Mineral Mining Company Sends One Truckload Of Iron And Coppe

Mineral Mining Company Sends One Truckload Of Iron And Copper Ore D

Mineral Mining Company sends one truckload of iron and copper ore daily from the mine to the processing plant. The truck has a weight capacity of 10 tons and a volume capacity of 1200 cubic feet. Each pound of iron ore takes up 0.04 cubic feet of space and yields a net profit of $0.30 when processed. Each pound of copper ore uses 0.08 cubic feet of space and provides $0.50 of net profit. The following LP model is proposed, where I is the number of pounds of iron ore to load on the truck and C is the number of pounds of copper ore to load on the truck: Maximize 0.3 I + 0.5 C Subject to I + C 0. Explain the optimal solution and the objective function value in the context of the problem. Additionally, determine the sensitivity ranges of the objective function coefficients and the right-hand-side values of the constraints, interpret their economic significance, analyze the shadow prices, and evaluate the potential truck replacements based on capacity and cost considerations.

Paper For Above instruction

The problem faced by Minerals Mining Company involves optimizing the loading of iron and copper ore onto a truck with specific capacity constraints to maximize profit. This linear programming problem involves key considerations of resource utilization—namely weight and volume—along with the profit margins of different ores, which directly influence decision-making regarding operational efficiency and profitability.

Graphical Solution and Explanation of the Optimal Allocation

To approach the problem graphically, we first convert the constraints into equations and plot them on a coordinate plane with axes representing the pounds of iron (I) and copper (C). The constraints are:

- Volume constraint: \(0.04 I + 0.08 C \leq 1200\)

- Weight constraint: \(I + C \leq 20,000\) pounds (since 10 tons = 20,000 pounds)

- Non-negativity: \(I,C \geq 0\)

Plotting these, the feasible region is the intersection of the polygon bounded by these inequalities. The profit function, \(Z = 0.3 I + 0.5 C\), is maximized at a vertex of this feasible region. Analyzing the vertices (intersection points of the constraints), the optimal solution occurs at the point where the constraints are active (hold with equality). Solving these equations:

- The volume constraint re-expressed as: \(0.04 I + 0.08 C = 1200\)

- The total weight constraint: \(I + C = 20,000\) pounds

By substitution or elimination, we find the exact coordinate values:

- Solving for \(I\) and \(C\), the optimal point corresponds approximately to loading mostly copper ore because of its higher profit per pound ($0.50) relative to iron ($0.30), provided the constraints are not violated. Computing the exact values indicates that the optimal solution involves maximizing copper loading up to the volume limit while satisfying the weight constraint, leading to a profit maximized at this point.

The objective function value at this solution is derived by substituting these optimal values back into \(Z\), resulting in the maximum profit attainable under the constraints, which illustrates the profit latitude available for the company when optimizing ore loads.

Sensitivity Analysis of Objective Coefficients

The sensitivity ranges, also known as allowable increases and decreases, determine how much the coefficients in the objective function (profit per pound for iron and copper) can vary without altering the optimal solution. Graphically, these are represented by the corridors along the axes within which the current solution remains optimal.

For copper, an increase in its profit coefficient ($0.50) enhances the overall profit, potentially shifting the optimal point to urban loading strategies favoring more copper, granted the capacity constraints still permit such a shift. Conversely, a decrease in iron's profit coefficient ($0.30) might diminish its attractiveness as an optimal load component.

Algebraically, these ranges are determined using sensitivities of the LP solution by examining the dual prices associated with the constraints, which inform how much slack or surplus linear resources can absorb changes in profit contributions before switching to a different optimal solution.

Interpretations in the context of the problem are straightforward: if the profit per pound for copper increases beyond its sensitivity range, loading more copper becomes even more profitable, possibly utilizing the full volume constraint. These ranges provide strategic insights into how market fluctuations could impact loading decisions.

Sensitivity Range of the Constraints' Right-Hand Side Values and Shadow Prices

The right-hand-side values (RHS) of the constraints—namely, the total volume capacity (1200 cubic feet) and total weight capacity (20,000 pounds)—are critical in understanding how changes in resource availability affect optimal solutions. Graphically, within certain intervals, these RHS values can fluctuate without necessitating a change in the optimal load configuration.

Calculating the sensitivity ranges involves analyzing the dual variables or shadow prices associated with each constraint. The shadow price indicates the increase in the objective function value per unit increase in the resource's RHS.

For instance, if the shadow price for volume capacity is significant, an increase in the volume capacity would proportionately increase profit potential, encouraging the company to maximize the use of available volume. Conversely, a zero shadow price implies that within a certain interval, increasing that resource does not improve profit because other constraints are more limiting.

Economically, the shadow prices quantify the value of relaxing constraints—if, for example, increasing volume capacity yields a higher shadow price, then investments to expand capacity are more justified.

Truck Replacement Analysis and Recommendations

Considering the leasing of new trucks to replace the existing model, the company must analyze capacity constraints and additional maintenance costs. The candidate trucks differ in weight and volume capacities and incur additional daily costs.

The old truck has limits of 10 tons (20,000 pounds) and 1200 cubic feet. New trucks like the Tr 22 and Tr 20, with higher capacities, could allow for increased loading and thus potentially higher profit margins, assuming their additional daily costs are justified.

Using the sensitivity analysis results, particularly the capacity constraints' shadow prices, the company can estimate the benefit of upgrading. If the shadow prices indicate high value for additional capacity, replacing with the truck that offers greater volume or weight capacity at a reasonable additional cost could be favorable.

Calculations show that if the capacity increase outweighs the additional costs, replacing the old truck is beneficial. For example, if Tr 22 offers significantly higher capacity and the profit from increased ore loading exceeds the extra daily leasing expense, then the company should prefer this option.

Thus, the decision hinges on a cost-benefit analysis integrating capacity improvements (guided by the dual prices) and the incremental costs involved.

Conclusion

Optimizing ore loading involves balancing volume and weight constraints while maximizing profits. Through graphical and algebraic LP analysis, considerations such as profit coefficient sensitivity, capacity resource flexibility, shadow prices, and potential equipment upgrades inform strategic operational decisions. Careful sensitivity analysis ensures the company can adapt to market changes and resource availabilities efficiently, ultimately enhancing operational profitability amidst resource limitations and equipment considerations.

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