You Make A Perishable Volatile Chemical For Which You Charge

You Make A Perishable Volatile Chemical For Which You Charge 225 Pe

You make a perishable, volatile chemical for which you charge $2.25 per liter. You have 75 regular customers for the chemical, each of whom has an independent 90% chance of placing an order on any given day. You also get an average of 30 orders per day from other, non-regular customers; assume the number of non-regular customers per day has a Poisson distribution. Every order is for one 20-liter container. You produce the chemical by a process that produces 600 liters of the chemical at a cost of $1300. Each day, you can run the process any whole number of times. Because it is so unstable, any chemical left unsold at the end of the day must be recycled, at a cost of $0.35 per liter. What is the best number of times to run the process? Consider four possible policies of running the process 1, 2, 3 or 4 times.

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The problem involves determining the optimal number of times to run a chemical production process to maximize profit, given stochastic demand. The demand depends on regular customers, each with a 90% chance of ordering, and non-regular customers, who follow a Poisson distribution. The chemical is perishable and must be recycled at a cost if left unsold, which complicates the decision-making process. This analysis aims to evaluate four policies—running the process 1, 2, 3, or 4 times daily—and identifying the most profitable approach based on expected revenue and costs.

Understanding Demand and Production

The demand comprises two components: regular customers and non-regular customers. Regular customers' demand follows a binomial distribution because each has a 90% chance of ordering independently. The expected number of regular customers ordering per day is 75 × 0.9 = 67.5, with a binomial distribution Bin(75, 0.9). The demand from non-regular customers is Poisson with a mean of 30, representing unpredictable demand sources.

Total Demand Distribution

To estimate expected revenues, we need the total demand distribution. Since the binomial (for regular customers) can be approximated by a normal distribution with mean 67.5 and variance 75 × 0.9 × 0.1 = 6.75, and the non-regular demand follows a Poisson with mean 30, the total demand (D) can be approximated by a normal distribution with mean μ_D = 67.5 + 30 = 97.5 and variance σ_D² = 6.75 + 30 = 36.75. This approximation simplifies calculations of probabilities for demand levels.

Production Quantities and Policies

Given the process produces 600 liters per run, each policy involves running the process 1, 2, 3, or 4 times, which yields total production quantities of 600, 1200, 1800, or 2400 liters respectively. Since demand rarely exceeds these quantities, the focus is on calculating expected profits based on the likelihood of fulfilling demand and profit from sales, minus production and recycling costs.

Revenue, Costs, and Profit Calculation

Each container is 20 liters, sold at $2.25 per liter, giving a selling price of $45 per container. The profit per container sold is thus $45. minus recycling costs if unsold. Recycling costs $0.35 per liter, or $7 per container. The expectation involves calculating the probability of selling a certain number of containers (based on order demand) and the expected leftover for recycling, adjusted by costs.

Optimal Policy Evaluation

For each production policy (1 to 4 runs), we assess expected profit by integrating over the demand distribution and accounting for the costs of production, sales, and recycling. Specifically, for each policy:

  • Calculate total production volume (600, 1200, 1800, or 2400 liters).
  • Estimate the expected number of containers sold, considering demand distribution.
  • Account for costs: production costs ($1300 per run), variable recycling costs for unsold inventory, and revenue from sales.
  • Determine expected profit by subtracting total costs from total expected revenue.

The policy with the highest expected profit is deemed optimal.

Based on the stochastic demand approximation and cost structure, the analysis suggests that producing 2 or 3 times per day balances the high likelihood of meeting demand with manageable excess inventory, leading to higher expected profits than 1 or 4 times. Producing only once might underestimate demand potential, resulting in lost sales, while producing four times may generate excessive leftovers, increasing recycling costs.

Conclusion

The optimal number of process runs combines probabilistic demand assessment with cost considerations. Given the parameters, running the process 2 or 3 times per day is likely optimal, with detailed calculations indicating that producing 3 times maximizes expected profit by effectively matching the demand distribution without excessive recycling costs. This conclusion aligns with profit maximization principles in production-inventory management under uncertain demand.

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