# Python Pulp - LpMinimize Not Returning Expected Output - Scheduling Problem

I'm working on my first pulp model and something isn't going quite right. It looks like it's close, but for some reason it is not respecting one of my constraints...

It's a scheduling problem and I am trying to minimize the number of labor hours used whilst meeting the workforce requirements for each time window. The model returns an "optimal" result, but the staffing figures are far short of the workforce requirements for each window.

There are 120 potential shifts to choose from ("n"), and 84 time windows (t). Each time window has its own workforce requirement (number of staff, "d"). Not all shifts will be filled.

"d" is the number of workers required in each of the 84 time windows, it is as follows:

I want to minimize the number of labor hours required in total ("w") whilst meeting the workforce requirements for each time window. "w" looks like this, in hours per shift - one entry for each of the 120 potential shifts:

The final solution does not satisfy the workforce requirements for the various time windows, it's substantially short. Only 5 shifts are filled in total:

Status: Optimal
The number of workers needed for shift 0 is 1 workers
The number of workers needed for shift 90 is 2 workers
The number of workers needed for shift 91 is 1 workers
The number of workers needed for shift 92 is 1 workers



For reference, just to show that the needed amount of labor isn't in the solution, these are the full schedules for each of the 120 shifts (asterisks on the shifts that the solver filled (1, 89 (2 employees), 90, and 91):

Shift_Num Start_Shift First_Break_Start First_Break_End Lunch_Start Lunch_End Second_Break_Start Second_Break_End End_Shift
***1 5:00:00 AM 6:40:00 AM 6:50:00 AM 9:00:00 AM 10:00:00 AM 11:40:00 AM 11:50:00 AM 2:00:00 PM
2 5:00:00 AM 6:50:00 AM 7:00:00 AM 9:00:00 AM 10:00:00 AM 11:50:00 AM 12:00:00 PM 2:00:00 PM
3 5:00:00 AM 7:00:00 AM 7:10:00 AM 9:00:00 AM 10:00:00 AM 12:00:00 PM 12:10:00 PM 2:00:00 PM
4 5:00:00 AM 7:10:00 AM 7:20:00 AM 9:00:00 AM 10:00:00 AM 12:10:00 PM 12:20:00 PM 2:00:00 PM
5 5:00:00 AM 7:20:00 AM 7:30:00 AM 9:00:00 AM 10:00:00 AM 12:20:00 PM 12:30:00 PM 2:00:00 PM
6 5:00:00 AM 6:40:00 AM 6:50:00 AM 9:15:00 AM 10:15:00 AM 11:55:00 AM 12:05:00 PM 2:00:00 PM
7 5:00:00 AM 6:50:00 AM 7:00:00 AM 9:15:00 AM 10:15:00 AM 12:05:00 PM 12:15:00 PM 2:00:00 PM
8 5:00:00 AM 7:00:00 AM 7:10:00 AM 9:15:00 AM 10:15:00 AM 12:15:00 PM 12:25:00 PM 2:00:00 PM
9 5:00:00 AM 7:10:00 AM 7:20:00 AM 9:15:00 AM 10:15:00 AM 12:25:00 PM 12:35:00 PM 2:00:00 PM
10 5:00:00 AM 7:20:00 AM 7:30:00 AM 9:15:00 AM 10:15:00 AM 12:35:00 PM 12:45:00 PM 2:00:00 PM
11 5:00:00 AM 6:40:00 AM 6:50:00 AM 9:30:00 AM 10:30:00 AM 12:00:00 PM 12:10:00 PM 2:00:00 PM
12 5:00:00 AM 6:50:00 AM 7:00:00 AM 9:30:00 AM 10:30:00 AM 12:10:00 PM 12:20:00 PM 2:00:00 PM
13 5:00:00 AM 7:00:00 AM 7:10:00 AM 9:30:00 AM 10:30:00 AM 12:20:00 PM 12:30:00 PM 2:00:00 PM
14 5:00:00 AM 7:10:00 AM 7:20:00 AM 9:30:00 AM 10:30:00 AM 12:30:00 PM 12:40:00 PM 2:00:00 PM
15 5:00:00 AM 7:20:00 AM 7:30:00 AM 9:30:00 AM 10:30:00 AM 12:30:00 PM 12:40:00 PM 2:00:00 PM
16 6:00:00 AM 7:40:00 AM 7:50:00 AM 10:00:00 AM 11:00:00 AM 12:40:00 PM 12:50:00 PM 3:00:00 PM
17 6:00:00 AM 7:50:00 AM 8:00:00 AM 10:00:00 AM 11:00:00 AM 12:50:00 PM 1:00:00 PM 3:00:00 PM
18 6:00:00 AM 8:00:00 AM 8:10:00 AM 10:00:00 AM 11:00:00 AM 1:00:00 PM 1:10:00 PM 3:00:00 PM
19 6:00:00 AM 8:10:00 AM 8:20:00 AM 10:00:00 AM 11:00:00 AM 1:10:00 PM 1:20:00 PM 3:00:00 PM
20 6:00:00 AM 8:20:00 AM 8:30:00 AM 10:00:00 AM 11:00:00 AM 1:20:00 PM 1:30:00 PM 3:00:00 PM
21 6:00:00 AM 7:40:00 AM 7:50:00 AM 10:15:00 AM 11:15:00 AM 12:55:00 PM 1:05:00 PM 3:00:00 PM
22 6:00:00 AM 7:50:00 AM 8:00:00 AM 10:15:00 AM 11:15:00 AM 1:05:00 PM 1:15:00 PM 3:00:00 PM
23 6:00:00 AM 8:00:00 AM 8:10:00 AM 10:15:00 AM 11:15:00 AM 1:15:00 PM 1:25:00 PM 3:00:00 PM
24 6:00:00 AM 8:10:00 AM 8:20:00 AM 10:15:00 AM 11:15:00 AM 1:25:00 PM 1:35:00 PM 3:00:00 PM
25 6:00:00 AM 8:20:00 AM 8:30:00 AM 10:15:00 AM 11:15:00 AM 1:35:00 PM 1:45:00 PM 3:00:00 PM
26 6:00:00 AM 7:40:00 AM 7:50:00 AM 10:30:00 AM 11:30:00 AM 1:00:00 PM 1:10:00 PM 3:00:00 PM
27 6:00:00 AM 7:50:00 AM 8:00:00 AM 10:30:00 AM 11:30:00 AM 1:10:00 PM 1:20:00 PM 3:00:00 PM
28 6:00:00 AM 8:00:00 AM 8:10:00 AM 10:30:00 AM 11:30:00 AM 1:20:00 PM 1:30:00 PM 3:00:00 PM
29 6:00:00 AM 8:10:00 AM 8:20:00 AM 10:30:00 AM 11:30:00 AM 1:30:00 PM 1:40:00 PM 3:00:00 PM
30 6:00:00 AM 8:20:00 AM 8:30:00 AM 10:30:00 AM 11:30:00 AM 1:30:00 PM 1:40:00 PM 3:00:00 PM
31 7:00:00 AM 8:40:00 AM 8:50:00 AM 11:00:00 AM 12:00:00 PM 1:40:00 PM 1:50:00 PM 4:00:00 PM
32 7:00:00 AM 8:50:00 AM 9:00:00 AM 11:00:00 AM 12:00:00 PM 1:50:00 PM 2:00:00 PM 4:00:00 PM
33 7:00:00 AM 9:00:00 AM 9:10:00 AM 11:00:00 AM 12:00:00 PM 2:00:00 PM 2:10:00 PM 4:00:00 PM
34 7:00:00 AM 9:10:00 AM 9:20:00 AM 11:00:00 AM 12:00:00 PM 2:10:00 PM 2:20:00 PM 4:00:00 PM
35 7:00:00 AM 9:20:00 AM 9:30:00 AM 11:00:00 AM 12:00:00 PM 2:20:00 PM 2:30:00 PM 4:00:00 PM
36 7:00:00 AM 8:40:00 AM 8:50:00 AM 11:15:00 AM 12:15:00 PM 1:55:00 PM 2:05:00 PM 4:00:00 PM
37 7:00:00 AM 8:50:00 AM 9:00:00 AM 11:15:00 AM 12:15:00 PM 2:05:00 PM 2:15:00 PM 4:00:00 PM
38 7:00:00 AM 9:00:00 AM 9:10:00 AM 11:15:00 AM 12:15:00 PM 2:15:00 PM 2:25:00 PM 4:00:00 PM
39 7:00:00 AM 9:10:00 AM 9:20:00 AM 11:15:00 AM 12:15:00 PM 2:25:00 PM 2:35:00 PM 4:00:00 PM
40 7:00:00 AM 9:20:00 AM 9:30:00 AM 11:15:00 AM 12:15:00 PM 2:35:00 PM 2:45:00 PM 4:00:00 PM
41 7:00:00 AM 8:40:00 AM 8:50:00 AM 11:30:00 AM 12:30:00 PM 2:00:00 PM 2:10:00 PM 4:00:00 PM
42 7:00:00 AM 8:50:00 AM 9:00:00 AM 11:30:00 AM 12:30:00 PM 2:10:00 PM 2:20:00 PM 4:00:00 PM
43 7:00:00 AM 9:00:00 AM 9:10:00 AM 11:30:00 AM 12:30:00 PM 2:20:00 PM 2:30:00 PM 4:00:00 PM
44 7:00:00 AM 9:10:00 AM 9:20:00 AM 11:30:00 AM 12:30:00 PM 2:30:00 PM 2:40:00 PM 4:00:00 PM
45 7:00:00 AM 9:20:00 AM 9:30:00 AM 11:30:00 AM 12:30:00 PM 2:30:00 PM 2:40:00 PM 4:00:00 PM
46 8:00:00 AM 9:40:00 AM 9:50:00 AM 12:00:00 PM 1:00:00 PM 2:40:00 PM 2:50:00 PM 5:00:00 PM
47 8:00:00 AM 9:50:00 AM 10:00:00 AM 12:00:00 PM 1:00:00 PM 2:50:00 PM 3:00:00 PM 5:00:00 PM
48 8:00:00 AM 10:00:00 AM 10:10:00 AM 12:00:00 PM 1:00:00 PM 3:00:00 PM 3:10:00 PM 5:00:00 PM
49 8:00:00 AM 10:10:00 AM 10:20:00 AM 12:00:00 PM 1:00:00 PM 3:10:00 PM 3:20:00 PM 5:00:00 PM
50 8:00:00 AM 10:20:00 AM 10:30:00 AM 12:00:00 PM 1:00:00 PM 3:20:00 PM 3:30:00 PM 5:00:00 PM
51 8:00:00 AM 9:40:00 AM 9:50:00 AM 12:15:00 PM 1:15:00 PM 2:55:00 PM 3:05:00 PM 5:00:00 PM
52 8:00:00 AM 9:50:00 AM 10:00:00 AM 12:15:00 PM 1:15:00 PM 3:05:00 PM 3:15:00 PM 5:00:00 PM
53 8:00:00 AM 10:00:00 AM 10:10:00 AM 12:15:00 PM 1:15:00 PM 3:15:00 PM 3:25:00 PM 5:00:00 PM
54 8:00:00 AM 10:10:00 AM 10:20:00 AM 12:15:00 PM 1:15:00 PM 3:25:00 PM 3:35:00 PM 5:00:00 PM
55 8:00:00 AM 10:20:00 AM 10:30:00 AM 12:15:00 PM 1:15:00 PM 3:35:00 PM 3:45:00 PM 5:00:00 PM
56 8:00:00 AM 9:40:00 AM 9:50:00 AM 12:30:00 PM 1:30:00 PM 3:00:00 PM 3:10:00 PM 5:00:00 PM
57 8:00:00 AM 9:50:00 AM 10:00:00 AM 12:30:00 PM 1:30:00 PM 3:10:00 PM 3:20:00 PM 5:00:00 PM
58 8:00:00 AM 10:00:00 AM 10:10:00 AM 12:30:00 PM 1:30:00 PM 3:20:00 PM 3:30:00 PM 5:00:00 PM
59 8:00:00 AM 10:10:00 AM 10:20:00 AM 12:30:00 PM 1:30:00 PM 3:30:00 PM 3:40:00 PM 5:00:00 PM
60 8:00:00 AM 10:20:00 AM 10:30:00 AM 12:30:00 PM 1:30:00 PM 3:30:00 PM 3:40:00 PM 5:00:00 PM
61 9:00:00 AM 10:40:00 AM 10:50:00 AM 1:00:00 PM 2:00:00 PM 3:40:00 PM 3:50:00 PM 6:00:00 PM
62 9:00:00 AM 10:50:00 AM 11:00:00 AM 1:00:00 PM 2:00:00 PM 3:50:00 PM 4:00:00 PM 6:00:00 PM
63 9:00:00 AM 11:00:00 AM 11:10:00 AM 1:00:00 PM 2:00:00 PM 4:00:00 PM 4:10:00 PM 6:00:00 PM
64 9:00:00 AM 11:10:00 AM 11:20:00 AM 1:00:00 PM 2:00:00 PM 4:10:00 PM 4:20:00 PM 6:00:00 PM
65 9:00:00 AM 11:20:00 AM 11:30:00 AM 1:00:00 PM 2:00:00 PM 4:20:00 PM 4:30:00 PM 6:00:00 PM
66 9:00:00 AM 10:40:00 AM 10:50:00 AM 1:15:00 PM 2:15:00 PM 3:55:00 PM 4:05:00 PM 6:00:00 PM
67 9:00:00 AM 10:50:00 AM 11:00:00 AM 1:15:00 PM 2:15:00 PM 4:05:00 PM 4:15:00 PM 6:00:00 PM
68 9:00:00 AM 11:00:00 AM 11:10:00 AM 1:15:00 PM 2:15:00 PM 4:15:00 PM 4:25:00 PM 6:00:00 PM
69 9:00:00 AM 11:10:00 AM 11:20:00 AM 1:15:00 PM 2:15:00 PM 4:25:00 PM 4:35:00 PM 6:00:00 PM
70 9:00:00 AM 11:20:00 AM 11:30:00 AM 1:15:00 PM 2:15:00 PM 4:35:00 PM 4:45:00 PM 6:00:00 PM
71 9:00:00 AM 10:40:00 AM 10:50:00 AM 1:30:00 PM 2:30:00 PM 4:00:00 PM 4:10:00 PM 6:00:00 PM
72 9:00:00 AM 10:50:00 AM 11:00:00 AM 1:30:00 PM 2:30:00 PM 4:10:00 PM 4:20:00 PM 6:00:00 PM
73 9:00:00 AM 11:00:00 AM 11:10:00 AM 1:30:00 PM 2:30:00 PM 4:20:00 PM 4:30:00 PM 6:00:00 PM
74 9:00:00 AM 11:10:00 AM 11:20:00 AM 1:30:00 PM 2:30:00 PM 4:30:00 PM 4:40:00 PM 6:00:00 PM
75 9:00:00 AM 11:20:00 AM 11:30:00 AM 1:30:00 PM 2:30:00 PM 4:30:00 PM 4:40:00 PM 6:00:00 PM
76 10:00:00 AM 11:40:00 AM 11:50:00 AM 2:00:00 PM 3:00:00 PM 4:40:00 PM 4:50:00 PM 7:00:00 PM
77 10:00:00 AM 11:50:00 AM 12:00:00 PM 2:00:00 PM 3:00:00 PM 4:50:00 PM 5:00:00 PM 7:00:00 PM
78 10:00:00 AM 12:00:00 PM 12:10:00 PM 2:00:00 PM 3:00:00 PM 5:00:00 PM 5:10:00 PM 7:00:00 PM
79 10:00:00 AM 12:10:00 PM 12:20:00 PM 2:00:00 PM 3:00:00 PM 5:10:00 PM 5:20:00 PM 7:00:00 PM
80 10:00:00 AM 12:20:00 PM 12:30:00 PM 2:00:00 PM 3:00:00 PM 5:20:00 PM 5:30:00 PM 7:00:00 PM
81 10:00:00 AM 11:40:00 AM 11:50:00 AM 2:15:00 PM 3:15:00 PM 4:55:00 PM 5:05:00 PM 7:00:00 PM
82 10:00:00 AM 11:50:00 AM 12:00:00 PM 2:15:00 PM 3:15:00 PM 5:05:00 PM 5:15:00 PM 7:00:00 PM
83 10:00:00 AM 12:00:00 PM 12:10:00 PM 2:15:00 PM 3:15:00 PM 5:15:00 PM 5:25:00 PM 7:00:00 PM
84 10:00:00 AM 12:10:00 PM 12:20:00 PM 2:15:00 PM 3:15:00 PM 5:25:00 PM 5:35:00 PM 7:00:00 PM
85 10:00:00 AM 12:20:00 PM 12:30:00 PM 2:15:00 PM 3:15:00 PM 5:35:00 PM 5:45:00 PM 7:00:00 PM
86 10:00:00 AM 11:40:00 AM 11:50:00 AM 2:30:00 PM 3:30:00 PM 5:00:00 PM 5:10:00 PM 7:00:00 PM
87 10:00:00 AM 11:50:00 AM 12:00:00 PM 2:30:00 PM 3:30:00 PM 5:10:00 PM 5:20:00 PM 7:00:00 PM
88 10:00:00 AM 12:00:00 PM 12:10:00 PM 2:30:00 PM 3:30:00 PM 5:20:00 PM 5:30:00 PM 7:00:00 PM
***89 10:00:00 AM 12:10:00 PM 12:20:00 PM 2:30:00 PM 3:30:00 PM 5:30:00 PM 5:40:00 PM 7:00:00 PM
***90 10:00:00 AM 12:20:00 PM 12:30:00 PM 2:30:00 PM 3:30:00 PM 5:30:00 PM 5:40:00 PM 7:00:00 PM
***91 6:00:00 AM 7:40:00 AM 7:50:00 AM 10:00:00 AM
92 6:00:00 AM 7:50:00 AM 8:00:00 AM 10:00:00 AM
93 6:00:00 AM 8:00:00 AM 8:10:00 AM 10:00:00 AM
94 6:00:00 AM 8:10:00 AM 8:20:00 AM 10:00:00 AM
95 6:00:00 AM 8:20:00 AM 8:30:00 AM 10:00:00 AM
96 8:00:00 AM 10:10:00 AM 10:20:00 AM 1:00:00 PM
97 8:00:00 AM 10:20:00 AM 10:30:00 AM 1:00:00 PM
98 8:00:00 AM 10:30:00 AM 10:40:00 AM 1:00:00 PM
99 8:00:00 AM 10:40:00 AM 10:50:00 AM 1:00:00 PM
100 8:00:00 AM 10:50:00 AM 11:00:00 AM 1:00:00 PM
101 9:00:00 AM 11:10:00 AM 11:20:00 AM 2:00:00 PM
102 9:00:00 AM 11:20:00 AM 11:30:00 AM 2:00:00 PM
103 9:00:00 AM 11:30:00 AM 11:40:00 AM 2:00:00 PM
104 9:00:00 AM 11:40:00 AM 11:50:00 AM 2:00:00 PM
105 9:00:00 AM 11:50:00 AM 12:00:00 PM 2:00:00 PM
106 12:00:00 PM 1:40:00 PM 1:50:00 PM 4:00:00 PM
107 12:00:00 PM 1:50:00 PM 2:00:00 PM 4:00:00 PM
108 12:00:00 PM 2:00:00 PM 2:10:00 PM 4:00:00 PM
109 12:00:00 PM 2:10:00 PM 2:20:00 PM 4:00:00 PM
110 12:00:00 PM 2:20:00 PM 2:30:00 PM 4:00:00 PM
111 12:00:00 PM 2:10:00 PM 2:20:00 PM 5:00:00 PM
112 12:00:00 PM 2:20:00 PM 2:30:00 PM 5:00:00 PM
113 12:00:00 PM 2:30:00 PM 2:40:00 PM 5:00:00 PM
114 12:00:00 PM 2:40:00 PM 2:50:00 PM 5:00:00 PM
115 12:00:00 PM 2:50:00 PM 3:00:00 PM 5:00:00 PM
116 2:00:00 PM 4:10:00 PM 4:20:00 PM 7:00:00 PM
117 2:00:00 PM 4:20:00 PM 4:30:00 PM 7:00:00 PM
118 2:00:00 PM 4:30:00 PM 4:40:00 PM 7:00:00 PM
119 2:00:00 PM 4:40:00 PM 4:50:00 PM 7:00:00 PM
120 2:00:00 PM 4:50:00 PM 5:00:00 PM 7:00:00 PM

This is the code that I'm using, I know I'm missing something, I'm just not sure what. It's not lost on me that the solver scheduled 5 shifts and the max workforce needed in any timeframe is 5, but it's not enough to cover the workforce needs for the entire day.

# create shift matrix
df = num_workers_t_window.fillna(0).applymap(lambda x: 1 if x == "x" else x)
df = df.drop(columns = ["startTime"])

a = df.drop(columns =["T_Window","num_workers_required"]).values

# number of shifts
n = a.shape[1]

# number of time windows
T = a.shape[0]

# of workers required per time window
d = df["num_workers_required"].values

# labor hours per shift
w = shift_hours["Shift_Hours"].values.astype(int)

# Decision Variables
y = LpVariable.dicts("num_workers", list(range(n)), lowBound=0, cat="Integer")

# Create Problem
prob = LpProblem("Schedule_Creation", LpMinimize)

# Minimize Labor Hours
prob += lpSum([w[j] * y[j] for j in range(n)])

# Constraint for workers required in each time window
for t in range(T):
prob += lpSum([a[t, j] * y[j] for j in range (n)]) >= d[t]

prob.solve()

print("Status:", LpStatus[prob.status])

for shift in range(n):
print(
f"The number of workers needed for shift {shift} is {(y[shift].value())} workers"
)


One of the constraints, per request in comments:

Adding the LP Model, source for this code is from this project: https://towardsdatascience.com/how-to-solve-a-staff-scheduling-problem-with-python-63ae50435ba4

Minimize function

Workforce requirement constraint

Adding a sample of data frame for shifts, workers required (last column), and associated time windows:

T_Window Shift_1 Shift_2 Shift_16 Shift_17 Shift_31 num_workers_required
5:00:00 AM 1 1 0 0 0 1
5:10:00 AM 1 1 0 0 0 1
5:20:00 AM 1 1 0 0 0 1
5:30:00 AM 1 1 0 0 0 1
5:40:00 AM 1 1 0 0 0 1
5:50:00 AM 1 1 0 0 0 1
6:00:00 AM 1 1 1 1 0 2
6:10:00 AM 1 1 1 1 0 2
6:20:00 AM 1 1 1 1 0 2
6:30:00 AM 1 1 1 1 0 2
6:40:00 AM 0 1 1 1 0 2
6:50:00 AM 1 0 1 1 0 2
7:00:00 AM 1 1 1 1 1 3
7:10:00 AM 1 1 1 1 1 3
7:20:00 AM 1 1 1 1 1 3
7:30:00 AM 1 1 1 1 1 3
7:40:00 AM 1 1 0 1 1 3
7:50:00 AM 1 1 1 0 1 3
8:00:00 AM 1 1 1 1 1 4
8:10:00 AM 1 1 1 1 1 4
8:20:00 AM 1 1 1 1 1 4
8:30:00 AM 1 1 1 1 1 4
8:40:00 AM 1 1 1 1 0 4
8:50:00 AM 1 1 1 1 1 4
9:00:00 AM 0 0 1 1 1 4
9:10:00 AM 0 0 1 1 1 4
9:20:00 AM 0 0 1 1 1 4
9:30:00 AM 0 0 1 1 1 4
9:40:00 AM 0 0 1 1 1 4
9:50:00 AM 0 0 1 1 1 4
10:00:00 AM 1 1 0 0 1 4
10:10:00 AM 1 1 0 0 1 4
10:20:00 AM 1 1 0 0 1 4
10:30:00 AM 1 1 0 0 1 4
10:40:00 AM 1 1 0 0 1 4
10:50:00 AM 1 1 0 0 1 4
11:00:00 AM 1 1 1 1 0 4
11:10:00 AM 1 1 1 1 0 4
11:20:00 AM 1 1 1 1 0 4
11:30:00 AM 1 1 1 1 0 4
11:40:00 AM 0 1 1 1 0 4
11:50:00 AM 1 0 1 1 0 4
12:00:00 PM 1 1 1 1 1 4
12:10:00 PM 1 1 1 1 1 4
12:20:00 PM 1 1 1 1 1 4
12:30:00 PM 1 1 1 1 1 4
12:40:00 PM 1 1 0 1 1 4
12:50:00 PM 1 1 1 0 1 4
1:00:00 PM 1 1 1 1 1 5
1:10:00 PM 1 1 1 1 1 5
1:20:00 PM 1 1 1 1 1 5
1:30:00 PM 1 1 1 1 1 5
1:40:00 PM 1 1 1 1 0 5
1:50:00 PM 1 1 1 1 1 5
2:00:00 PM 1 1 1 1 1 4
2:10:00 PM 0 0 1 1 1 4
2:20:00 PM 0 0 1 1 1 4
2:30:00 PM 0 0 1 1 1 4
2:40:00 PM 0 0 1 1 1 4
2:50:00 PM 0 0 1 1 1 4
3:00:00 PM 0 0 1 1 1 4
3:10:00 PM 0 0 0 0 1 4
3:20:00 PM 0 0 0 0 1 4
3:30:00 PM 0 0 0 0 1 4
3:40:00 PM 0 0 0 0 1 4
3:50:00 PM 0 0 0 0 1 4
4:00:00 PM 0 0 0 0 1 3
4:10:00 PM 0 0 0 0 0 3
4:20:00 PM 0 0 0 0 0 3
4:30:00 PM 0 0 0 0 0 3
4:40:00 PM 0 0 0 0 0 3
4:50:00 PM 0 0 0 0 0 3
5:00:00 PM 0 0 0 0 0 2
5:10:00 PM 0 0 0 0 0 2
5:20:00 PM 0 0 0 0 0 2
5:30:00 PM 0 0 0 0 0 2
5:40:00 PM 0 0 0 0 0 2
5:50:00 PM 0 0 0 0 0 2
6:00:00 PM 0 0 0 0 0 1
6:10:00 PM 0 0 0 0 0 1
6:20:00 PM 0 0 0 0 0 1
6:30:00 PM 0 0 0 0 0 1
6:40:00 PM 0 0 0 0 0 1
6:50:00 PM 0 0 0 0 0 1
• In your constraint, what does a[t,j] look like ? Commented Feb 15, 2022 at 8:53
• How would I grab just that portion of the constraint? I added a snip of one of the constraint iterations above.
– t25
Commented Feb 15, 2022 at 17:38
• It would help if you show your LP Commented Feb 15, 2022 at 18:04
• What is the best way to pull that? Sorry, new to Pulp. Happy to put it up, just not sure how to grab it.
– t25
Commented Feb 15, 2022 at 18:15
• It is your mathematical model. Preferably as Text in latex format. If you dont know how to use latex then as image Commented Feb 15, 2022 at 18:19

First, it is confusing when you say

The final solution does not satisfy the workforce requirements for the various time windows, it's substantially short. Only 5 shifts are filled in total.

You do not want to fill in shifts, but time windows. You select shifts in order to satisfy workforce requirements for each time slot.

I suspect you have some issues with your shifts, when you translate them from the format

Shift_Num Start_Shift First_Break_Start First_Break_End Lunch_Start Lunch_End Second_Break_Start Second_Break_End End_Shift

to the format with $$10$$ min time slots. For example, shift $$1$$ should be (if I am not mistaken) :

With the table of shifts added by the OP, removing the set of time slots which cannot be covered at all (after 4:00 PM), and with $$w_j=1$$ for $$j=1,2,16,17,31$$, I get a total of 9 shifts (shift $$2$$, shift $$17$$ (4 times), and shift $$31$$ (4 times)):

Here is the code I used (shifts.csv is a csv file with shifts OP posted in the last table):

import pulp

a = df.drop(columns=["num_workers_required"]).values

# number of shifts
n = a.shape[1]

# number of time windows
T = a.shape[0]

# of workers required per time window
d = df["num_workers_required"].values

# labor hours per shift
# w = shift_hours["Shift_Hours"].values.astype(int)

# Decision Variables
y = pulp.LpVariable.dicts("num_workers", list(range(n)), lowBound=0, cat="Integer")

# Create Problem
prob = pulp.LpProblem("Schedule_Creation", pulp.LpMinimize)

# Minimize Labor Hours
prob += pulp.lpSum([y[j] for j in range(n)])

# Constraint for workers required in each time window
for t in range(T):
prob += pulp.lpSum([a[t, j] * y[j] for j in range(n)]) >= d[t]

prob.solve()

print("Status:", pulp.LpStatus[prob.status])

for shift in range(n):
print(
f"The number of workers needed for shift {shift} is {(y[shift].value())} workers"
)


And here is the output:

With this data (and $$w_j=1$$ for every shift $$j$$), I get the following optimal solution: shifts 15, 36, 91 (twice), 94, i.e., 5 shifts. This solution is not unique. And this solution is feasible, as the orange line always lies above the blue line in the Figure below:

I think part of the confusion may come from the fact that in the data, the index of the first shift is $$1$$, while it is $$0$$ in the python code. So when the solver selects shifts 14, 35, 90, 93, it is actually pointing to shifts 15, 36, 91, 94 in the file with the raw data.

• Comments are not for extended discussion; this conversation has been moved to chat. Commented Feb 21, 2022 at 21:04
• Marking this as the answer. This helped me resolve a number of issues. A significant source of my issue was a problem with the source shift file, which this analysis helped identify.
– t25
Commented Feb 23, 2022 at 7:38

You expect $$y_j > 0$$ for every $$j$$. Looking at the LP however, this does not need to be the case. Just the sum $$y_j * a_{jt}$$ over every $$j$$ needs to fulfill your demand for every $$t$$ (i.e. be at least 1). Your pulp code seems to be right and also the solution it gives is (most likely) valid.

• y(j) needs to be >= 0 for every j (can have 0 workers, but can't have negative workers). Unfortunately, the solution is not valid, I've manually validated it and it's missing substantial headcount.
– t25
Commented Feb 16, 2022 at 15:10
• I'm thinking the problem may be that there are shifts that cover the same t period with different break schedules, which is why y(j) could potentially be equal to zero if it is not optimal to schedule workers on that shift to meet the demand for every t. Something in that logic is not working correctly, it may even be wrong in my LP and I'm just not accounting for that correctly.
– t25
Commented Feb 16, 2022 at 15:24
• That might be true. I can just state that the pulp implementation of your LP is correct. Commented Feb 16, 2022 at 15:46