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Test Bank Introduction to Operations Research 11th Edition by Frederick S. Hillier, Gerald J. Lieberman Chapter 3-28|Complete Guide A $10.98
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Test Bank Introduction to Operations Research 11th Edition by Frederick S. Hillier, Gerald J. Lieberman Chapter 3-28|Complete Guide A

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Test Bank Introduction to Operations Research 11th Edition by Frederick S. Hillier, Gerald J. Lieberman Chapter 3-28|Complete Guide A

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  • June 5, 2022
  • February 11, 2024
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Test Bank for Chapter 3



Problem 3-1:

The Weigelt Corporation has three branch plants with excess production capacity. Fortunately,
the corporation has a new product ready to begin production, and all three plants have this
capability, so some of the excess capacity can be used in this way. This product can be made in
three sizes--large, medium, and small--that yield a net unit profit of $420, $360, and $300,
respectively. Plants 1, 2, and 3 have the excess capacity to produce 750, 900, and 450 units per
day of this product, respectively, regardless of the size or combination of sizes involved.

The amount of available in-process storage space also imposes a limitation on the
production rates of the new product. Plants 1, 2, and 3 have 13,000, 12,000, and 5,000 square
feet, respectively, of in-process storage space available for a day's production of this product.
Each unit of the large, medium, and small sizes produced per day requires 20, 15, and 12 square
feet, respectively.

Sales forecasts indicate that if available, 900, 1,200, and 750 units of the large, medium,
and small sizes, respectively, would be sold per day.

At each plant, some employees will need to be laid off unless most of the plant’s excess
production capacity can be used to produce the new product. To avoid layoffs if possible,
management has decided that the plants should use the same percentage of their excess capacity
to produce the new product.

Management wishes to know how much of each of the sizes should be produced by each
of the plants to maximize profit.

Formulate a linear programming model for this problem.




Solution for Problem 3.1:


Copyright 2020 © McGraw-Hill Education. All rights reserved. No reproduction or distribution
without the prior written consent of McGraw-Hill Education.

,The decision variables can be denoted and defined as follows:




xP1L = number of large units produced per day at Plant 1,

xP1M = number of medium units produced per day at Plant 1,

xP1S = number of small units produced per day at Plant 1,

xP2L = number of large units produced per day at Plant 2,

xP2M = number of medium units produced per day at Plant 2,

xP2S = number of small units produced per day at Plant 2,

xP3L = number of large units produced per day at Plant 3,

xP3M = number of medium units produced per day at Plant 3,

xP3S = number of small units produced per day at Plant 3.




Also letting P (or Z) denote the total net profit per day, the linear programming model for this
problem is




Maximize P = 420 xP1L + 360 xP1M + 300 xP1S + 420 xP2L + 360 xP2M + 300 xP2S

+ 420 xP3L + 360 xP3M + 300 xP3S,

subject to

xP1L + xP1M + xP1S  750
Copyright 2020 © McGraw-Hill Education. All rights reserved. No reproduction or distribution
without the prior written consent of McGraw-Hill Education.

, xP2L + xP2M + xP2S  900

xP3L + xP3M + xP3S  450

20 xP1L + 15 xP1M + 12 xP1S  13000

20 xP2L + 15 xP2M + 12 xP2S  12000

20 xP3L + 15 xP3M + 12 xP3S  5000

xP1L + xP2L + xP3L  900

xP1M + xP2M + xP3M  1200

xP1S + xP2S + xP3S  750

1 1
( xP1L + xP1M + xP1S ) - ( xP2L + xP2M + xP2S ) = 0
750 900

1 1
( xP1L + xP1M + xP1S ) - ( xP3L + xP3M + xP3S ) = 0
750 450

and




xP1L  0, xP1M  0, xP1S  0, xP2L  0, xP2M  0, xP2S  0,

xP3L  0, xP3M  0, xP3S  0.




The above set of equality constraints also can include the following constraint:



1
( x P2 L + x P2 M + x P2S ) - 1 (x P3 L + xP3M + x P3S ) = 0.
900 450


Copyright 2020 © McGraw-Hill Education. All rights reserved. No reproduction or distribution
without the prior written consent of McGraw-Hill Education.

, However, any one of the three equality constraints is redundant, so any one (say, this one) can be
deleted.




Problem 3-2:

Comfortable Hands is a company which features a product line of winter gloves for the entire
family — men, women, and children. They are trying to decide what mix of these three types of
gloves to produce.

Comfortable Hands’ manufacturing labor force is unionized. Each full-time employee
works a 40-hour week. In addition, by union contract, the number of full-time employees can
never drop below 20. Nonunion, part-time workers can also be hired with the following union-
imposed restrictions: (1) each part-time worker works 20 hours per week, and (2) there must be
at least 2 full-time employees for each part-time employee.

All three types of gloves are made out of the same 100% genuine cowhide leather.
Comfortable Hands has a long term contract with a supplier of the leather, and receives a 5,000
square feet shipment of the material each week. The material requirements and labor
requirements, along with the gross profit per glove sold (not considering labor costs) is given in
the following table.



Material Required Labor Required Gross Profit
Glove (square feet) (minutes) (per pair)
Men’s 2 30 $8
Women’s 1.5 45 $10
Children’s 1 40 $6


Each full-time employee earns $13 per hour, while each part-time employee earns $10
per hour. Management wishes to know what mix of each of the three types of gloves to produce
Copyright 2020 © McGraw-Hill Education. All rights reserved. No reproduction or distribution
without the prior written consent of McGraw-Hill Education.

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