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A manufacturing facility uses a series of identical machines to produce a sing...

May 5, 2024

A manufacturing facility uses a series of identical machines to produce a single type of product. Under a contract, the plant is obligated to manufacture and deliver 4,500 units each day. When operating at full capacity, each machine has the ability to produce 1,000 units per day. Should a machine operate below its maximum capacity, its output adjusts accordingly. For instance, at 50% capacity, it will produce 500 products daily. Machines are inspected each morning and are classified according to their condition: I – Excellent working condition – can produce at 100% capacity at a daily running cost of

550. II – Good working condition – can produce at 80% capacity at a daily running cost of

600. III – Poor working condition – can produce at 60% capacity at a daily running cost of

650. IV – Very poor working condition – cannot produce at all. Assume that if a machine is deemed in condition X at the start of the day, unless it undergoes repair during the day it will operate at the above rates and costs for the whole day. Its condition may only change at the start of the following day. If a machine does not undergo repairing and is operated its condition remains unchanged or deteriorates according to the following pattern: If a machine is in Condition I this morning, it has probabilities of 0.8, 0.1, 0.05, and 0.05 of being in Conditions I, II, III and IV tomorrow, respectively. If a machine is in Condition II this morning, it has probabilities of 0.8, 0.15, and 0.05 of being in Conditions II, III and IV tomorrow, respectively. If a machine is in Condition III this morning, it has probabilities of 0.9 and 0.1 of being in Conditions III and IV tomorrow, respectively. If a machine is in Condition IV this morning, it will be in Condition IV tomorrow. Machines in Conditions II, III, and IV can be repaired at a cost of

2,500,

3,000 and

4,000, respectively. Machines that are being repaired cannot be used for production during the whole day and return on the following morning in Condition I. The company aims to develop an optimal long-term strategy that balances meeting production demands with minimising costs. To achieve this, the company plans to establish a fixed number of machines to operate daily and implement a consistent repair policy for each machine, maintaining this approach every day. Assume the following: 1. It is clear that if a machine is in Condition IV it has to be repaired. You are required to decide at which Condition II, III or IV repair work should be undertaken. Don’t consider options that don’t make sense (repair in Conditions II and IV, but not in Condition III, etc.) 2. The initial purchase of the machines is not taken into account. Since we are interested in long-term strategies the setup cost is not important. 3. You cannot operate a machine for a portion of a day. For example, if you require to meet a demand of 800 products and have two machines with daily production capacities of 500 products each, then you have to operate both machines for the whole day. 4. Assume that you are required to meet an average daily demand of 4,500. Moreover, you are neither rewarded nor penalized for exceeding the demand. Provide a complete analysis of the decision problem and suggest a long-term policy for the company. Base your recommendation on calculations of the expected costs for the different policies. Include a short discussion at the end of the report.

Solution

Identifying the optimal number of machines: To meet the daily demand of 4,500 units, the minimum number of machines needed at full capacity (1,000 units per machine) is 5. However, since machines can operate below full capacity and may be out of service due to repairs, we need to calculate the expected output and determine the optimal number of machines to operate daily

Calculating expected daily output per machine: We need to calculate the expected output for a machine in each condition, considering the probability of it being in that condition the next day. For example, the expected output for a machine in Condition I is

0.8 \times 1000 + 0.1 \times 800 + 0.05 \times 600 + 0.05 \times 0 = 920

units

Determining repair policy: We need to compare the expected costs of repairing a machine in Conditions II and III versus the costs of operating it in a deteriorated condition. This involves calculating the expected cost savings from repairing a machine and comparing it to the additional operating costs incurred by not repairing it

Establishing a consistent repair policy: Based on the cost calculations, we can determine at which condition (II, III, or IV) it is most cost-effective to repair the machines. This policy will be applied consistently to minimize costs while meeting production demands

Answer

The company should operate a sufficient number of machines to meet the daily demand, taking into account the expected output per machine in various conditions and the probabilities of these conditions. A consistent repair policy should be established based on the condition that minimizes the long-term expected costs.

Key Concept

Optimal long-term strategy for machine operation and repair

Explanation

The optimal strategy involves calculating the expected daily output and costs for machines in different conditions, determining the most cost-effective condition for repair, and establishing a fixed number of machines to operate daily to meet production demands at minimal costs.