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海湾城市搬家公司的管理
海湾城市搬家公司是当地的一家公司,专门从事城际搬家。在提交给其赞助商的商业计划中,海湾城市的货运能力至少为36吨。公司正在更换所有的卡车，包括1吨接卡车和2.5吨搬运车类卡车。1吨的卡车将由一名工人负责,而大货车将有四个人员来完成大型的搬家行动。海湾城搬家公司雇佣48个工人，目前设施了40辆卡车。接卡车公司花费了24000美元,移动面包车耗资60000美元。公司希望在卡车的货运能力保持至少36吨的情况下供应最低的投资,并且不需要任何新员工或运输设施。
虽然连续性假设是不严谨的(因为每个购买卡车的数量必须是整数),但是使用一个线性规划模型来确定海湾城市搬家公司的最优卡车和货车购买量。你会发现选择最佳的解决方案是可行的。
准备一份报告,列出这些选项中的几个,并讨论各自的利弊。
Management at bay city movers
Bay City Movers is a local company that specializes in intercity moves. In the business plan submitted to its backers, Bay City has committed itself to a total trucking capacity of at least 36 tons. The company is in the process of replacing its entire fleet of trucks with 1 ton pick up trucks and 2.5 ton moving van type trucks. The 1 ton pick up trucks will be manned by one worker, whereas the large vans will utilize a total of four personnel for larger moves. Bay City Movers currently employs 48 workers and has facilities for 40 trucks. Pick up trucks cost the company $24,000 and the moving vans cost $60,000. The company wishes to make a minimum investment in trucks that will provide a trucking capacity of at least 36 tons while not requiring any new hires or trucking facilities.
Although the continuity assumption is violated (since the number of each truck purchased must be integer), use a linear programming model to determine the optimal purchase of pick up trucks and vans for Bay City Movers. You will find that alternative optimal solutions are possible.
Prepare a report detailing several of these options and discuss the pros and cons of each. Among the alternatives, you should present in your report are the following:
Purchasing only one type of truck.
Purchasing the same number of pick up trucks as moving vans.
Purchasing the minimum total number of trucks.
Also include in your report pertinent sensitivity information you feel would be interest to management at Bay City Movers.
Decision Variables
X1 = No. of Pick up Trucks
X2 = No. of Moving Vans
Constraints
In Addition to Non-negativity Constraint for the decision variables, there are three functional constraints:
No. of workers
Trucking Capacity.
Facilities
No. of Workers:
The pick up trucks uses only one worker, while the large vans utilize 4 workers. Currently Bay City movers has maximum of 48 workers.
X1 + 4X2 ≤ 48
Trucking Capacity:
Pick up Trucks has a Capacity of 1 ton while moving vans has capacity of 2.5 ton. Company wants its trucking Capacity to be atleast 36 tons.
X1 + 2.5X2 ≥ 36
Facilities:
Bay City movers have facilities of 40 trucks, including both pick up trucks and moving vans.
X1 + X2 ≤ 40
Non-negativity of Decision Variables:
No. of pick up trucks and moving vans cannot be negative and it’s impossible. Therefore,
X1, X2 ≥ 0
The Mathematical Model
Min 24000X1 + 60000X2 (Objective Function)
Subject to:
X1 + 4X2 ≤ 48 (No. of workers)
X1 + 2.5X2 ≥ 36 (Trucking Capacity)
X1 + X2 ≤ 40 (Facilities)
X1, X2 ≥ 0 (Non-Negativity)
Detailed Step by Step Description of the Problem and The Graphical Solution Algorithm:
For solving the Bay City Movers Problem, we first should have all the limitation’s understanding. Further, we should know how these limitations will affect the optimization. In this case, Company’s total trucking capacity and no. of workers are necessary for understanding the optimal solution. Then only it is possible to write a linear equation that represents these limitations.
Variable X1 will symbolize no. of Pick up Trucks and variable X2 will embody no. of Moving Vans. It is also seen that 1 ton pick up trucks will be manned by one worker. Therefore 1X1 represents the no. of workers required for 1 ton pick up truck. Similarly, 4X2 represents the no. of workers for large moving vans. The total no. of workers available is 48 and therefore total workers for both trucks and vans must be less than or equal to 48. The linear equation X1 + 4X2 ≤ 48 places right in all the conditions.
Next constraint we look at is the trucking capacity. By looking at the problem, we see that Bay City has committed itself to a total trucking capacity of at least 36. Trucks have the capacity to carry 1 ton only while moving vans have capacity of 2.5 tons. So the total capacity of trucks and moving vans must be more than or equal to 36 tons and therefore the linear equation, X1 + 2.5X2 ≥ 36 satisfies the conditions.
Considering the next constraint, are the facilities of the company. Bay City Movers has total facilities of 40 trucks. Company has a limit of 40 trucks including only pick up trucks and moving vans. Therefore Total no. of trucks must be less than or equal to total trucking facility i.e., 40. The linear equation that represents these constraints is X1 + X2 ≤ 40.
The final constraint and most important one is the non- negative constraint. There cannot be negative no. of trucks and X1 and X2 must be greater than zero.
After understanding the constraints, we must remember the goal. In this case it is minimization in the investments in trucks. We want to minimize the investment in trucks with minimum capacity of 36 tons. We can see that the Cost of Pick up truck is $24000 and the cost of moving vans is $60000. We combine these costs to the variables that symbolize pick up trucks and moving vans. This can be seen in the equation Min 24000X1 + 60000X2. This is known as the Objective Function of the problem.
While making a graph of the problem, the horizontal axis represents No. of trucks and the vertical axis represents no. of vans. First we graph the no. of workers constraint. According to the constraint equation, if we put X2 equal to 0, than X1 would be 48. So we draw a point of X-axis that represents X1. Similarly, we put X1 equal to 0, then X2 value comes out to be 12. So we plot the point at Y-Axis which represents X2. When a line is then drawn across the graph joining these two points, it will represent the time constraint.
Now we draw the trucking capacity constraint. This is determined by finding the point in the X-axis that represents the case of Pick up trucks capacity. This point is 36. Now we draw a point on the Y-axis that represents moving vans capacity. This point is 14.4. Joining these two dots with a line, we get the Trucking capacity constraint line.
Finally we look at the Facility constraint. This is represented by the point in X axis, where X2 is 0 and X1 is 40, and the point in Y axis, where X1 is 0 and X2 is 40. Line joining these two points will represent the facility constraint equation.
All the constraints will be graphed just till the limit of those constraints. Such as, No. of workers is limited to 48, trucking capacity has a minimum limit of 36 and facilities are limited to 40. When all of these lines are drawn on graph and the areas excluded are removed, including the negative areas, the area that is left is called the feasible region. Any of the points of the feasible region is represented as solution to the given constraints. In order to minimize investment, a point where two constraint lines cross must be taken into consideration.
As we know, a Formulation of Bay City Movers problem is:
X1 = the number of Pick up trucks
X2 = the number of Moving vans
Max Objective Function 24000X1+60000X2
Subject to:
C1 = X1 + 4X2 ≤ 48 (constraint of No. of workers) |
