Dynamic Programming Question
Problem Statement
The sales manager for a pharmaceutical company has six traveling
salespeople of a medical ramp team to assign 3 cities namely; Lahore,
Karachi and Multan. He has decided that each city should be assigned
at least one salesperson and that each individual salesperson should be
restricted to one of the cities, but now he wants to determine how
many salespeople of a team should be assigned to the respective cities
in order to maximize sales.
Dynamic Programming Question
Cities
Sales person
Lahore Karachi Multan
1
35
21
28
2
48
42
41
3
70
56
63
4
89
70
75
Dynamic Programming Question
Objective
Maximize the sale in the three cities
3
𝑃𝑖 (𝑥𝑖 )
𝑖=1
Constraints
We have constraints of the sales person in medical ramp team. Total
members were 6 in the team if each member was restricted to each
city then for each city we were available with extra three sales persons.
So then each city can have minimum one team or maximum 4 teams.
Dynamic Programming Question
Constrained
3
(1 ≤ 𝑥𝑖 ≤ 4)
𝑖=1
Formulation by backward recursive
𝑓𝑛 (𝑆𝑛 , 𝑥𝑛 ) = 𝑃𝑛 (𝑥𝑛 ) + 𝑓𝑛+1 (𝑆𝑛 − 𝑥𝑛 )
𝑆𝑛 = number of sales persons still available for allocation to remaining
countries
𝑥𝑛 = available teams
Dynamic Programming Question
n=3
𝑆3
𝑃3 (𝑥3 )
𝑥3 (𝐷𝑒𝑐𝑖𝑠𝑖𝑜𝑛)
1
28
1
2
41
2
3
63
3
4
75
4
Dynamic Programming Question
𝑥2
𝑠2
1
2
1
21
2
49
42
3
62
70
4
84
83
3
𝑃2 (𝑥2 )
𝑥2 Decision
21
1
49
1
56
70
2
84
84
1 or 3
Dynamic Programming Question
𝑥1
𝑠1
1
4 105
2
3
97
91
𝑃1 (𝑥1 )
𝑥1 Decision
105
. X1= 1 medical ramp sales officer for city n=1
• X2= 2 medical ramp sales officers for city n=2
• X3= 3 medical ramp sales officers for city n=3
1