# DEL (dynamic economic lotsize) model

The dynamic economic lot size (DEL) model is an analogue of the EOQ model, where the demand and the purchase cost can vary over time in any manner [1].

The central managerial dilemma is to find the right balance between the cost of holding inventory and the cost of ordering, including scale economies.

## Assumptions

Demand for a product occurs at each of several time points. All demand must be satisfied; no backorders or lost sales are permitted. One can order or produce supplies at each point, and carry inventory from one point to the next. Replenishment decisions take effect immediately; there is no order leadtime.

Each order incurs a fixed cost regardless of the size of the order, and also a variable cost proportional to the order quantity. There is a cost to hold inventory between any two successive points. These costs as well as the demands may change over time. The goal is to determine a feasible ordering plan which minimizes the total cost over all time points.

To formulate this problem, denote

$\,T$ = time horizon

$\,t$ = index for time points, $\,t=0,...,T$

The horizon $\,T$ is finite. The data of the problem include

$\,d(t)$ = demand at time

These are arbitrary nonnegative constants. The decision variables are

$\,x(t)$ = inventory at time

$\,z(t)$ = order size at time

The starting inventory at time $\,t=0$ is the known constant $\,x_{0}$.

The precise sequence of events at each time point t < T is as follows:

1. We observe the inventory $\,x(t)$.

2. We decide the order size $\,z(t)$.

The cost parameters are the following:

$\,k(t)$ = fixed order cost at time $\,t$

$\,c(t)$ = variable order cost at time $\,t$

$\,h(t)$ = inventory holding cost at time $\,t$.

These are all nonnegative.

Consider also

$\,D(t)$ -- cumulative demand through time $\,t$,

$\,D(t)=\sum _{s=0}^{t}d(s)$

$\,D[t,u)$ -- demand from time $\,t$ through $\,u-1$

$\,D[t,u)=D(u-1)-D(t-1),\, \, \, \, t\le u$

$\,\tilde{c}[t,u)$ -- variable cost to order a unit at $\,t$ and hold it until $\,u$

$\,\tilde{c}[t,u)=c(t)+\sum _{s=t+1}^{u}h(s),\, \, \, \, t\le u$

Mathematical statement of the DEL model

$\,v(t)\in \{ 0,1\}$

$\,z(t)\le D[t,T)v(t)\, \, \, \, t=1,...,T-1$

Minimize

$\,\sum _{t=0}^{T-1}[k(t)v(z(t))+c(t)z(t)]+\sum _{t=1}^{T}h(t)x(t)$

Thus, the DEL model can be expressed as a linear mixed-integer program.

## Network Representation and Solution

To solve the DEL model, then, the primary problem is to select the order times. Focus on the case $\,x_{0} =0$. This selection problem can be expressed in terms of a network: The nodes correspond to the time points $\,t=0,...,T$. There is a directed arc from t to u for every node pair $\,(t,u)$ with $\,t. Any specific choice of order times corresponds to a path in this network from node 0 to node $\,T$, and vice versa.

The nodes encountered along the path (excluding $\,T$ itself) identify the order times.

Now, consider any one path and a particular arc $\,(t,u)$ along the path. Choosing this arc means that $\,t$ is an order time, and we order just enough at $\,t$ to meet the demands at points $\,t$ through $\,u-1$. Let $\,k[t,u)$ denote the total cost incurred by this decision.

The dynamics imply that

$\,z(t)=D[t,u),$

$\,x(t+1)=z(t)-d(t)=D[t+1,u),$

and

$\,x(s+1)=x(s)-d(s)=D[s+1,u)\, \, for\, \, t

Thus,

$\,k[t,u)=k(t)+c(t)z(t)+\sum _{s=t+1}^{u}h(s)x(s) =k(t)+\sum _{s=t}^{u-1}\tilde{c}[t,s)d(s)$

Now add the over all arcs encountered on the path. This sum accounts for the all costs in all periods - it is the cost of choosing the order times along the path.

Optimal solution

So, compute $\,k[t,u)$ for every arc in the network, and think of $\,k[t,u)$ as the travel cost along arc $\,(t,u)$. To determine an optimal sequence of order times, find a path of minimum total cost. In sum, the DEL model reduces to a shortest-path calculation (see Network models).

## References

1. Zipkin P. (2000) Foundations of inventory management; The McGraw-Hill Companies, Inc.