Army rations (Minimization problem):
Army administration is seeking to buy battle rations for the troops from two suppliers.
Supplier 1 sells the ration for $0.6 containing 20% of the minimum daily requirement
(MDR) of vitamin A, 25% of the MDR of vitamin D and 50% of the MDR of iron.
Supplier 2 sells the ration for $0.5 containing 50% of the MDR of vitamin A, 25% of
the MDR of vitamin D and 10% of the MDR of iron.
Model the problem for minimizing the total cost of the combination of both suppliers’
rations while maintaining the three nutrients’ MDRs.
X2
Solution:
Let X1 = number of rations from supplier 1.
X2 = number of rations from supplier 2. 10
Slopes:
OF Minimize 0.6X1 + 0.5X2 -0.6/0.5
s.t.
Vitamin A: 20X1 + 50X2 ≥ 100 (0, 2) (5, 0) -20/50 Feasible area
Vitamin D: 25X1 + 25X2 ≥ 100 (0, 4) (4, 0) -25/25 Iron
Iron: 50X1 + 10X2 ≥ 100 (0, 10) (2, 0) -50/10
X1, X2 ≥ 0
4
The slope of the objective function (-1.2) is trapped
between that of the vitamin D (-1) and that of Iron 2
Vit D
(-5) making them the binding constraints and their
Vit A
intersection the optimal point:
Vitamin D: X2 = 4 - X1 plug into X1
Iron: 5X1 + 4 - X1 = 10 X1 = 1.5 X2 = 2.5 2 4 5
OFV = 0.6(1.5) + 0.5(2.5) = $2.15. OF
Solver Solution:
A survey of 2000 investors for the financial advising firm of William
and Ryde is conducted to determine service satisfaction. The
investors are divided into four groups:
Group I: Large investors with William and Ryde
Group II: Small investors with William and Ryde
Group III: Large investors with other firms
Group IV: Small investors with other firms
The groups are further subdivided into those that will be contacted
by phone and those that will be visited in person. The estimated cost
of taking a survey depends on the group and method of survey
collection. These are detailed in the following table.
Determine the number of investors that should be surveyed from
each group by phone and in person to minimize Gladstone and
Associates’ overall total estimated cost if:
• At least half of those surveyed invest with William and Ryde.
• At least one-fourth are surveyed in person.
• At least half of the large William and Ryde investors surveyed
are contacted in person.
At most 40% of those surveyed are small investors.
• At least 10% and no more than 50% of the investors surveyed
are from each group.
• At most 25% of the small investors surveyed are contacted in
person.
Sensitivity Analysis:
A company is planning weekly production of washers, dryers, and refrigerators to maximize the total profit. The production is limited
by the number of hours available in the departments of molding, assembly, and packing. To solve the problem, an LP model has
been developed and it has been solved using Excel Solver (assume integer constraints are not necessary). The LP model, the Excel
model with the optimal solution, the answer report, and the Sensitivity Report are shown below.
Mathematical Model and Sensitivity Report:
(a) How many washers, dryers, and refrigerators are produced per week and what is the total weekly profit?
(b) Which departments are limiting the production?
(c) How would the optimal solution and the total profit be affected, if the unit profit from refrigerators decreased from $130 to
$100?
(d) The company discovered that it could assign a new worker to one of the three departments, which would increase the available
hours in this department by 100. To which department would you assign the new worker to maximize the increase in the total profit?
(e) The company is concerned with the fact that dryers are not produced. By how much should the unit profit from dryers increase in
order for dryers to become profitable?
(f) In addition to washers, dryers, and refrigerators, the company would like to produce stoves. Each stove would give a profit of
$100 and would require 4 hours of molding, 12 hours of assembly, and 2 hours of packing. Would it be profitable to introduce the
production of stoves?
