5 Managing a warehouse

5.1 Introduction5.2 Warehouse design5.3 Tactical decisions for warehouse logistics planning5.4 Operational decisions for warehouse logistics management5.5 Case study: Performance evaluation of an AS/RS

system conducted by Wert Consulting5.6 Case study: Inventory management at Wolferine5.7 Case study: Airplane loading at FedEx5.8 Questions and problems

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 1 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Wert Consulting

- Dutch company that specializes in the design,manufacturing and selling of shelvings, stacker cranes forAS/RS warehouses and other storage systems;

- in February 2012, Wert acted as consultant of a primaryfood company;

- aim: evaluate the performance of a palletized AS/RSwarehouse, in which each aisle will be served by a newstacker crane.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 2 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Wert Consulting

- Dutch company that specializes in the design,manufacturing and selling of shelvings, stacker cranes forAS/RS warehouses and other storage systems;

- in February 2012, Wert acted as consultant of a primaryfood company;

- aim: evaluate the performance of a palletized AS/RSwarehouse, in which each aisle will be served by a newstacker crane.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 2 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Wert Consulting

- Dutch company that specializes in the design,manufacturing and selling of shelvings, stacker cranes forAS/RS warehouses and other storage systems;

- in February 2012, Wert acted as consultant of a primaryfood company;

- aim: evaluate the performance of a palletized AS/RSwarehouse, in which each aisle will be served by a newstacker crane.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 2 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Operating modes of the stacker crane

- single cycle: storage and retrieval operations are performedone at a time;

- dual cycle: pairs of storage and retrieval operations aremade in sequence in an attempt to reduce the overall traveltime.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 3 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Operating modes of the stacker crane

- single cycle: storage and retrieval operations are performedone at a time;

- dual cycle: pairs of storage and retrieval operations aremade in sequence in an attempt to reduce the overall traveltime.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 3 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (1/4)

- M: number of storage aisles;- 0: I/O point of the stacker crane;- N: total number of storage positions in the rack;- Lx : length of the rack;- Ly : height of the rack;- vx : (constant) speed of the stacker crane along the x axis;- vy : (constant) speed of the stacker crane along the y axis;- t0i, i = 1, . . . ,N: one-way travel time between the I/O point

and the storage position i (t0i = ti0);- tij, i, j = 1, . . . ,N: one-way travel time between the storage

positions i and j (tij = tji);- TSC : round-trip travel (random) time of a single command;- TDC : round-trip travel (random) time of a dual command.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 4 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (2/4)- Performance evaluation of the AS/RS system: expected

single- and dual-command travel times of each stackercrane:

E[TSC ] =

1N

N∑

i=12t0i;

E[TDC ] =

1N(N−1)

N−1∑

i=1

N∑

j=i+12(t0i + tij + tj0);

- closed-form expressions by Bozer and White: storing andpicking times associated with load handling assumednegligible (independent of the rack shape and of the travelspeed of the stacker crane);

- I/O point: assumed located at the lower left-hand corner.G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 5 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (2/4)- Performance evaluation of the AS/RS system: expected

single- and dual-command travel times of each stackercrane:

E[TSC ] =

1N

N∑

i=12t0i;

E[TDC ] =

1N(N−1)

N−1∑

i=1

N∑

j=i+12(t0i + tij + tj0);

- closed-form expressions by Bozer and White: storing andpicking times associated with load handling assumednegligible (independent of the rack shape and of the travelspeed of the stacker crane);

- I/O point: assumed located at the lower left-hand corner.G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 5 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (2/4)- Performance evaluation of the AS/RS system: expected

single- and dual-command travel times of each stackercrane:

E[TSC ] =

1N

N∑

i=12t0i;

E[TDC ] =

1N(N−1)

N−1∑

i=1

N∑

j=i+12(t0i + tij + tj0);

- closed-form expressions by Bozer and White: storing andpicking times associated with load handling assumednegligible (independent of the rack shape and of the travelspeed of the stacker crane);

- I/O point: assumed located at the lower left-hand corner.G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 5 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (2/4)- Performance evaluation of the AS/RS system: expected

single- and dual-command travel times of each stackercrane:

E[TSC ] =

1N

N∑

i=12t0i;

E[TDC ] =

1N(N−1)

N−1∑

i=1

N∑

j=i+12(t0i + tij + tj0);

- closed-form expressions by Bozer and White: storing andpicking times associated with load handling assumednegligible (independent of the rack shape and of the travelspeed of the stacker crane);

- I/O point: assumed located at the lower left-hand corner.G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 5 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (2/4)- Performance evaluation of the AS/RS system: expected

single- and dual-command travel times of each stackercrane:

E[TSC ] =

1N

N∑

i=12t0i;

E[TDC ] =

1N(N−1)

N−1∑

i=1

N∑

j=i+12(t0i + tij + tj0);

- closed-form expressions by Bozer and White: storing andpicking times associated with load handling assumednegligible (independent of the rack shape and of the travelspeed of the stacker crane);

- I/O point: assumed located at the lower left-hand corner.G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 5 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (2/4)- Performance evaluation of the AS/RS system: expected

single- and dual-command travel times of each stackercrane:

E[TSC ] =

1N

N∑

i=12t0i;

E[TDC ] =

1N(N−1)

N−1∑

i=1

N∑

j=i+12(t0i + tij + tj0);

- closed-form expressions by Bozer and White: storing andpicking times associated with load handling assumednegligible (independent of the rack shape and of the travelspeed of the stacker crane);

- I/O point: assumed located at the lower left-hand corner.G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 5 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (3/4)

- Tx : horizontal travel time required to go to the farthestcolumn from the I/O station:

Tx = Lx/vx ;

- Ty : vertical travel time required to go to the farthest row(level):

Ty =Ty/vy .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 6 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (3/4)

- Tx : horizontal travel time required to go to the farthestcolumn from the I/O station:

Tx = Lx/vx ;

- Ty : vertical travel time required to go to the farthest row(level):

Ty =Ty/vy .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 6 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (3/4)

- Tx : horizontal travel time required to go to the farthestcolumn from the I/O station:

Tx = Lx/vx ;

- Ty : vertical travel time required to go to the farthest row(level):

Ty =Ty/vy .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 6 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (3/4)

- Tx : horizontal travel time required to go to the farthestcolumn from the I/O station:

Tx = Lx/vx ;

- Ty : vertical travel time required to go to the farthest row(level):

Ty =Ty/vy .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 6 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (4/4)

- Stacker crane travels simultaneously and independently inthe horizontal and vertical directions:

T =max(Tx ,Ty);

- T : time required to reach the farthest storage location fromthe I/O point;

b =min(Tx/T ,Ty/T);

- Assumptions: T =Tx and b =Ty/T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 7 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (4/4)

- Stacker crane travels simultaneously and independently inthe horizontal and vertical directions:

T =max(Tx ,Ty);

- T : time required to reach the farthest storage location fromthe I/O point;

b =min(Tx/T ,Ty/T);

- Assumptions: T =Tx and b =Ty/T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 7 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (4/4)

- Stacker crane travels simultaneously and independently inthe horizontal and vertical directions:

T =max(Tx ,Ty);

- T : time required to reach the farthest storage location fromthe I/O point;

b =min(Tx/T ,Ty/T);

- Assumptions: T =Tx and b =Ty/T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 7 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

AS/RS system (4/4)

- Stacker crane travels simultaneously and independently inthe horizontal and vertical directions:

T =max(Tx ,Ty);

- T : time required to reach the farthest storage location fromthe I/O point;

b =min(Tx/T ,Ty/T);

- Assumptions: T =Tx and b =Ty/T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 7 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (1/2)

- (x ,y): Cartesian coordinate of a storage location;- txy =max(tx , ty): time required by the stacker crane to reach(x ,y) (tx = x/vx and ty = y/vy );

- hypothesis. Each storage location is accessible with thesame probability:

> coordinates x and y are independent random variables:

Ftxy (t)=Pr(txy ≤ t)=Pr(tx ≤ t , ty ≤ t)=Pr(tx ≤ t)Pr(ty ≤ t);

> coordinates x and y are uniformly distributed:

Pr(tx ≤ t) = t/T , 0≤ t ≤T ;

Pr(ty ≤ t) =

{

t/bT , if 0≤ t ≤ bT1, if bT < t ≤T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 8 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (1/2)

- (x ,y): Cartesian coordinate of a storage location;- txy =max(tx , ty): time required by the stacker crane to reach(x ,y) (tx = x/vx and ty = y/vy );

- hypothesis. Each storage location is accessible with thesame probability:

> coordinates x and y are independent random variables:

Ftxy (t)=Pr(txy ≤ t)=Pr(tx ≤ t , ty ≤ t)=Pr(tx ≤ t)Pr(ty ≤ t);

> coordinates x and y are uniformly distributed:

Pr(tx ≤ t) = t/T , 0≤ t ≤T ;

Pr(ty ≤ t) =

{

t/bT , if 0≤ t ≤ bT1, if bT < t ≤T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 8 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (1/2)

- (x ,y): Cartesian coordinate of a storage location;- txy =max(tx , ty): time required by the stacker crane to reach(x ,y) (tx = x/vx and ty = y/vy );

- hypothesis. Each storage location is accessible with thesame probability:

> coordinates x and y are independent random variables:

Ftxy (t)=Pr(txy ≤ t)=Pr(tx ≤ t , ty ≤ t)=Pr(tx ≤ t)Pr(ty ≤ t);

> coordinates x and y are uniformly distributed:

Pr(tx ≤ t) = t/T , 0≤ t ≤T ;

Pr(ty ≤ t) =

{

t/bT , if 0≤ t ≤ bT1, if bT < t ≤T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 8 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (1/2)

- (x ,y): Cartesian coordinate of a storage location;- txy =max(tx , ty): time required by the stacker crane to reach(x ,y) (tx = x/vx and ty = y/vy );

- hypothesis. Each storage location is accessible with thesame probability:

> coordinates x and y are independent random variables:

Ftxy (t)=Pr(txy ≤ t)=Pr(tx ≤ t , ty ≤ t)=Pr(tx ≤ t)Pr(ty ≤ t);

> coordinates x and y are uniformly distributed:

Pr(tx ≤ t) = t/T , 0≤ t ≤T ;

Pr(ty ≤ t) =

{

t/bT , if 0≤ t ≤ bT1, if bT < t ≤T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 8 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (1/2)

- (x ,y): Cartesian coordinate of a storage location;- txy =max(tx , ty): time required by the stacker crane to reach(x ,y) (tx = x/vx and ty = y/vy );

- hypothesis. Each storage location is accessible with thesame probability:

> coordinates x and y are independent random variables:

Ftxy (t)=Pr(txy ≤ t)=Pr(tx ≤ t , ty ≤ t)=Pr(tx ≤ t)Pr(ty ≤ t);

> coordinates x and y are uniformly distributed:

Pr(tx ≤ t) = t/T , 0≤ t ≤T ;

Pr(ty ≤ t) =

{

t/bT , if 0≤ t ≤ bT1, if bT < t ≤T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 8 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (1/2)

- (x ,y): Cartesian coordinate of a storage location;- txy =max(tx , ty): time required by the stacker crane to reach(x ,y) (tx = x/vx and ty = y/vy );

- hypothesis. Each storage location is accessible with thesame probability:

> coordinates x and y are independent random variables:

Ftxy (t)=Pr(txy ≤ t)=Pr(tx ≤ t , ty ≤ t)=Pr(tx ≤ t)Pr(ty ≤ t);

> coordinates x and y are uniformly distributed:

Pr(tx ≤ t) = t/T , 0≤ t ≤T ;

Pr(ty ≤ t) =

{

t/bT , if 0≤ t ≤ bT1, if bT < t ≤T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 8 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (1/2)

- (x ,y): Cartesian coordinate of a storage location;- txy =max(tx , ty): time required by the stacker crane to reach(x ,y) (tx = x/vx and ty = y/vy );

- hypothesis. Each storage location is accessible with thesame probability:

> coordinates x and y are independent random variables:

Ftxy (t)=Pr(txy ≤ t)=Pr(tx ≤ t , ty ≤ t)=Pr(tx ≤ t)Pr(ty ≤ t);

> coordinates x and y are uniformly distributed:

Pr(tx ≤ t) = t/T , 0≤ t ≤T ;

Pr(ty ≤ t) =

{

t/bT , if 0≤ t ≤ bT1, if bT < t ≤T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 8 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (1/2)

- (x ,y): Cartesian coordinate of a storage location;- txy =max(tx , ty): time required by the stacker crane to reach(x ,y) (tx = x/vx and ty = y/vy );

- hypothesis. Each storage location is accessible with thesame probability:

> coordinates x and y are independent random variables:

Ftxy (t)=Pr(txy ≤ t)=Pr(tx ≤ t , ty ≤ t)=Pr(tx ≤ t)Pr(ty ≤ t);

> coordinates x and y are uniformly distributed:

Pr(tx ≤ t) = t/T , 0≤ t ≤T ;

Pr(ty ≤ t) =

{

t/bT , if 0≤ t ≤ bT1, if bT < t ≤T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 8 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (2/2)

- Cumulative probability distribution function of txy :

Ftxy (t)={

t2/bT2, if 0≤ t ≤ bTt/T , if bT < t ≤T ;

- associated probability density function:

ftxy (t)={

2t/bT2, if 0≤ t ≤ bT1/T , if bT < t ≤T ;

- Expected single-command travel time:

E[TSC ]= 2∫ T

0tftxy (t)dx =

4bT2

∫ bT

0t2 dx +

2T

∫ T

bTt dx = (b2/3+1)T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 9 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (2/2)

- Cumulative probability distribution function of txy :

Ftxy (t)={

t2/bT2, if 0≤ t ≤ bTt/T , if bT < t ≤T ;

- associated probability density function:

ftxy (t)={

2t/bT2, if 0≤ t ≤ bT1/T , if bT < t ≤T ;

- Expected single-command travel time:

E[TSC ]= 2∫ T

0tftxy (t)dx =

4bT2

∫ bT

0t2 dx +

2T

∫ T

bTt dx = (b2/3+1)T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 9 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (2/2)

- Cumulative probability distribution function of txy :

Ftxy (t)={

t2/bT2, if 0≤ t ≤ bTt/T , if bT < t ≤T ;

- associated probability density function:

ftxy (t)={

2t/bT2, if 0≤ t ≤ bT1/T , if bT < t ≤T ;

- Expected single-command travel time:

E[TSC ]= 2∫ T

0tftxy (t)dx =

4bT2

∫ bT

0t2 dx +

2T

∫ T

bTt dx = (b2/3+1)T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 9 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (2/2)

- Cumulative probability distribution function of txy :

Ftxy (t)={

t2/bT2, if 0≤ t ≤ bTt/T , if bT < t ≤T ;

- associated probability density function:

ftxy (t)={

2t/bT2, if 0≤ t ≤ bT1/T , if bT < t ≤T ;

- Expected single-command travel time:

E[TSC ]= 2∫ T

0tftxy (t)dx =

4bT2

∫ bT

0t2 dx +

2T

∫ T

bTt dx = (b2/3+1)T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 9 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (2/2)

- Cumulative probability distribution function of txy :

Ftxy (t)={

t2/bT2, if 0≤ t ≤ bTt/T , if bT < t ≤T ;

- associated probability density function:

ftxy (t)={

2t/bT2, if 0≤ t ≤ bT1/T , if bT < t ≤T ;

- Expected single-command travel time:

E[TSC ]= 2∫ T

0tftxy (t)dx =

4bT2

∫ bT

0t2 dx +

2T

∫ T

bTt dx = (b2/3+1)T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 9 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected single-command travel time (2/2)

- Cumulative probability distribution function of txy :

Ftxy (t)={

t2/bT2, if 0≤ t ≤ bTt/T , if bT < t ≤T ;

- associated probability density function:

ftxy (t)={

2t/bT2, if 0≤ t ≤ bT1/T , if bT < t ≤T ;

- Expected single-command travel time:

E[TSC ]= 2∫ T

0tftxy (t)dx =

4bT2

∫ bT

0t2 dx +

2T

∫ T

bTt dx = (b2/3+1)T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 9 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Expected dual-command travel time

E[TDC ]= (4/3+b2/2−b3/30)T .

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 10 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: data

- M = 3;- Lx = 95 m;- Ly = 20 m;- vx = 2.5 m/sec;- vy = 0.7 m/sec.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 11 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: data

- M = 3;- Lx = 95 m;- Ly = 20 m;- vx = 2.5 m/sec;- vy = 0.7 m/sec.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 11 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: data

- M = 3;- Lx = 95 m;- Ly = 20 m;- vx = 2.5 m/sec;- vy = 0.7 m/sec.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 11 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: data

- M = 3;- Lx = 95 m;- Ly = 20 m;- vx = 2.5 m/sec;- vy = 0.7 m/sec.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 11 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: data

- M = 3;- Lx = 95 m;- Ly = 20 m;- vx = 2.5 m/sec;- vy = 0.7 m/sec.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 11 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: computed parameters

- Tx = L/vx = 38.00 sec;- Ty = Ly/vy = 28.57 sec;- T =max(Tx ,Ty)= 38.00 sec;- b =min(Tx/T ,Ty/T)= 0.752.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 12 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: computed parameters

- Tx = L/vx = 38.00 sec;- Ty = Ly/vy = 28.57 sec;- T =max(Tx ,Ty)= 38.00 sec;- b =min(Tx/T ,Ty/T)= 0.752.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 12 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: computed parameters

- Tx = L/vx = 38.00 sec;- Ty = Ly/vy = 28.57 sec;- T =max(Tx ,Ty)= 38.00 sec;- b =min(Tx/T ,Ty/T)= 0.752.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 12 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: computed parameters

- Tx = L/vx = 38.00 sec;- Ty = Ly/vy = 28.57 sec;- T =max(Tx ,Ty)= 38.00 sec;- b =min(Tx/T ,Ty/T)= 0.752.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 12 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: conclusions

Since the stacker crane will operate only single commands:- expected travel time: (b2/3+1)× T = 45.16 sec;- throughput of the stacker crane: approximately equal to

3600/45.16;- about 80×3= 240 pallets per hour for the whole warehouse.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 13 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: conclusions

Since the stacker crane will operate only single commands:- expected travel time: (b2/3+1)× T = 45.16 sec;- throughput of the stacker crane: approximately equal to

3600/45.16;- about 80×3= 240 pallets per hour for the whole warehouse.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 13 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: conclusions

Since the stacker crane will operate only single commands:- expected travel time: (b2/3+1)× T = 45.16 sec;- throughput of the stacker crane: approximately equal to

3600/45.16;- about 80×3= 240 pallets per hour for the whole warehouse.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 13 / 13

5 Managing a warehouse Case study: Performance evaluation of an AS/RS system

Test of the stacker crane: conclusions

Since the stacker crane will operate only single commands:- expected travel time: (b2/3+1)× T = 45.16 sec;- throughput of the stacker crane: approximately equal to

3600/45.16;- about 80×3= 240 pallets per hour for the whole warehouse.

G. Ghiani, G. Laporte, R. Musmanno Introduction to Logistics System Management © John Wiley & Sons, Ltd 13 / 13