Question

Consider an inventory system that fits the model for a serial two-echelon system, where K1 = $5,000, K2 = $200, h1 = $10, h2 = $11, and d = 100. Fill the values in the given table that shows the results from performing both separate optimization of the installations and simultaneous optimization of the installations

hen calculate the percentage change in the total variable cost per unit time if the results from performing separate optimization were to be used instead of the results from the valid approach of performing simultaneous optimization

Answer 1

Consider the following deterministic multi-echelon inventory model

To determine the optimal total variable cost, the optimal order quantity at both the installation and optima

For the installation1,

The setup cost

The holding cost

The constant demand rate

For installation2,

The setup cost

The holding cost

For installation2, the optimal order quantity

Then

  

For installation2, the optimal total variable cost

Then

  

and

Then

And

since,

Then,

  

Now, for installation 1 the optimal order quantity

Then,

For installation1, the optimal total variable cost

Then

Therefore, the optimal total variable cost for both the installations

Then

Now, for simultaneous optimization

  

and

Then,

  

Then

And

Since,

Then,

And, at installation2 the optimal order quantity

Then

  

At installation1 the optimal order quantity

Then

Therefore, the optimal total variable cost for both the installations

Then

The table for serial two-echelon system is

Quantity	Separate Optimization	Simultaneous Optimization
	60.3022	161.245
	5.244	1.58
	6	2
	361.81	322.49
	2459.823	994.987

The biggest difference is that the order quantity at installation 2 is nearly trice large to the order quantity in seperable optimization and the increase in total variable cost per unit time is nearly 0.09 percent because there is minor difference between these two variable costs.