Product Mix Problem and Simulation in a Restaurant Kitchen

Product Mix Problem

The manager of the company needs to determine the product mix for the next month in order to maximize the total profit of the company for the next month.

a) Formulate an Integer Programming model for this product mix problem so that the Total Profit of the company in the next month is maximized (Total Profit = Total Revenue – Raw Material Cost – Labour Cost). Describe clearly each of the following parts of your mathematical model with ONE or TWO sentences: objective function, decision variables, constraints and decision variable domain.

b) Model this problem in Excel and use Solver to find the optimal solution. Provide a description of the optimal solution found in the context of the application with ONE or TWO sentences. A hard copy screenshot of your Excel model should accompany your description.

c) Generate the Answer Report and Sensitivity Analysis Report by dropping the integer requirement on decision variables in a). Use the Answer Report and Sensitivity Analysis Report to find out the most limiting resource or activity to the furniture company. Analyse what if the availability of the most limiting resource changes. Are the Total Profit and/or production plan going to change? How are they going to change and why?

d) Model the following situations by adding extra variables and revising objectives and constraints of the Integer programme model in a). Each question is independent of the others. No Excel implementation is needed.

i) All production of Products A, B, and C needs to be made in 1000’s. For example, the production of Product A needs to be 1000, 2000, 3000 etc.

ii) Production of Product A and production of Product B cannot exceed the estimated demand at the same time.

iii) If Product A is made more than 20% in excess of the estimated demand, Product B must be made no more than 20% in excess of the estimated demand.

iv) If a third party offers to produce product A for the company with a fixed charge of £500,000 and they also charge £1000 per Product A. Should the company manufacture Product A on them own or the company should buy it from the third party? Or should the company do both?

Problem 2: Simulation

In a restaurant kitchen, orders arrive randomly from 9am – 10pm; the time between two consecutive orders arriving follows an exponential distribution with an average of 10 minutes between 9-12am, 5 minutes between 12-2pm, 20 minutes between 2-5pm, and 8 minutes between 5-10pm. 15 percent of  all orders are cold meals, 60 percent are hot meals, and 25 percent are fish meals; these percentagesdo not significantly change during the day.

• In a first operation, the ingredients for the order are prepared; preparation time is exponentially distributed with an average of 5 minutes for a cold plate, 10 minutes for a hot meal, and 15 minutes for a fish dish.
• In a second operation, the meal is cooked; the time of this operation is exponentially distributed with an average of 4 minutes for a cold plate, 12 minutes for a hot meal, and 23 minutes for a fish dish.
• In a third operation, the orders are arranged; the time is exponentially distributed with an average of 4 minutes for a cold plate, 6 minutes for a hot meal, and 9 minutes for a fish dish.

The restaurant opens 7 days a week. Build a simulation model that can test the following two queue priority rules: First-In-First-Out, Shortest Expected Processing Time First. The objective is to find the processing rule that will minimise the average throughput time of an order. Let the model report the throughput time of each order.

1. a) Conceptual Model. Develop a conceptual model for the problem, by describing the objectives, inputs, outputs, content (including a logic flow diagram) assumptions and simplifications of the model, in approximately 500 words.
2. b) Computer Model. Build a simulation computer model in

i) Set Result Collection Period = 2000 weeks. Set number of runs in trial = 5. Halt the simulation at limit: 1000, 5000, 11,000, 12,000, and 13,000 orders leaving the system. Create a table and a graph to observe if the system reaches steady state with the increasing limit of the number of orders leaving the system. Explain with a couple of sentences.

ii) Select ONE of the limits from i) to halt the simulation so that the system reaches steady state. Start with “number of runs in trial = 5”. Calculate the number ofruns for a trial, so that 95% confidence interval of the average throughput time of an order is reduced to ±10 minutes?

iii) Use the parameter settings in ii), change the number of runs in trial to the value calculated in ii). Run the trial to determine the following:

o Average throughput time of an order of all three types.

o Average throughput time of an order for cold plates.

o Average throughput time of an order for hot meals.

o Average throughput time of an order for fish dishes.

c) Experimentation. Experiment with the simulation model you built into find average throughput time of an order for a cold meal. Make a comparison of the system following two queue priority rules: First-In-First-Out, Shortest Expected Processing Time First. Provide detailed calculation and your conclusions and recommendations based on your statistical analysis.