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In some construction projects, hiring and firing are exercised to maintain a labor force that meets the needs of the project.

**Work-Force Size Model**

In some
construction projects, hiring and firing are exercised to maintain a labor
force that meets the needs of the project. Given that the activities of hiring
and firing both incur additional costs, how should the labor force be
maintained throughout the life of the project?

Let us
assume that the project will be executed over the span of *n* weeks and that the minimum labor force required in week *i* is *b _{i}*
laborers. Theoretically, we can use hiring and firing to keep the work-force in week

Given
that *x _{i}* is the actual
number of laborers employed in week

The
elements of the DP model are defined as follows:

*a. **Stage i *is represented by week* i, i *=*
*1, 2, ….. ,* **n.*

b.
The *alternatives*
at stage *i* are *x _{i},* the number
of laborers in week

c.
The *state*
at stage *i* is represented by the
number of laborers available at stage (week) *i* - 1, *x _{i-1}.*

The DP
recursive equation is given as

The
computations start at stage *n* with *x* * _{n}* =

**Example
10.3-2**

A
construction contractor estimates that the size of the work force needed over
the next 5 weeks to be 5, 7, 8.4, and 6 workers, respectively. Excess labor
kept on the force will cost $300 per worker per week, and new hiring in any
week will incur a fixed cost of $400 plus $200 per worker per week.

The data
of the problem are summarized as

**PROBLEM SET 10.3B**

1. Solve Example 10.3.2 for each of the following minimum labor
requirements:

2. In Example 10.3-2, if a severance pay of $100 is incurred for each
fired worker, determine the optimum solution.

*3. Luxor Travel arranges I-week tours to southern Egypt. The agency is
contracted to pro-vide tourist groups with 7,4,7, and 8 rental cars over the
next 4 weeks, respectively. Luxor Travel subcontracts with a local car dealer
to supply rental needs. The dealer charges a rental fee of $220 per car per
week, plus a flat fee of $500 for any rental transaction. Luxor, however, may
elect not to return the rental cars at the end of the week, in which case the
agency will be responsible only for the weekly rental ($220). What is the best
way for Luxor Travel to handle the rental situation?

4. GECO is contracted for the next 4 years to supply aircraft engines at
the rate of four engines a year. Available production capacity and production
costs vary from year to year. GECO can produce five engines in year 1, six in
year 2, three in year 3, and five in year 4.
The corresponding production costs per engine over the next 4 years are
$300,000, $330,000, $350,000, and $420,000, respectively. GECO can elect to
produce more than it needs in a certain year, in which case the engines must be properly stored until shipment date. The
storage cost per
engine also varies from year to year, and is estimated
to be $20,000 for year 1, $30,000 for year 2, $40,000 for year 3, and $50,000
for year 4. Currently, at the start of year 1, GECO has one engine ready for
shipping. Develop an optimal production plan for GECO.

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Operations Research: An Introduction : Deterministic Dynamic Programming : Work Force Size Model- Dynamic Programming(DP) Applications |

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