Simple Model
Overview
The simple model assumes a constant product demand, where the entire inventory is depleted before anything is re-ordered. Re-ordering
is by a fixed quantity and gets filled immediately. This model is overly simplistic but is valuable in examining the tradeoff between
order size and inventory costs.
Total Costs
Well define the total units of inventory held as {% Q %}. Then the total holding costs are (excluding fixed costs)
{% Total\: Costs = h \times Average\: Q + k(D/Q) + c_T D%}
In the case where we assume constant product demand, the average Q is just Q/2.
{% Total\: Costs = h \times Q/2 + k(D/Q) + c_T D%}
Transaction Costs
Transacation costs can likewise be categorized as fixed and variable. Here the fixed cost is more important to
keep track of because it is a fixed cost per transaction, and the number of transactions will vary based on the
inventory policy chosen.
{% k %} is usually defined as the fixed cost per transaction and {% c_T %} is the variable transaction cost.
If we assume a simple model of constant product demand of {% D %} per period, where we reorder {% Q %} units everytime
inventory goes to zero, then our per period transaction costs are
{% k (D/Q) + c_T D %}
This assumes that once an inventory order is issued, it is fulfilled instantaneously (so that the constant demand
is continuously fulfilled). The cost curve mapped against {% Q %} is given below.
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Analytical Solution
The simple model can be solved by taking the derivative of the cost equation with respect to {% Q %}, and setting to zero.
{% \partial Costs/\partial Q = 0 %}
{% h/2 - k (D/Q^2) = 0 %}
{% Q_{opt} = \sqrt{2kD/h} %}
This is the amount of inventory order that minimizes total costs.
