Reep Construction late won a contract for the digging and site readying of a new remainder country on the Pennsylvania Turnpike. The chief job is that the house wants to minimise cost of run intoing the monthly hauling demands for this undertaking but besides follows a no-layoff policy.

The restraints of the job are as follows:

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The occupation requires four months to finish. with 10. 12. 14 and 8 trucks needed in months 1 through 4. severally.

The house signed a long-run rental with PennState Leasing last twelvemonth for trucks where one of these trucks will be available for usage on the new undertaking in month 1.

two for month 2. three for month 3 and one for month 4. The long-run leasing contract has a monthly cost of $ 600 per truck. Reep building pays its truck drivers $ 20 an hr and day-to-day fuel costs are besides about $ 100 per truck. which leads to a monthly cost of approximately $ 2000 since each truck used on the new undertaking will be runing about eight hours a twenty-four hours.

five yearss a hebdomad for four hebdomads a month.

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

The determination variables in this job can be divided into two classs: the figure of trucks obtained through long-run leasing and the figure of trucks obtained through short-run leasing.

For the trucks obtained through long-run leasing. there is a fit figure of trucks that can be used for each several month stated earlier in the restraints.

Therefore allow L1= # long-run leased trucks in month 1L2= # long-run leased trucks in month 2L3= # long-run leased trucks in month 3L4= # long-run leased trucks in month 4

For the trucks obtained through short-run leasing. there are more options. A truck’s rental can get down in any of the four months for any length of clip between 1-4 months within the entire four-month time-span. For illustration. a truck can be leased get downing in the first month for any length of clip between 1-4 months. but a truck that is leased get downing in the 3rd month can merely be leased for either one or two months. and a truck leased get downing in the 4th month can merely be leased for one month. Therefore each short-run leased truck has two features: the month its rental began and the lease’s length of clip. In any one month. the short-run trucks leased can hold any combination of these two features.

Therefore allow Sij= # short-run leased trucks wherei=the month the rental beganj=the lease’s length of clip

With the determination variables defined. the nonsubjective map and restraints 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-run Lease Truck Availability| L1?1|Month 2 Long-run Lease Truck Availability| L2?2|Month 3 Long-run Lease Truck Availability| L3?3|Month 4 Long-run Lease Truck Availability| L4?1|

Since the driver’s rewards are fixed costs ( they are paid irrespective of how many trucks are leased ) . they are non included in the nonsubjective map. For long-run chartered trucks. the $ 2000 per month fuel costs are the lone outgo. For short-run chartered trucks. PennState Leasing’s monthly fees include both driver’s rewards and the cost of renting the truck and are given as 4000. 3700. 3225 and 3040 for length of rental of 1-4 months. severally.

The entire cost for the long-run chartered trucks is calculated by adding together the monthly fuel. truck and driver costs and so multiplying this figure by the figure of months the truck is leased.

From Solver. we see that the optimum leasing program is to utilize all of the trucks available from the long-run leasing program. rent three trucks for three months and six trucks for four months in the first month. one truck for three months in the 2nd month. and one truck for one month in the 3rd month. This program would take to a minimum cost of $ 151. 660.

The costs are split up amongst the type of trucks in this manner: Type of Truck| Cost|Long-run trucks ( 7 ) | 2000*7 = 14000|S13 ( 3 ) | 11675*3 = 35025|S14 ( 6 ) | 14160*6 = 84960|S23 ( 1 ) | 11675|S31 ( 1 ) | 6000|Entire Cost| 151660|

The decreased cost for S11. S41. S12. S22. and S32 is above nothing because in the solution these values are zero. so increasing the “final value” of these trucks or renting one of any of these trucks would take to an addition in cost of 3515. 3515. 3725. 210 and 915. severally. This besides means that the cost would hold to diminish by those several Numberss in order for the optimum solution to include those variables.

The shadow monetary values for each of the restraints show how much the nonsubjective map would acquire better or worse by if the right manus side was increased by one unit. For case if the entire figure of trucks needed for month 1 increased from 10 to 11. the cost would acquire better by $ 2485 or diminish by $ 2485 ( since the shadow monetary value is the negative of the double monetary value ) . The positive double values for the long-run trucks show that utilizing the long-run trucks alternatively of the short-run trucks really worsens costs.

Without the lay-off policy. the driver’s costs becomes a relevant instead than a fixed cost. and hence must be included in the nonsubjective map. The nonsubjective map alterations in footings of the long-run determination variable’s coefficients. The cost of drivers’ rewards is calculated to be 3200 because they are paid 20 dollars an hr. and work for about eight hours a twenty-four hours for five yearss a hebdomad. for four hebdomads a month. Thus the L1 through L4 coefficients alteration from lone 2000 ( which is merely the cost of fuel ) to 5200 ( after adding 3200 to 2000 ) .

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-run Lease Truck Availability| L1?1|Month 2 Long-run Lease Truck Availability| L2?2|Month 3 Long-run Lease Truck Availability| L3?3|Month 4 Long-run Lease Truck Availability| L4?1|

The consequences show that the new optimum leasing program with driver’s rewards as a relevant cost would give a minimized cost of $ 165. 410. The program would affect utilizing merely two of the available long-run leasing trucks in month three. and four short-run renting trucks in the first month for three months. six short-run trucks in the first month for four months and twoshort-run trucks in the 2nd month for three months.

Under the no-layoff program. all of the trucks available from the long-run leasing program must be used so that the drivers will all hold occupations. 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 extra $ 8650 from $ 165. 410.

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