delivered via this vehicle fleet. After solving this problem, one can verify that vehicle 1 will visit depots 2 and 3 and return to the home depot 0, while vehicle 2 will make an out-and-back route to depot 1. In this paper, the repair process is not explicitly modeled, but there is a repair capability at each depot that supplements (and in fact lowers) its stored inventory (McGrath 1999). The repair capacity is expressed in terms of a range of repairable units that can be handled. For example, the repair capability at depot 1 is between 0 and 6 items. Being a mixed- integer stochastic-program with both a nonlinear objective function and constraints, we solve the model by generalized Benders’ decomposition. The decomposition allows us to linearize the nonlinear constraints, solve large problems, and use gen- eral failure and repair functions familiar to logistics analysis. Analytical and com- putational experience via generalized Benders’ cuts suggests that substantial cost efficiency is achieved by considering delivery, inventory and repair simultaneously.

In the beginning of the period, each depot has an initial inventory. For example, the initial inventory at depot 1 is one unit of spare part. There is a minimum and maximum allowable inventory level at each depot, corresponding to the threshold and the depot storage-capacity respectively. Again, for depot 1 the range is between 0 and 4. Any part can fail with a probability distribution at depots i = 0, 1, 2, 3. Once failed, we need to replace the part from inventory or to repair the part.

At each depot, maintaining the inventory entails holding and shortage costs. We will make vehicle delivery, for example, only when a depot inventory level poten- tially falls below its minimum threshold—in order to lower inventory and transpor- tation costs. There is a delivery cost for shipping serviceable parts from any depot i

For node-1

Inial level: 1 Z₁₁=7

X (d)=1

Vehicle-1 (9.3 Hours) Vehicle-2 (6 Hours)

Minimum level: 0 Maximum level: 4 Repair capability: [0-6]

Failure: [0-9]

Cosˆ uncon:

X11₁₁(u)=6 Z₀₁=6 X²₀₁(u)=4 X²₀₁(d)=2

1

Z₀₀=7

3

X₀₀(u)=7 For node-2

Inial level: 2 Minimum level: 0

Z₂₂=11 X₂₂(d)=2 X₂₂(u)=9 q₂(z₂)=48.75-3.75z₀₁-3.75z₂₁-3.75z₃₁

0 2

2 3.8

3.5

For node-0

Maximum level: 5 Repair capability: [0-10]

Failure: [0-10]

Cost funcon:

q (z )=51 4z 4z 4z Z₀₂=4 X¹₀₂(u)=4

3

Inial level: 2 Minimum level: 0 Maximum level: 10 Repair capability: [0-15]

Cosˆ uncon:

For node-3 Inial level: 1 Minimum level: 0 Maximum level: 5

Z₃₃=10 X₃₃(d)=1 X₃₃(u)=9

2 2 02- ₁₂- ₃₂

q₀(z₀)=18+0.5z₁₀+0.5z₂₀+0.5z₃₀ Repair capability: [0-9]

Failure: [0-9]

Cost funcon:

q₃(z₃)=39-2z₀₃-2z₁₃-2z₂₃ Fig. 4.2  Example problem

to depot j, where i, j = 0, 1, 2, 3. Figure 4.2 shows a transportation cost of three units between the home depot 0 and depot 1.

We can provide lateral re-supply from any depot (including the main depot). This way, lateral re-supply can come expediently from any proximal depot, instead of just the main depot (Herer and Rashit 1999). The shipping cost from depot i to its adjacent depot j is known (where i, j = 0, 1, 2, 3). Figure 4.2 shows that re-supplying from depot 2 to depot 3 laterally will cost 3.8 units. Now we decide how many parts to repair, to hold, to deliver, and which vehicle to use for delivery―considering both available repaired and inventoried items at all depots. Supposed the problem is solved, operational cost and resource allocation are determined, including the real- ized demand. For example, z₀₁ = 6 means that six spare parts are delivered from the home depot 0 to depot 1, supplementing one unit of stored inventory to address the shortage. This concludes one planning cycle. Consumption of the supplies begins after delivery, and a new cycle begins.

As will be shown in the modeling part, solving such a mixed-integer stochastic- problem even in simpler terms as in FZ model requires using some special algo- rithms/techniques to come up with the optimal solutions. Computational experience confirms that Benders’ decomposition algorithm is very effective in solving this type of problems.