Inventory and Variability

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1 Inventory and Variability 1/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Inventory and Variability Stanley B. Gershwin gershwin@mit.edu Massachusetts Institute of Technology December 13, 21

2 Inventory and Variability 2/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Inventory Inventory is a necessary evil, and we do our best to minimize it. Variability makes inventory necessary. Variability is the enemy of manufacturing.

3 Inventory and Variability 3/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Inventory Costs Financial: raw materials have been paid for, but no revenue comes in until the finished product is sold. Demand risk: a product may not be sold due to (for example): time value (newspaper) obsolescence going out of fashion spoilage (food) The longer an item is in storage, the more likely that one of these will occur.

4 Inventory and Variability 4/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Inventory Costs Holding cost (warehouse space) Damage/theft/loss Reduction of yield when inspection is not immediate. Concealment of causes of problems the sources of variability.

5 Inventory and Variability 5/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Inventory Benefits and Motivation The purpose of storage is to allow systems to survive even when important events are unsynchronized. Storage improves system performance by decoupling parts of the system from one another. Storage reduces the propagation of disturbances, and thus reduces instability and the fragility of complex, expensive systems. Storage allows production systems (of energy or goods) to be built with capacity less than the peak demand.

6 Inventory and Variability 6/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Inventory Benefits and Motivation Reduces the propagation of disturbances (eg, machine failures). Allows economies of scale: volume purchasing set-ups Helps manage seasonality and limited capacity. Helps manage uncertainty: Short term: random arrivals of customers or orders. Long term: Total demand for a product next year.

7 Inventory and Variability 7/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Variability Causes Random (uncontrollable) random demand random supply machine failures human variability Deterministic (controllable) setups batches order costs MRP nervousness

8 Inventory and Variability 8/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Variability How variability causes inventory 75 1 B 1 No variability: small pipe 75 1 B 2 Assume the small pipe is the bottleneck. There is no accumulation of water in either tank. large pipes 75 1 The outflow rate in the third pipe is always the same as the rate of flow into the first pipe. Call this Case 1.

9 Inventory and Variability 9/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Variability How variability causes inventory M 1 B 1 all valves closed M 2 large pipes Variable valves, Case 2 In this figure, all valves are closed. B 2 M 3 While the system is in this state, the water level does not change in any tank, and there is no inflow or outflow. But it does not stay in this state forever.

10 Inventory and Variability 1/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Variability How variability causes inventory M 1 B 1 M 2 1 B 2 M 3 Variable valves, Case 2 Now the second valve is opened. While the system is in this state, the first tank loses water and the second tank gains water, and there is no inflow or outflow. If it stays in this state long enough, the first tank becomes empty or the second tank becomes full. But it does not stay in this state forever either.

11 Inventory and Variability 11/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Variability How variability causes inventory M 1 B 1 Variable valves, Case 2 Now the second and third valves are opened. M 2 1 B 2 M 3 1 While the system is in this state, the first tank loses water. The level second tank stays constant. There is no inflow but there is outflow at maximum rate... until the first tank becomes empty, and then the second tank loses water until it becomes empty. But it leaves this state eventually also.

12 Inventory and Variability 12/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Variability How variability causes inventory M 1 B 1 Variable valves, Case 2 1 M 2 There are a total of eight valve states. For each state, the flow rate in each pipe is either or 1. 1 B 2 M 3 Think of an open valve as operational and a closed valve as failed. Then the valves are characterized by MTTF, MTTR. For some set of MTTFs and MTTRs, the average flow rate of Case 2 is the same as the flow rate of Case but Case 2 has inventory and Case 1 does not.

13 Inventory and Variability 13/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Variability How variability causes inventory Conclusions: Variability causes inventory. The tanks in Case 1 are always empty. The tanks in Case 2 gain or lose material due to the unsynchronized valve behavior. Variability makes inventory necessary. We would get the same flow rate in Case 1 if the tanks were not there. The flow rate in Case 2 is a function of the valve behavior (MTTF and MTTR) and the sizes of the tanks. To make the average flow rate of Case 2 the same as the flow rate of Case 1, tanks may be necessary.

14 Inventory and Variability 14/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. What is an Operation? Machine Operator Part Consumable Operation Part Waste Nothing happens until everything is present.

15 Inventory and Variability 15/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. What is an Operation? Waiting Whatever does not arrive last must wait. Inventory: parts waiting. Under-utilization: machines waiting. Idle work force: operators waiting.

16 Inventory and Variability 16/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. What is an Operation? Waiting Operator Machine Part Consumable Operation Part Waste Reductions in the availability, or... Variability in the availability of any one of these items causes waiting in the rest of them and reduces performance of the system.

17 Inventory and Variability 17/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Variability Causes and Cures Creation Causes random demand random supply machine failures human variability setups batches order costs Cures manage demand, find a more reliable supplier improve machines improve human management reduce setup and order costs increase capacity and flexibility

19 Inventory and Variability 19/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Variability Causes and Cures Amplification Causes MRP nervousness changes in schedule in the presence of batching Cures reduce setup and order costs develop improved scheduling methods

20 Inventory and Variability 2/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. Variability Causes and Cures Conclusion: Therefore, factories should be designed and operated to minimize the creation, propagation, and amplification of uncertainty, variability, and randomness.

21 Inventory and Variability 21/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. How inventory and variability interact Simulations: Buffer Level vs. time n(t) n(t) t MTTR i = 1,MTTF i = 1,i = 1,2;N = 1 Small buffer t MTTR i = 1,MTTF i = 1,i = 1,2;N = 1 Large buffer

22 Inventory and Variability 22/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. How inventory and variability interact Simulations: Buffer Level vs. time n(t) n(t) t MTTR i = 1,i = 1,2, MTTF 1 = 5, MTTF 2 = 1;N = 1 First Machine is Bottleneck t MTTR i = 1,i = 1,2, MTTF 1 = 1, MTTF 2 = 5;N = 1 Second Machine is Bottleneck

23 Inventory and Variability 23/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. How inventory and variability interact Production Rate vs. Buffer Size.92.9 P Increasing a small buffer may pay off; increasing a large buffer will not. N

24 Inventory and Variability 24/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. How inventory and variability interact Average Inventory vs. Buffer Size n Bottleneck at second machine. Bottleneck at first machine There will be less inventory if the bottleneck is upstream. N

25 Inventory and Variability 25/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. WIP and Buffer Size in Long lines All buffer sizes = 23, total buffer space = average inventory Production Rate =.886

27 Inventory and Variability 27/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. WIP and Buffer Size in Long lines All buffer sizes = 37, total buffer space = average inventory Production Rate =.884

29 Inventory and Variability 29/29 Copyright c 21 Stanley B. Gershwin. All rights reserved. WIP and Buffer Size in Long lines Conclusions Buffer space is not needed where inventory is greatest; Buffer space is needed where inventory variability is greatest.