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.
The Term Paper on Solve Business Problems 2
1.1 Describe ways of recognising when a business problem exists In most organizations recognition of a business problem typically relies on measures and management reporting of measures against expectations. Usually each performance area measures quality, cost, and speed of service – plus possibly customer satisfaction. If one of these measures starts to vary from expectations then ...
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 Essay on Short Term And Long Term Memory
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The total cost for the long-term leased trucks is calculated by adding together the monthly fuel, truck and driver costs and then multiplying this number by the number of months the truck is leased.
From Solver, we see that the optimal leasing plan is to use all of the trucks available from the long-term leasing plan, lease three trucks for three months and six trucks for four months in the first month, one truck for three months in the second month, and one truck for one month in the third month. This plan would lead to a minimal cost of $151,660.
The costs are split up amongst the type of trucks in this way: Type of Truck| Cost|
Long-term trucks (7)| 2000*7 = 14000|
S13 (3)| 11675*3 = 35025|
S14 (6)| 14160*6 = 84960|
S23 (1)| 11675|
S31 (1)| 6000|
Total Cost| 151660|
The reduced cost for S11, S41, S12, S22, and S32 is above zero because in the solution these values are zero, so increasing the “final value” of these trucks or leasing one of any of these trucks would lead to an increase in cost of 3515, 3515, 3725, 210 and 915, respectively. This also means that the cost would have to decrease by those respective numbers in order for the optimal solution to include those variables.
The shadow prices for each of the constraints show how much the objective function would get better or worse by if the right hand side was increased by one unit. For instance if the total number of trucks needed for month 1 increased from 10 to 11, the cost would get better by $2485 or decrease by $2485 (since the shadow price is the negative of the dual price).
The positive dual values for the long-term trucks show that using the long-term trucks instead of the short-term trucks actually worsens costs.
Without the lay-off policy, the driver’s costs becomes a relevant rather than a fixed cost, and therefore must be included in the objective function. The objective function changes in terms of the long-term decision variable’s coefficients. The cost of drivers’ wages is calculated to be 3200 because they are paid 20 dollars an hour, and work for approximately eight hours a day for five days a week, for four weeks a month. Thus the L1 through L4 coefficients change from only 2000 (which is only the cost of fuel) to 5200 (after adding 3200 to 2000).
The Research paper on Cost Terms And Concepts
In this module you will have an opportunity to demonstrate your understanding of cost terms and their application in the aviation industry. For this Case Study complete the four requirements below: 1. ABC Airlines has determined both the fixed and variable costs per flying hour associated with flying each of the 10 different types of aircraft in their fleet. How might this type of information be ...
Objective Function| min2000L1+ 2000L2+ 2000L3+ 2000L4+4000S11+ 4000S21+ 4000S31+ 4000S41+ 3700S12 + 3700S22 +3700S32+ 3225S13+ 3225S23+ 3040S14| 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|
The results show that the new optimal leasing plan with driver’s wages as a relevant cost would yield a minimized cost of $165,410. The plan would involve using only two of the available long-term leasing trucks in month three, and four short-term leasing trucks in the first month for three months, six short-term trucks in the first month for four months and two
short-term trucks in the second month for three months.
Under the no-layoff plan, all of the trucks available from the long-term leasing plan must be used so that the drivers will all have jobs. In other words L1 must be 1, L2 must be 2, L3 must be 3 and L4 must be 1.
The minimized cost is now $174,060, which means that because of the no-layoff policy costs are an additional $8650 from $165,410.
