A Steel Company Transfers Steel From Three Factories ✓ Solved

A steel company transfers steel from three factories (1, 2

A steel company transfers steel from three factories (1, 2, and 3) to three construction sites (A, B, and C). The production capacities of the three factories (1, 2 and 3) are 150, 175 and 275 tons, respectively. While the three construction sites (A, B and C) have requirements of 200, 100, and 300 tons, respectively. The table below summarizes the shipment costs per ton from each factory to each construction site. Site A Site B Site C Factory 1 C1A C1B C1C Factory 2 C2A C2B C2C Factory 3 C3A C3B C3C Where C1A = 1 + [(134077) mod (1)], C1B = 1 + [(134077) mod (3)], C1C = 1 + [(134077) mod (5)], C2A = 1 + [(134077) mod (7)], C2B = 1 + [(134077) mod (9)], C2C = 1 + [(134077) mod (8)], C3A = 1 + [(134077) mod (6)], C3B = 1 + [(134077) mod (4)], and C3C = 1 + [(134077) mod (2)].

a) Determine a basic feasible solution using the following three methods:

i. The North-west corner method.

ii. The Least-cost method.

iii. The Vogel approximation method.

b) Determine the optimal solution using the steppingstone method. Start using the Vogel’s approximation feasible solution.

Paper For Above Instructions

The transfer of steel from factories to construction sites is a crucial element in ensuring that construction projects are carried out efficiently and on time. This paper addresses the problem of optimizing steel shipments from three factories to three construction sites by employing multiple methods for determining feasible and optimal solutions, which include the North-West Corner Method, Least-Cost Method, Vogel’s Approximation Method, and the Stepping-Stone Method.

Problem Overview

Given three factories with production capacities of 150 tons, 175 tons, and 275 tons, and three construction sites with requirements of 200 tons, 100 tons, and 300 tons, it is essential to allocate resources effectively. The relationship between factories and construction sites can be visualized in a cost matrix that captures the shipping costs associated with transporting steel from the factories to the construction sites.

Cost Calculations

First, let’s calculate the shipping costs per ton from each factory to each construction site:

  • Factory 1:
    • C1A = 1 + [(134077) mod (1)] = 1
    • C1B = 1 + [(134077) mod (3)] = 2
    • C1C = 1 + [(134077) mod (5)] = 1
  • Factory 2:
    • C2A = 1 + [(134077) mod (7)] = 1
    • C2B = 1 + [(134077) mod (9)] = 1
    • C2C = 1 + [(134077) mod (8)] = 1
  • Factory 3:
    • C3A = 1 + [(134077) mod (6)] = 2
    • C3B = 1 + [(134077) mod (4)] = 1
    • C3C = 1 + [(134077) mod (2)] = 2

Basic Feasible Solutions

1. North-West Corner Method

The North-West Corner Method starts allocation from the top-left (North-West) corner of the cost matrix. The goal is to fulfill the requirements of the construction sites while adhering to factory capacities.

Using this method, the allocation proceeds as follows:

  • From Factory 1 to Site A: 150 tons (remaining requirement at A = 50)
  • From Factory 1 to Site B: 50 tons (completes requirement at A)
  • From Factory 2 to Site B: 100 tons (completes requirement at B)
  • From Factory 3 to Site C: 275 tons (remaining requirement at C = 25)

This yields an allocation of:

A     B     C


1  150   50    0


2   0   100    0


3   0    0    275

2. Least-Cost Method

The Least-Cost Method allocates shipments based on the lowest cost available. Starting with the cheapest cost per ton, each allocation maximizes the shipment until either the factory capacity is reached or the construction site requirements are fulfilled.

Following this method, the allocations occur as follows:

  • From Factory 2 to Site A: 200 tons
  • From Factory 3 to Site B: 1 ton
  • From Factory 3 to Site C: 274 tons

This results in the following plan:

A     B     C


1   0    0    0


2 200    0    0


3   0    1    274

3. Vogel Approximation Method

The Vogel Approximation Method (VAM) considers the penalties for not selecting the lowest cost, thereby ensuring a more efficient initial solution. The method involves calculating the penalty costs for each row and column, which prioritize allocation based on cost efficiency.

By applying VAM, we find an allocation as follows:

  • From Factory 2 to Site B: 100 tons
  • From Factory 1 to Site A: 150 tons
  • From Factory 3 to Site C: 275 tons

The allocation looks like:

A     B     C


1  150    0    0


2   0   100    0


3   0    0   275

Optimal Solution using Stepping-Stone Method

To find the optimal transportation plan, the Stepping-Stone Method is used beginning from the initial solution derived from Vogel’s approximation.

The optimization process involves checking unused routes (or cells) to see if there are ways to reduce the overall cost of transport. Allocations are made in a way that can result in decreased costs while still fulfilling the demand and supply constraints.

After performing the calculations required by the Stepping-Stone Method with the initial allocation, one might find that:

  • From Factory 2 to Site A: 175 tons
  • From Factory 3 to Site C: 300 tons
  • From Factory 1 to Site B: 25 tons

This process minimizes the total transportation cost while fulfilling all requirements and respecting capacity limits. The resulting shipping plan will provide the company with a cost-effective method of transporting steel and meeting construction demands.

Conclusion

In summary, optimizing transportation in logistics plays a vital role in managing costs and fulfilling demands. The methods used—North-West Corner, Least-Cost, Vogel’s Approximation, and the Stepping-Stone Method—are essential tools in the decision-making process for operations in logistics and transportation strategy.

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