Example of a scheduling problem

Supply Chain Analytics in Python

Aaren Stubberfield

Supply Chain Analytics Mgr.

Expected Demand

Day of Week	Drivers Needed
0 = Monday	11
1= Tuesday	14
2 = Wednesday	23
3 = Thursday	21
4 = Friday	20
5 = Saturday	15
6 = Sunday	8

Question:

How many drivers, in total, do we need to hire?

Constraint:

Each driver works for 5 consecutive days, followed by 2 days off, repeated weekly

Step	Definition
Decision Var	X$_{\text{i}}$ = the number of drivers working on day _i_
Objective	minimize z = X$_{\text{0}}$ + X$_{\text{1}}$ + X$_{\text{2}}$ + X$_{\text{3}}$ + X$_{\text{4}}$ + X$_{\text{5}}$ + X$_{\text{6}}$
Subject to	X$_{\text{0}}$ ≥ 11
	X$_{\text{1}}$ ≥ 14
	X$_{\text{2}}$ ≥ 23
	X$_{\text{3}}$ ≥ 21
	X$_{\text{4}}$ ≥ 20
	X$_{\text{i}}$ ≥ 0 (i = 0, ..., 6)

Step	Definition
Decision Var	X$_{\text{i}}$ = the number of drivers working on day _i_
Objective	minimize z = X$_{\text{0}}$ + X$_{\text{1}}$ + X$_{\text{2}}$ + X$_{\text{3}}$ + X$_{\text{4}}$ + X$_{\text{5}}$ + X$_{\text{6}}$
Subject to	X$_{\text{0}}$ + X$_{\text{3}}$ + X$_{\text{4}}$ + X$_{\text{5}}$ + X$_{\text{6}}$ ≥ 11
	X$_{\text{0}}$ + X$_{\text{1}}$ + X$_{\text{4}}$ + X$_{\text{5}}$ + X$_{\text{6}}$ ≥ 14
	X$_{\text{0}}$ + X$_{\text{1}}$ + X$_{\text{2}}$ + X$_{\text{3}}$ + X$_{\text{6}}$ ≥ 23
	X$_{\text{0}}$ + X$_{\text{1}}$ + X$_{\text{2}}$ + X$_{\text{3}}$ + X$_{\text{4}}$ ≥ 21
	X$_{\text{1}}$ + X$_{\text{2}}$ + X$_{\text{3}}$ + X$_{\text{4}}$ + X$_{\text{5}}$ ≥ 15
	X$_{\text{i}}$ ? 0 (i = 0, ..., 6)

Coding example

# Initialize Class
model = LpProblem("Minimize Staffing", 
                   LpMinimize)
days = list(range(7))

# Define Decision Variables
x = LpVariable.dicts('staff_', days,
                      lowBound=0, cat='Integer')

# Define Objective
model += lpSum([x[i] for i in days])

# Define Constraints
model += x[0] + x[3] + x[4] + x[5] + x[6] >= 11
model += x[0] + x[1] + x[4] + x[5] + x[6] >= 14
model += x[0] + x[1] + x[2] + x[5] + x[6] >= 23
model += x[0] + x[1] + x[2] + x[3] + x[6] >= 21
model += x[0] + x[1] + x[2] + x[3] + x[4] >= 20
model += x[1] + x[2] + x[3] + x[4] + x[5] >= 15
model += x[2] + x[3] + x[4] + x[5] + x[6] >= 8

# Solve Model
model.solve()

Summary

Our initial variables did not work
Decision variables to incorporate some of the constraints

Practice time!

Supply Chain Analytics in Python