# Truck Leasing Problem

ESE 204

Truck Leasing Strategy

Reep Construction recently won a contract for the excavation and site preparation of a new rest area on the Pennsylvania Turnpike. The main problem is that the firm wants to minimize cost of meeting the monthly trucking requirements for this project but also follows a no-layoff policy.

The constraints of the problem are as follows:

The job requires four months to complete, with 10, 12, 14 and 8 trucks needed in months 1 through 4, respectively.

The firm signed a long-term lease with PennState Leasing last year for trucks where one of these trucks will be available for use on the new project in month 1, two for month 2, three for month 3 and one for month 4. The long-term leasing contract has a monthly cost of $600 per truck. Reep construction pays its truck drivers $20 an hour and daily fuel costs are also approximately $100 per truck, which leads to a monthly cost of about $2000 since each truck used on the new project will be operating approximately eight hours a day, five days a week for four weeks a month.

PennState Leasing also offers a short-term lease that includes the cost of both a truck and a driver, where a short-term lease of one month costs $4000, of two months costs $3700, three months costs $3225 and four months costs $3040.

The decision variables in this problem can be divided into two categories: the number of trucks obtained through long-term leasing and the number of trucks obtained through short-term leasing.

For the trucks obtained through long-term leasing, there is a set number of trucks that can be used for each respective month stated earlier in the constraints.

Thus let L1=# long-term leased trucks in month 1

L2=# long-term leased trucks in month 2

L3=# long-term leased trucks in month 3

L4=# long-term leased trucks in month 4

For the trucks obtained through short-term leasing, there are more options. A truck’s lease can start in any of the four months for any length of time between 1-4 months within the total four-month time-span. For example, a truck can be leased starting in the first month for any length of time between 1-4 months, but a truck that is leased starting in the third month can only be leased for either one or two months, and a truck leased starting in the fourth month can only be leased for one month. Thus each short-term leased truck has two characteristics: the month its lease began and the lease’s length of time. In any one month, the short-term trucks leased can have any combination of these two characteristics.

Thus let Sij=# short-term leased trucks

where

i=the month the lease began

j=the lease's length of time

With the decision variables defined, the objective function and constraints can now be defined:

Objective Function| min2000L1+ 2000L2+ 2000L3+ 2000L4+6000S11+ 6000S21+ 6000S31+ 6000S41+ 9400S12 + 9400S22 +9400S32+ 11675S13+ 11675S23+ 14160S14| First Month Constraint| L1+S11+S12+S13+S14=10|

Second Month Constraint| L2+S12+S13+S14+ S21+ S22+S23=12| Third Month Constraint| L3++S13+S14+ S22+S23+ S31+S32=14| Fourth Month Constraint| L4+S14+S23+S32+ S41=8|

Month 1 Long-term Lease Truck Availability| L1≤1|

Month 2 Long-term Lease Truck Availability| L2≤2|

Month 3 Long-term Lease Truck Availability| L3≤3|

Month 4 Long-term Lease Truck Availability| L4≤1|

Since the driver’s wages are fixed costs (they are paid regardless of how many trucks are leased), they are not included in the objective function. For long-term leased trucks, the $2000 per month fuel costs are the only expenditure. For short-term leased trucks, PennState Leasing’s monthly fees include both driver’s wages and the cost of leasing the truck and are given as 4000, 3700, 3225 and 3040 for length of lease of 1-4 months, respectively.

The total cost for the long-term leased trucks is calculated by adding together the monthly fuel, truck and driver costs and then multiplying...

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