1. Capacity analysis of complex materials handling systems

2. Capacity calculation of a single conveyor section Capacity The capacity of a material handling machine can be calculated from the follow distance of two incoming unit loads (L) and the speed of the moving unit load (v). v L Q = v / L [piece/sec]

3. Relevant features Reducing effect by mechanical interlocking Turntable A Driven roller conveyors v unit load time ω turntable time A B C B C t cycle t cycle = t moving in + t turning right +t moving out + t turning left Capacity = 3600/t cycle [piece/hour]

4. Relevant features Reducing effect by presence of multiple unit loads Empty roller conveyor sections due to traffic control v i+2 th unit load i+1 th unit load i th unit load S follow Capacity = 3600*v/S follow [piece/hour] Traffic control’s principle: If the unit load reaches end of the i th section, close down the i-1 th section and free the i-2 th section

5. Relevant features Reducing effect by presence of branches A B1 Definition: Partial capacity limits for different relations: μ AB1 = 3600 / t AB1 μ AB2 = 3600 / t AB2 B2 If the actual material flows for different relations are: AB1 and AB2 then the following equation is valid: AB1 / μ AB1 + AB2 / μ AB2  1 AB1, μ AB1 AB2, μ AB2 μ AB1 μ AB2

6. Relevant features Partial capacity limit example – opening branch A B1 Transfer table for roller conveyor μ AB2 = 3600 / t AB2 B2 3 m2 m3 m 2,5 m v unit load time v transfer table time A t cycle t cycle = t moving in + t transfer t. down +t moving out + t transfer t. up Speed and acceleration data: v unit load : 0,3 m/s a unit load : 0,2 m/s 2 v transfer t. : 0,2 m/s a transfer t. : 0,1 m/s 2 4,6/0,3+0,3/0,2 2,5/0,2+0,2/0,1 3,4/0,3+0,3/(2*0,2) 2,5/0,2+0,2/0,1

7. Fundamentals of Production Logistic systems

8. Relation of materials handling and production logistics Materials handling Provision of materials’ transport services Production logistic Requirements on timing and relations Operational characteristics of a production company Operation in increasing market competition Decreasing product life cycles Continuous adaptive behavior required Increasing product diversity Flexible manufacturing systems Necessity of continuously improving products and services Requirements on price delivery time and reliability Flexible materials handling and logistic services Low stock amount and WIP rates Economical operation required

9. Objectives of production logistic

11. Throughput time’s effect on the final value of a product Production costs total raw material workforce amortization techn. time throughput time wait time value production cost logistic cost profit value function production cost function modified throughput time loss value total production cost

12. Effect of stock amount on the production costs Delivery costStorage cost Total cost Optimal delivered amount:

13. Representation of logistic objectives in throughput diagrams

14. Modelling possibilities of complex materials handling systems

15. Model types 1. Logistic operation function models 2. Queuing models 3. Graph models 4. Petri net models 5. Simulation models Characteristics: examination of production related logistic features such as WIP, throughput time, using throughput diagrams Characteristics: examination of equipment related features such as length of the waiting queue, average waiting time, statistical distribution of the incoming arrivals, necessary number of servers. Characteristics: modeling of static topology and examination of material flow related parameters. Characteristics: modeling and visualizing of actual states and state changes Characteristics: modeling of static topology, state changes. Widespread visual, analytic and optimization features.

16. Logistic operation function models Characteristics: examination of production related logistic features such as WIP, throughput time, using throughput diagrams

17. Queuing models Construction of queuing models: Characteristics: examination of equipment related features such as length of the waiting queue, average waiting time, statistical distribution of the incoming arrivals, necessary number of servers. arrivals (unlimited or limited length) Server 1 Server N.:.: Model types: 1. Queuing model with unlimited arrivals and a single server 2. Queuing model with limited arrivals and a single server 3. Queuing model with unlimited arrivals and multiple servers 4. Queuing model with limited arrivals and multiple servers Applicability of explicite equations: Arrival rate – Poisson distribution Service time – Exponential distribution (Markov model)

18. Queuing models M/M/1 queuing model: arrivals (unlimited length) Server Calculation of optimal service rate: Cost components

19. Queuing models M/M/S queuing model: arrivals (unlimited length) Calculation of the number of optimal servers: Server 1 Server N.:.:

20. Graph models Directed graphs – for modelling continuously operating materials handling machines Undirected graphs – for modelling non-continuously operating materials handling machines Graphs: set of objects, where some pairs are connected by links. Characteristics: modeling of static topology and examination of material flow related parameters.

21. Graph models Mathematical representation of graphs using adjacency matrix. 5 2 3 4 6 1 A = 1 2 5 3 4 1 2 3 4 5 6 1 -1 1 2 1 -1 1 3 -1 -1 -1 4 1 -1 5 1 1 Material flow related parameters: Material flow intensity vector: µ [piece/time unit] Nodal material flow intensity vector: µ n [piece/time unit] Relationship between µ and µ n : µ n = A µ Example: roller conveyor system for unit loads having different traveling order.