# operational research

Pages: 7 (1019 words) Published: April 22, 2014
Ricerca operativa 8/ed
Frederick S. Hillier, Gerald J. Lieberman

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.

McGraw-Hill

Solution for Problem 3.1:

The decision variables can be denoted and defined as follows:

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xP1L
xP1M
xP1S
xP2L
xP2M
xP2S
xP3L
xP3M
xP3S

=
=
=
=
=
=
=
=
=

number of large units produced per day at Plant 1,
number of medium units produced per day at Plant 1,
number of small units produced per day at Plant 1,
number of large units produced per day at Plant 2,
number of medium units produced per day at Plant 2,
number of small units produced per day at Plant 2,
number of large units produced per day at Plant 3,
number of medium units produced per day at Plant 3,
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

Ricerca operativa 8/ed
Frederick S. Hillier, Gerald J. Lieberman

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
( xP2L + xP2M + xP2S ) = 0
( xP1L + xP1M + xP1S ) 900
750
1
1
( xP3L + xP3M + xP3S ) = 0
( xP1L + xP1M + xP1S ) 450
750

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 P2L + x P2M + x P2S ) − 1 (x P3L + xP3M + x P3S ) = 0.
900
450

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

McGraw-Hill

Problem 3-2:

Comfortable Hands is a company which features a product line of winter gloves for the entire Tutti i diritti riservati
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...