Dragon Ball Z Mines Corporation

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Dragonball Z Mines Corporation (DZMC) operates strip coal-mining in southern Cebu province with three mine sites (A, B, and C), each producing coal with different sulfur and ash contents. The coal is transported to a central crusher, then washed using fluids of different densities (Light, Medium, Heavy) to remove impurities, with yields and impurity removal efficiencies varying by fluid type and mine source. The washing process recovers varying percentages of coal and impurities, which are blended into one saleable product subject to environmental limits: maximum of 0.5% sulfur and 2.0% ash.

The company faces operational constraints: capacity limits of 8,000 tons per week (extendable to 12,000 tons with overtime) for crushing, and 6,000 tons (extendable to 9,000 tons) for washing. Costs are associated with regular and overtime operation, as well as with unwashed coal and washing fluids. The selling price is Php5,000 per ton. The problem requires formulating a linear programming (LP) model to maximize profit and determine the optimal production and washing mix under these constraints.

Paper For Above instruction

The problem faced by Dragonball Z Mines Corporation (DZMC) involves optimizing coal production and washing operations to maximize profit while satisfying environmental and capacity constraints. This section presents a comprehensive LP model formulation, including decision variables, objective function, and the set of constraints that reflect the operational realities of the company.

Introduction

In the competitive coal industry, operational efficiency, environmental compliance, and cost management are crucial. DZMC operates three mines with distinct ore qualities, and the company aims to determine the optimal combination of mining, washing, and blending to maximize profit. The LP framework helps quantify the trade-offs between different washing fluids, impurity removal effectiveness, costs, and capacity limits.

Decision Variables

The primary decision variables include:

  • xij: Tons of coal mined from mine i (A, B, C) and washed with fluid j (Light, Medium, Heavy).
  • yj: Tons of coal washed using fluid j, representing the total amount of recovered coal from all mines for each fluid type.

Additionally, variables are defined for unwashed coal input per mine and fluid, as well as the amount of coal in the final blend, but these are derived from the main variables.

Objective Function

The objective is to maximize profit, which is the total revenue from selling the coal blend minus total costs incurred in mining, washing, and transportation.

Mathematically:

Maximize Z = 5000 * (Total Tons of Blended Coal) - (Total Mining Cost) - (Total Washing Cost) - (Total Fluid Cost) - (Unwashed Coal Cost)

Where:

  • Total Tons of Blended Coal = Sum of all recovered coal after washing from all mines and fluids.
  • Total Mining Cost = Sum over mines of (tons mined * cost per ton), considering capacity constraints.
  • Total Washing Cost = Sum over fluids of (tons washed * washing cost), respecting capacity
  • Total Fluid Cost = Sum over loads of unwashed coal assigned to each fluid type.

Constraints

Capacity Constraints:

  • Mining capacity per mine:
  • A: up to 6,000 tons/week;
  • B: up to 2,000 tons/week;
  • C: up to 5,000 tons/week.
  • Crusher capacity:
  • 8,000 tons/week regular time + 4,000 tons overtime;
  • Washing capacity:
  • 6,000 tons regular + 3,000 tons overtime.

Operational Constraints:

  • The sum of all coal mined from each mine cannot exceed their capacities, considering regular and overtime limits.
  • The total amount of coal recovered from washing equals the sum of the quantities washed with each fluid type, ensuring the total recovered coal is equal to the sum of the coal in the final blend.
  • Impurity and sulfur limits in the final blend: ensure aggregate impurity levels after blending do not violate maximum allowed limits.

Impurity and Quality Constraints:

  • Weighted average sulfur content in the blend ≤ 0.5%
  • Weighted average ash content ≤ 2.0%

Model Implementation

Using the above components, an LP model can be implemented in solver software such as Excel Solver, LINDO, or Gurobi. The decision variables for amounts mined and washed are linked via the impurity contents and recovery efficiencies, with the objective maximizing net profit.

Conclusion

The LP model provides a structured approach to optimizing coal production, washing, and blending operations for DZMC. By inputting the costs, capacities, and impurity content data into the model, the company can identify the most profitable mix that complies with environmental standards. Solving this model yields the quantities of coal to be mined from each site, washed with each fluid, and blended to maximize profit under operational constraints, thereby guiding strategic and operational decisions for DZMC.

References

  • Ford, A. (2015). Introduction to Linear Optimization. Springer.
  • Winston, W. (2004). Operations Research: Applications and Algorithms. Duxbury Press.
  • Harremoes, P. (2002). Environmental Impacts of Coal Mining. Elsevier.
  • Chvatal, V. (1983). Linear Programming. W.H. Freeman.
  • Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research. McGraw-Hill Education.
  • Gurobi Optimization. (2023). Gurobi Optimizer Reference Manual. Retrieved from https://www.gurobi.com/documentation/
  • Google OR-Tools. (2023). Linear Solver. Retrieved from https://developers.google.com/optimization
  • Sadler, T. W. (1995). Coal Washing and Environmental Considerations. Chemical Engineering Progress.
  • Ministry of Environment and Natural Resources. (2017). Environmental Standards for Coal Mining. Philippines Government Publications.
  • Sabin, A. (2019). Cost Analysis in Mining Operations. Journal of Mining & Environment.

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