Linear Programming Problems: Formulation and Solver Solutions Analysis

This document presents solutions to two linear programming problems. The first problem focuses on optimizing the cost of consultant staffing, considering both full-time and part-time consultants across different time slots, aiming to minimize overall labor costs while meeting minimum staffing requirements. It includes the formulation of the objective function and constraints, followed by the solver solution. The second problem addresses cargo plane optimization, maximizing profit by determining the optimal amount of different cargo types to stow in three compartments with weight and space limitations. The solution involves formulating the objective function, constraints related to cargo weight, space, and compartment capacity, and also includes the solver solution for the optimization problem.

LINEAR PROGRAMMING PROBLEM
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Problem: 3.15
Formulation of linear programming problem
Time of Day Minimum Number of
Consultants Required to be
on Duty
8 am – noon 6
Noon – 4 pm 8
4 pm- 8 pm 12
8 pm -midnight 6
Full time consultants are paid = $17.50 per hours
Part time consultants are paid = $15 per hours
Let
X1 = Number of full-time consultants working in 8 am to 4 pm
X2 = Number of full-time consultants working in noon to 8 pm
X3 = Number of full-time consultants working in 4 pm to midnight
X4 = Number of part-time consultants working in 8 am to noon
X5 = Number of part-time consultants working in noon to 4 pm
X6 = Number of part-time consultants working in 4 pm to 8 pm
X7 = Number of part-time consultants working in 8 pm to mid night
Per day cost of part-time workers = 4 * $15 = $60
Per day cost of full-time workers = 8*$17.50 = $140
The aim is to minimize the cost.
Objective function
Min Z = 140 (X1+X2+X3) + 60 (X4+X5+X6+X7)
Subject to constraints
X1+X4 >= 6
X1+X2+X5 >= 8
X2+X3+X6 >= 12
X3+X7>= 6
X1- 2X4>=0
X1+X2-2X5 >=0
X2+X3-2X6>=0
1

X3-2X7>=0
X1, X2, X3, X4, X5, X6, X7 >0
Non-negativity constraint.
Solution of linear programming problem
(Solver Solution)
2

Problem 3.22
Formulation of linear programming problem
A cargo plane has 3 compartments for storing the cargo: front, center, and back.
These compartments have capacity limits in terms of weight and space and is summarized
below:
The four cargoes have been offered for shipment as flight space as shown below:
The aim is to maximize the profit.
xij =the number of tons of cargo type
i=1,2,3,4 stowed in compartment
j= 1,2,3 [front, center, back]
Objective function
Max Z = 320 (x11 + x12+ x13) + 400 (x21 + x22+ x23) + 360 (x31 + x32+ x33) + 290 (x41 + x42+ x43)
Subject to constraints
x11 + x21 + x31 + x41 ≤ 12
x12 + x22 + x32 + x42 ≤ 18
x13 + x23 + x332+ x43 ≤ 10
x11 + x12+ x13 ≤ 20
x21 + x22+ x23 ≤16
x31 + x32+ x33 ≤25
3

x41 + x42+ x43 ≤13
500x11 +700 x21 + 600x31 + 400x41≤ 7000
500x12 + 700 x22 + 600 x32 +400 x42 ≤ 9000
500x13 +700 x23 +600 x332+ 400 x43 ≤ 5000
1/12 (x11 + x21 + x31 + x41) - 1/18 (x12 + x22 + x32 + x42) = 0
1/12 (x11 + x21 + x31 + x41) - 1/10 (x13 + x23 + x332+ x43) = 0
All xij ≥ 0
Non-negativity constraint.
Solution of linear programming problem
(Solver Solution)
4