This post outlines the generalized solution to Multi-Echelon Inventory Decisions by finding the lower, upper and average bound for an optimal solution. It then outlines a genetic algorithm approach to finding the optimal solution. The generalized solution procedure is applied to the Multi-Echelon Inventory Decisions at Jefferson Plumbing Supplies To Store or Not to Store? Case Study

# Introduction:

Alex, a NASCAR enthusiast, is the Inventory Manager of Jefferson Plumbing Supplies, and he faces an interesting quandary. He has been notified by one of his suppliers that there is an upcoming increase to their minimum order quantity. The following case study summary presents a description, a system model, and a summary and conclusion for the problem Alex faces.

# Problem:

Jefferson Plumbing has know demand of 100 units annually, from loyal customers, of a specialty faucet that is ordered from a long time supplier. Alex received a letter from the supplier informing him that they would be increasing their minimum order quantity. The new minimum order quantity is triple what would normally be ordered. Due to their storefront’s city location, inventory holding costs for the faucet are expensive. Jefferson Plumbing Supplies has a warehouse with a cheaper annual holding cost, and Alex would like to determine the optimum costs previous to, and after the policy change.

## Options:

1. convince the company to maintain relations to drop the minimum order amount

2. not stock the item -> lost revenue and customers

3. order and store in the store

4. order and store in the storage facility (multi-echelon system)

5. find alternative storage

Given that option 1, 2, and 5 are not feasible there is option 3 (the base case) and option 4.

# Constrained Optimization Solution

To improve upon the generalized approach and attempt to find the true optimal solution, the optimization problem can be solved using a non-linear programming tool such as Goal Seek in Excel, as demonstrated in this spreadsheet. Alternatively, the lower bound of the optimal solution can be estimated numerically

or using calculus. Then $\theta_j$ can be calculated and rounded to find the order quantities (as shown in the generalized solution).

The results from both methods are in agreement and the result in $Q_w^* = 34$, $Q_p^*= 136$ and an overall average annual cost of about $441.59 which is$5.62 dollars cheaper than before the policy change.