**Introduction**

All businesses have process flows in which a product is designed or manufactured or in which a service is rendered. An on-going goal is to achieve the maximum possible throughput at the lowest possible cost while meeting all the requirements of the product or service.

**Inventory Benefits**

A certain minimum amount of in-process inventory is always necessary. This level is defined by Little’s Law:

*I = R x T*

*where I = inventory, R = flow rate, and T = flow time, all of which are average values.*

*The actual amount of inventory in the process will be greater than the theoretical amount because some inventory always will be in-transit between different locations. Furthermore, the actual levels usually are planned to be even higher.*

*There are four possible reasons that firms intentionally plan excess inventory levels:*

*1. Economies of scale*

- Quantity discounts offered by suppliers.
- Fixed ordering costs and fixed setup costs are lower if spread across more units.

2. Production and capacity smoothing

- Rather than vary processing rate to match varying demand, it may be more economical to process at a constant rate and use inventory as a buffer.

3. Protection against supply disruptions and demand surges

- Supply disruptions may result in process starvation, downtime, and throughput reduction.
- Demand surges can result in delayed deliveries, lost sales, and customer dissatisfaction.

4. Profiting from price changes

- Speculative inventories can be used to protect and profit from sudden price changes.
- Inflows and outflows can be managed in order to optimize the financial value of the inventory.

**Inventory Costs **(Disadvantages of a large inventory)

- Increased response time to changes in market demand.
- Increased time to change to new products.
- Delay in detection of quality problems
- Decouples stages of the process flow, discouraging teamwork
- Holding costs (physical holding cost and cost of capital)
- Physical holding costs include operating costs and losses due to spoilage, obsolescence, pilferage, etc. Expressed as a fraction
*h*of the variable cost*C*of one flow unit of inventory. So the physical holding cost for one unit of inventory for one time period is equal to*h x C*.

- Cost of capital is the opportunity cost of foregone returns on the amount invested in inventory that could have been invested in other projects. Cost of capital per flow unit is expressed as
*r x C*where*r*is the cost of capital and*C*is the variable cost of one flow unit. - Total holding cost is:
*H = (h + r) x C*.

When counting average inventory in a process, the steps prior to the process bottleneck will be full, and those afterwards will be occupied by a ratio of the throughput rate of those steps to that of the bottleneck.

**Conveyor Belts**

When changing a production system from a long conveyor belt to a production cell system, with each cell producing the full product with fewer workers, one method to determine the optimal place to break the line is:

- Add the total labor time and divide by the number of workers in a cell.
- Add the tasks for the first worker’s steps until this average time is exceeded.
- Drop the last task and determine if the resulting average for the remaining workers is greater than or less than that for the first with that time included.

**Inventory in Queue with Variable Arrival**

*I _{I} = [ ρ x ρ^{sqrt(2(c+1)) – 1} / 1 – ρ ] x ( C_{i}^{2} + C_{p}^{2} ) / 2*

where *I _{I}*

ρ = capacity utilization

C = std. dev / mean

When calculating average wait time, divide average inventory in the queue by the arrival rate, not by the processing rate. This is because the system throughput is determined by the incoming rate, not the processing rate (capacity utilization < 1).

**Some Quality Experts**

Deming: statistical process control

Juran: modeled cost of quality as U-shaped curve

Crosby: quality is free