def get_gridworld():
gridworld = EnvBaseline(name='Sample Grid World',
s_hash_rowL=s_hash_rowL,
row_tickL=row_tickL,
x_axis_label=x_axis_label,
col_tickL=col_tickL,
y_axis_label=y_axis_label,
colorD={
'Goal': 'g',
'Pit': 'r',
'Start': 'b'
},
basic_color='skyblue')
gridworld.set_info('Sample Grid World showing basic MDP creation.')
# add actions from each state
# (note: a_prob will be normalized within add_action_dict)
gridworld.add_action_dict(actionD)
# for each action, define the next state and transition probability
# (here we use the layout definition to aid the logic)
for s_hash, aL in actionD.items():
for a_desc in aL:
sn_hash = get_next_state(s_hash, a_desc)
reward = rewardD.get(sn_hash, 0.0)
# for deterministic MDP, use t_prob=1.0
gridworld.add_transition(s_hash,
a_desc,
sn_hash,
t_prob=1.0,
reward_obj=reward)
# after the "add" commands, send all states and actions to environment
# (any required normalization is done here as well.)
gridworld.define_env_states_actions()
# If there is a start state, define it here.
gridworld.start_state_hash = 'Start'
# If a limited number of start states are desired, define them here.
gridworld.define_limited_start_state_list([(2, 0), (2, 2)])
# if a default policy is desired, define it as a dict.
gridworld.default_policyD = {
(0, 0): 'R',
(1, 0): 'U',
(0, 1): 'R',
(0, 2): 'R',
(1, 2): 'U',
'Start': 'U',
(2, 2): 'U',
(2, 1): 'R',
(2, 3): 'L'
}
return gridworld
def get_gambler(prob_heads=0.4):
gambler = EnvBaseline( name='Gamblers Coin Flip Problem',
s_hash_rowL=s_hash_rowL,
colorD={100:'g', 0:'r'},
basic_color='skyblue' )
gambler.set_info( 'Example 4.3 from Sutton & Barto 2nd Edition page 84.' )
for s in range(1, 100): # 1 to 99
s_max = min(s, 100-s)
for a_desc in range(1, s_max + 1):
gambler.add_action( s, a_desc, a_prob=1.0 )
# define reward for all states
def get_reward( sn ):
if sn==100:
return 1.0
else:
return 0.0
# define all possible transitions.
for s in range(1, 100): # 1 to 99
s_max = min(s, 100-s)
for a_desc in range(1, s_max + 1):
sn_hash = s - a_desc
rval = get_reward( sn_hash )
gambler.add_transition( s, a_desc, sn_hash, t_prob=1.0-prob_heads, reward_obj=rval)
sn_hash = s + a_desc
rval = get_reward( sn_hash )
gambler.add_transition( s, a_desc, sn_hash, t_prob=prob_heads, reward_obj=rval)
gambler.define_env_states_actions() # send all states and actions to environment
# If there is a start state, define it here.
gambler.start_state_hash = (50)
# define default policy (if any)
gambler.default_policyD = {}
return gambler
def get_gridworld(step_reward=-0.04):
gridworld = EnvBaseline(name='Sutton Ex4.1 5x5 Grid World',
s_hash_rowL=s_hash_rowL)
gridworld.set_info("""
Sutton 5x5 Gridworld
Book Answer from page 65 (linear eqn solve) for gamma=0.9
22.0 24.4 22.0 19.4 17.5
19.8 22.0 19.8 17.8 16.0
17.8 19.8 17.8 16.0 14.4
16.0 17.8 16.0 14.4 13.0
14.4 16.0 14.4 13.0 11.7
================================================= """)
def get_action_snext_reward(s_hash, action):
"""returns reward and state_next_hash"""
di = 0
dj = 0
reward = 0
if action == 'N':
di = 1
elif action == 'S':
di = -1
elif action == 'E':
dj = 1
elif action == 'W':
dj = -1
(i, j) = s_hash
i_next = i + di
j_next = j + dj
if (i == 4) and (j == 1):
i_next = 0
j_next = 1
reward = 10
elif (i == 4) and (j == 3):
i_next = 2
j_next = 3
reward = 5
elif (i_next < 0) or (i_next > 4) or (j_next < 0) or (j_next > 4):
i_next = i
j_next = j
reward = -1
state_next_hash = (i_next, j_next)
return reward, state_next_hash
# define default policy
gridworld.default_policyD = {
} #index=s_hash, value=list of equiprobable actions
for i in range(5):
for j in range(5):
s_hash = (i, j)
gridworld.default_policyD[s_hash] = ('N', 'S', 'E', 'W')
for a_desc in ['N', 'S', 'E', 'W']:
gridworld.add_action(s_hash, a_desc,
a_prob=1.0) # a_prob will be normalized
reward_val, sn_hash = get_action_snext_reward(s_hash, a_desc)
# add each event to transitions object
gridworld.add_transition(s_hash,
a_desc,
sn_hash,
t_prob=1.0,
reward_obj=reward_val)
gridworld.define_env_states_actions(
) # send all states and actions to environment
gridworld.start_state_hash = (0, 0)
return gridworld
def get_gridworld(step_reward=-1, height=7, goal=(3,7),
windT=(0,0,0,1,1,1,2,2,1,0)):
"""
Windy Gridworld with (0,0) at lower left
width is defined by length of windT tuple.
"""
gridworld = EnvBaseline( name='Windy Kings Gridworld' ) # GenericLayout set below
gridworld.set_info( """""" )
width = len( windT )
def get_action_snext( s_hash, action):
"""returns state_next_hash"""
di = 0
dj = 0
if 'N' in action:
di = 1
elif 'S' in action:
di = -1
if 'E' in action:
dj = 1
elif 'W' in action:
dj = -1
(i,j) = s_hash
wind_di = windT[ j ]
i_next = i + di
# constrain basic move to be inside the grid
i_next = max(0, min(height-1, i_next))
i_next += wind_di # add wind to constrained move.
j_next = j + dj
# constrain next position to be inside the grid
i_next = max(0, min(height-1, i_next))
j_next = max(0, min(width-1, j_next))
state_next_hash = (i_next, j_next)
if state_next_hash == goal:
state_next_hash = 'Goal'
return state_next_hash
# define default policy
gridworld.default_policyD = {} #index=s_hash, value=list of equiprobable actions
for i in range(height):
for j in range(width):
s_hash = (i,j)
if s_hash == goal:
pass # s_hash == 'Goal'
else:
gridworld.default_policyD[ s_hash ] = ('N','S','E','W', 'NE','SE','SW','NW')
for a_desc in ['N','S','E','W', 'NE','SE','SW','NW']:
gridworld.add_action( s_hash, a_desc, a_prob=1.0 ) # a_prob will be normalized
sn_hash = get_action_snext( s_hash, a_desc )
# add each event to transitions object
gridworld.add_transition( s_hash, a_desc, sn_hash, t_prob=1.0, reward_obj=step_reward)
gridworld.define_env_states_actions() # send all states and actions to environment
# --------------------
s_hash_rowL = [] # layout rows for makeing 2D output
for i in range(height): # put (0,0) at upper left
rowL = []
for j in range(width):
s_hash = (i,j)
if s_hash == goal:
s_hash = 'Goal'
rowL.append( s_hash )
# use insert to put (0,0) at lower left, append for upper left
s_hash_rowL.insert(0, rowL )# layout rows for makeing 2D output
gridworld.layout = GenericLayout( gridworld, s_hash_rowL=s_hash_rowL,
col_tickL=windT,
x_axis_label='Upward Wind Speed' )
gridworld.start_state_hash = (3,0)
return gridworld
def get_gridworld(step_reward=0.0):
gridworld = EnvBaseline(
name='Simple Grid World') # GenericLayout set below
gridworld.set_info('Simple Grid World Example.')
actionD = {
(0, 0): ('D', 'R'),
(0, 1): ('L', 'R'),
(0, 2): ('L', 'D', 'R'),
(1, 0): ('U', 'D'),
(1, 2): ('U', 'D', 'R'),
(2, 0): ('U', 'R'),
(2, 1): ('L', 'R'),
(2, 2): ('L', 'R', 'U'),
(2, 3): ('L', 'U')
}
rewardD = {(0, 3): 1, (1, 3): -1}
for state_hash, actionL in actionD.items():
for action_desc in actionL:
gridworld.add_action(state_hash, action_desc,
a_prob=1.0) # a_prob will be normalized
a = action_desc
s = state_hash
if a == 'U':
state_next_hash = (s[0] - 1, s[1])
elif a == 'D':
state_next_hash = (s[0] + 1, s[1])
elif a == 'R':
state_next_hash = (s[0], s[1] + 1)
elif a == 'L':
state_next_hash = (s[0], s[1] - 1)
reward_val = rewardD.get(state_next_hash, step_reward)
gridworld.add_transition(state_hash,
action_desc,
state_next_hash,
t_prob=1.0,
reward_obj=reward_val)
gridworld.define_env_states_actions(
) # send all states and actions to environment
gridworld.layout = GenericLayout(
gridworld) # uses default "get_layout_row_col_of_state"
# If there is a start state, define it here.
gridworld.start_state_hash = (2, 0)
gridworld.define_limited_start_state_list([(2, 0), (2, 2)])
# define default policy (if any)
# Policy Dictionary for: GridWorld
policyD = {} # index=state_hash, value=action_desc
# Vpi shown for gamma=0.9
policyD[(0, 0)] = 'R' # Vpi=0.81
policyD[(1, 0)] = 'U' # Vpi=0.729
policyD[(0, 1)] = 'R' # Vpi=0.9
policyD[(0, 2)] = 'R' # Vpi=1.0
policyD[(1, 2)] = 'U' # Vpi=0.9
policyD[(2, 0)] = 'U' # Vpi=0.6561
policyD[(2, 2)] = 'U' # Vpi=0.81
policyD[(2, 1)] = 'R' # Vpi=0.729
policyD[(2, 3)] = 'L' # Vpi=0.729
gridworld.default_policyD = policyD
return gridworld
def get_robot(step_reward=-0.04):
gridworld = EnvBaseline(name='Slipper Cleaning Robot',
s_hash_rowL=s_hash_rowL)
gridworld.set_info("""
Example taken from "Dissecting Reinforcement Learning-Part 1"
Dec 9, 2016 Massimiliano Patacchiola
https://mpatacchiola.github.io/blog/2016/12/09/dissecting-reinforcement-learning.html
""")
def get_right_angle_list(a):
if a == 'U':
raL = ['L', 'R']
elif a == 'D':
raL = ['L', 'R']
elif a == 'R':
raL = ['U', 'D']
elif a == 'L':
raL = ['U', 'D']
return raL
def get_move_s_next(a, s):
sn = s
if a == 'U':
sn = (s[0] + 1, s[1])
elif a == 'D':
sn = (s[0] - 1, s[1])
elif a == 'R':
sn = (s[0], s[1] + 1)
elif a == 'L':
sn = (s[0], s[1] - 1)
if sn == (2, 2): # can't move into block in the middle.
sn = s
# limit moves to inside the edges.
sn_hash = (clamp(sn[0], 1, 3), clamp(sn[1], 1, 4))
return sn_hash
non_termL = [(3, 1), (3, 2), (3, 3), (2, 1), (2, 3), (1, 1), (1, 2),
(1, 3), (1, 4)]
rewardD = {(3, 4): 1, (2, 4): -1}
# put in 80% and both 10% moves to target
for s_hash in non_termL:
for a_desc in ['U', 'D', 'L', 'R']: # normal move
gridworld.add_action(s_hash, a_desc, a_prob=0.25)
# 80%
sn_hash = get_move_s_next(a_desc, s_hash)
reward_val = rewardD.get(sn_hash, step_reward)
gridworld.add_transition(s_hash,
a_desc,
sn_hash,
t_prob=0.8,
reward_obj=reward_val)
# both 10%
right_angL = get_right_angle_list(a_desc)
for ar_desc in right_angL:
sn_hash = get_move_s_next(ar_desc, s_hash)
reward_val = rewardD.get(sn_hash, step_reward)
gridworld.add_transition(s_hash,
a_desc,
sn_hash,
t_prob=0.1,
reward_obj=reward_val)
gridworld.define_env_states_actions()
# If there is a start state, define it here.
gridworld.start_state_hash = (1, 1)
# define default policy (if any)
policyD = {} # index=s_hash, value=a_desc
policyD[(3, 1)] = 'R'
policyD[(3, 3)] = 'R'
policyD[(3, 2)] = 'R'
policyD[(2, 1)] = 'U'
policyD[(2, 3)] = 'U'
policyD[(1, 1)] = 'U'
policyD[(1, 2)] = 'L'
policyD[(1, 3)] = 'L'
policyD[(1, 4)] = 'L'
gridworld.default_policyD = policyD
return gridworld
def get_gridworld(step_reward=0.0,
width=9,
height=6,
goal=(0, 8),
start=(2, 0),
wallL=((1, 2), (2, 2), (3, 2), (0, 7), (1, 7), (2, 7), (4,
5))):
gridworld = EnvBaseline(
name='Sutton Ex8.1 Dyna Maze') # GenericLayout set below
gridworld.set_info("""Sutton Ex8.1 Dyna Maze""")
def get_action_snext_reward(s_hash, action):
"""returns reward and state_next_hash"""
di = 0
dj = 0
reward = 0
if action == 'U':
di = -1
elif action == 'D':
di = 1
elif action == 'R':
dj = 1
elif action == 'L':
dj = -1
(i, j) = s_hash
i_next = i + di
j_next = j + dj
if j_next >= width:
j_next = j
elif j_next < 0:
j_next = j
if i_next >= height:
i_next = i
elif i_next < 0:
i_next = i
if (i_next, j_next) in wallL:
i_next, j_next = i, j
state_next_hash = (i_next, j_next)
if state_next_hash == goal:
reward = 1.0
else:
reward = 0.0
return reward, state_next_hash
# define default policy
gridworld.default_policyD = {
} #index=s_hash, value=list of equiprobable actions
for i in range(height):
for j in range(width):
s_hash = (i, j)
if s_hash != goal:
gridworld.default_policyD[s_hash] = ('U', 'D', 'R', 'L')
for a_desc in ['U', 'D', 'R', 'L']:
gridworld.add_action(
s_hash, a_desc,
a_prob=1.0) # a_prob will be normalized
reward_val, sn_hash = get_action_snext_reward(
s_hash, a_desc)
# add each event to transitions object
gridworld.add_transition(s_hash,
a_desc,
sn_hash,
t_prob=1.0,
reward_obj=reward_val)
gridworld.define_env_states_actions(
) # send all states and actions to environment
# --------------------
s_hash_rowL = [] # layout rows for makeing 2D output
for i in range(height): # put (0,0) at upper left
rowL = []
for j in range(width):
s = (i, j)
if s in wallL:
rowL.append('"Wall"')
else:
rowL.append(s)
# use insert to put (0,0) at lower left
s_hash_rowL.append(rowL) # layout rows for makeing 2D output
named_s_hashD = {start: 'Start', goal: 'Goal'}
gridworld.layout = GenericLayout(gridworld,
s_hash_rowL=s_hash_rowL,
named_s_hashD=named_s_hashD)
gridworld.start_state_hash = start
return gridworld
def get_robot():
robot = EnvBaseline(name='Slow-Fast Fallen Robot', s_hash_rowL=s_hash_rowL)
robot.set_info("""
Sample 3 State Fallen, Standing, Moving Robot.
https://sandipanweb.wordpress.com/2017/03/23/some-reinforcement-learning-using-policy-value-iteration-and-q-learning-for-a-markov-decision-process-in-python-and-r/
Some Reinforcement Learning: Using Policy & Value Iteration and Q-learning for a Markov Decision Process in Python and R
""")
robot.add_action('Fallen', 'Slow', a_prob=1.0)
robot.add_action('Standing', 'Slow', a_prob=1.0)
robot.add_action('Moving', 'Slow', a_prob=1.0)
robot.add_action('Standing', 'Fast', a_prob=1.0)
robot.add_action('Moving', 'Fast', a_prob=1.0)
robot.add_transition('Fallen',
'Slow',
'Fallen',
t_prob=0.6,
reward_obj=-1.0)
robot.add_transition('Fallen',
'Slow',
'Standing',
t_prob=0.4,
reward_obj=1.0)
robot.add_transition('Standing',
'Slow',
'Moving',
t_prob=1.0,
reward_obj=1.0)
robot.add_transition('Moving',
'Slow',
'Moving',
t_prob=1.0,
reward_obj=1.0)
robot.add_transition('Standing',
'Fast',
'Moving',
t_prob=0.6,
reward_obj=2.0)
robot.add_transition('Standing',
'Fast',
'Fallen',
t_prob=0.4,
reward_obj=-1.0)
robot.add_transition('Moving',
'Fast',
'Moving',
t_prob=0.8,
reward_obj=2.0)
robot.add_transition('Moving',
'Fast',
'Fallen',
t_prob=0.2,
reward_obj=-1.0)
robot.define_env_states_actions(
) # send all states and actions to environment
robot.start_state_hash = 'Standing'
# define default policy (if any)
policyD = {} # index=state_hash, value=action_desc
policyD['Standing'] = 'Slow'
policyD['Fallen'] = 'Slow'
policyD['Moving'] = 'Slow'
robot.default_policyD = policyD
return robot
def get_gridworld(
step_reward=0.0,
N_mult=1, # N_mult must be an integer.
width=9,
height=6,
goal=(0, 8),
start=(2, 0),
wallL=((1, 2), (2, 2), (3, 2), (0, 7), (1, 7), (2, 7), (4, 5))):
gridworld = EnvBaseline(
name='Sutton Ex8.4 Priority Sweep Maze') # GenericLayout set below
gridworld.set_info("""Sutton Ex8.1 Dyna Maze""")
width_big = width * N_mult
height_big = height * N_mult
gridworld.characteristic_dim = width_big + height_big * 2
# get relaxed optimal length from Zhang.
gridworld.optimal_path_len = int(14 * N_mult * 1.2) + 1
def get_action_snext_reward(s_hash, action):
"""returns reward and state_next_hash"""
di = 0
dj = 0
reward = 0
if action == 'U':
di = -1
elif action == 'D':
di = 1
elif action == 'R':
dj = 1
elif action == 'L':
dj = -1
(i, j) = s_hash
i_next = i + di
j_next = j + dj
if j_next >= width_big:
j_next = j
elif j_next < 0:
j_next = j
if i_next >= height_big:
i_next = i
elif i_next < 0:
i_next = i
if (i_next, j_next) in wall_set:
i_next, j_next = i, j
state_next_hash = (i_next, j_next)
if state_next_hash in goal_set:
reward = 1.0
else:
reward = 0.0
return reward, state_next_hash
def make_big_set(pos):
"""Take an (i,j) position, pos, and expand to new, big size in x and y"""
pos_set = set()
ip, jp = pos
ip *= N_mult
jp *= N_mult
for ixn in range(N_mult):
for jxn in range(N_mult):
pos_set.add((ip + ixn, jp + jxn))
return pos_set
# define default policy
gridworld.default_policyD = {
} #index=s_hash, value=list of equiprobable actions
# redefine start
istart, jstart = start
start = (istart * N_mult, jstart * N_mult)
# make goal set
goal_set = make_big_set(goal)
# make wall set
wall_set = set()
for wall in wallL:
wall_set.update(make_big_set(wall))
# create state hash entries
for i in range(height_big):
for j in range(width_big):
s_hash = (i, j)
if (s_hash not in wall_set) and (s_hash not in goal_set):
gridworld.default_policyD[s_hash] = ('U', 'D', 'R', 'L')
for a_desc in ['U', 'D', 'R', 'L']:
gridworld.add_action(
s_hash, a_desc,
a_prob=1.0) # a_prob will be normalized
reward_val, sn_hash = get_action_snext_reward(
s_hash, a_desc)
# add each event to transitions object
gridworld.add_transition(s_hash,
a_desc,
sn_hash,
t_prob=1.0,
reward_obj=reward_val)
gridworld.define_env_states_actions(
) # send all states and actions to environment
# --------------------
s_hash_rowL = [] # layout rows for makeing 2D output
for i in range(height_big): # put (0,0) at upper left
rowL = []
for j in range(width_big):
s = (i, j)
if s in wall_set:
rowL.append('"Wall"')
else:
rowL.append(s)
# use insert to put (0,0) at lower left
s_hash_rowL.append(rowL) # layout rows for makeing 2D output
named_s_hashD = {}
named_s_hashD[start] = 'Start'
for g in goal_set:
named_s_hashD[g] = 'Goal'
gridworld.layout = GenericLayout(gridworld,
s_hash_rowL=s_hash_rowL,
named_s_hashD=named_s_hashD)
gridworld.start_state_hash = start
return gridworld
def get_gridworld(step_reward=-0.04):
gridworld = EnvBaseline(name='Sutton Ex4.1 Grid World',
s_hash_rowL=s_hash_rowL)
gridworld.set_info("""
Example 4.1 grid
Label for blank space is "0" (both blanks are the same actual state)
(i.e. upper left corner and lower right corner are state "0")
""")
for state_hash in range(1, 15): # states are numbered 1-14
for action_desc in ['U', 'D', 'R', 'L']:
gridworld.add_action(state_hash, action_desc,
a_prob=1.0) # a_prob will be normalized
a = action_desc
s = state_hash
if a == 'U':
sn = s - 4
elif a == 'D':
sn = s + 4
elif a == 'R':
if s not in [3, 7, 11]:
sn = s + 1
else:
sn = s
elif a == 'L':
if s not in [4, 8, 12]:
sn = s - 1
else:
sn = s
if sn < 0:
sn = s
elif sn > 15:
sn = s
elif sn == 15:
sn = 0
gridworld.add_transition(state_hash,
action_desc,
sn,
t_prob=1.0,
reward_obj=-1.0)
gridworld.define_env_states_actions(
) # send all states and actions to environment
gridworld.start_state_hash = 12
# define default policy (if any)
policyD = {} # index=state_hash, value=action_desc
for (s_hash, a_desc) in gridworld.iter_state_hash_action_desc():
if s_hash not in policyD:
policyD[s_hash] = []
policyD[s_hash].append((a_desc, 0.25))
# make policyD entries hashable for later use (i.e. tuple, not list)
for s_hash, aL in policyD.items():
policyD[s_hash] = tuple(aL)
gridworld.default_policyD = policyD
return gridworld
def get_env():
env = EnvBaseline( name="Jacks Car Rental (const rtn)" ) # GenericLayout set below
simplified_str ="""Shangtong Zhang's simplified model such that
the # of cars returned in daytime becomes constant
rather than a random value from poisson distribution, which will reduce calculation time
and leave the optimal policy/value state matrix almost the same"""
env.set_info( 'Example 4.2 from Sutton & Barto 2nd Edition page 81.\n' + simplified_str )
# define all possible actions.
saL = [] # a list of (s1, s2, adesc)
s_hash_rowL = [] # layout rows for makeing 2D output
for s1 in range( MAX_CARS + 1 ): # 20 cars max
rowL = [] # row of s_hash_rowL
for s2 in range( MAX_CARS + 1 ): # 20 cars max
s_hash = (s1, s2)
rowL.append( s_hash )
for a_desc in range(-5, 6): # -5 moves 5 cars from 2nd to 1st. +5 from 1st to 2nd.
if a_desc < 0: # can only move cars if they are present
if (abs(a_desc) <= s2):
env.add_action( s_hash, a_desc, a_prob=1.0 )
saL.append( (s1, s2, a_desc) )
else:
if (a_desc <= s1): # can only move cars if they are present
env.add_action( s_hash, a_desc, a_prob=1.0 )
saL.append( (s1, s2, a_desc) )
# use insert to put (0,0) at lower left
s_hash_rowL.insert(0, rowL )# layout rows for makeing 2D output
# ------------------------------
# figure out transition probabilities and rewards
for s1 in range( MAX_CARS + 1 ):
for s2 in range( MAX_CARS + 1 ):
for a_desc in range( -5, 6 ):
get_prob_reward( s1, s2, a_desc)
# ------------------------------
print('\nStarting to define car rental transitions')
# with all the probability figured out, define all transitions
for (s1, s2, a_desc, sn_hash), t_prob in total_probD.items():
txr = sum_prob_x_rewardD[ (s1, s2, a_desc, sn_hash) ]
rval = txr / t_prob
env.add_transition( (s1,s2), a_desc, sn_hash, t_prob=t_prob, reward_obj=rval)
#if s1==10 and s2==10:
# print('for (10,10) a_desc=',a_desc,' sn_hash=',sn_hash,' t_prob=',t_prob,' rval=',rval)
print('Calling: env.define_env_states_actions')
env.define_env_states_actions() # send all states and actions to environment
print('Environment Ready.')
# If there is a start state, define it here.
env.start_state_hash = (10,10)
# define default policy (if any)
env.default_policyD = {}
# --------------------
# define layout for output
env.layout = GenericLayout( env, s_hash_rowL=s_hash_rowL,
x_axis_label='#Cars at Second Location',
y_axis_label='#Cars at First Location')
return env