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Chapter: Operations Research: An Introduction - Deterministic Dynamic Programming

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Work Force Size Model- Dynamic Programming(DP) Applications

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 bi laborers. Theoretically, we can use hiring and firing to keep the work-force in week i exactly equal to bi. Alternatively, it may be more economical to maintain a labor force larger than the minimum requirements through new hiring. This is the case we will consider here.

 

Given that xi is the actual number of laborers employed in week i, two costs can be incurred in week i: C1(xi - bi), the cost of maintaining an excess labor force xi - bi, and C2(xi - xi-1, the cost of hiring additional laborers, xi - xi-1. It is assumed that no additional cost is incurred when employment is discontinued.

 

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 xi, the number of laborers in week i.

 

c.      The state at stage i is represented by the number of laborers available at stage (week) i - 1, xi-1.

 

The DP recursive equation is given as


The computations start at stage n with x n  =       bn and terminate at stage 1.

 

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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