Individual Assignment Operations Analytics, Spring 2021
1. Process Analysis (25 points)
You have a pizza shop that bakes fresh pizzas. That means that it is make-to-order system and no work is done on a pizza before receiving a customer order. Then, each customer receives a completely fresh, piping hot pizza.
The production process is as follows. The first step is to take the customer’s order and receive payment, which takes 4 minutes on average. The second step is to roll the dough into a flat circle, on which you can put toppings. Rolling the dough takes 6 minutes on average. The third step is to add tomato sauce and mozzarella cheese, which takes 2 minutes on average. The final step is to put the pizza in the only oven, and to bake the pizza. Your oven, which can hold 2 pizzas, bake pizzas for 5 minutes. The set-up for each baking takes 1 minute. You have ready-to-use supply of pizza dough, tomato sauce, and shredded cheese. Immediately after a pizza finishes baking, it goes to the customer.
There are 4 employees. Employee A is responsible for taking the customer’s order and receiving payment. Employees B and C are responsible for rolling the dough. Employee D is responsible for assembling the pizza (that is, adding the tomato sauce and cheese). D is also responsible setting up the oven, which takes 1 minute.
(a) (2 points) How long will it take you to fill a customer order (assume no waiting)?
(b) (5 points) What is the capacity of each resource? Where is the bottleneck? What is the system capacity?
You have decided to also offer pepperoni pizzas besides the current cheese pizzas. Now, there is an extra step in the production process: adding pepperoni. This is done after the pizza sauce and mozzarella cheese is added. Then, the production process for the pepperoni pizza is: (1) take order and receive payment; (2) roll dough; (3) add tomato sauce and cheese; (4) add pepperoni; (5) bake pizza in oven. The production process for cheese pizzas is unchanged.
You have hired another worker, employee E, to focus on adding pepperonis. Employee E puts great effort into artfully arranging the pepperonis, and it takes 5 minutes per pizza to do so.
(c) (2 points) Suppose that all customers order pepperoni pizzas. What is the process capacity, in pizzas per hour?
From now on, we suppose that 50% of your customers order cheese pizzas and 50% of them order pepperoni pizzas.
(d) (2 points) What is the system capacity, in pizzas (of either type) per hour?
(e) (3 points) Suppose you cross-train all the 5 employees, what is the capacity of the system, in pizzas (of either type) per hour?
(f) (5 points) Suppose Employee E’s job is artful and cannot be done be others. You also want Employee E to focus on pepperoni arrangement and do nothing else. You only cross-train all the other 4 employees, what is the capacity of the system, in pizzas (of either type) per hour?
Now we assume there are no cross-training and go back to initial job allocation among Employees A-E. Customers arrive in the system as a Poisson process. In average, there are 12 customers per hour. 50% of them order cheese pizzas and 50% of them order pepperoni pizzas. In average, there are 6 customers in the shop.
(g) (2 points) What is the flow time of the customers?
(h) (2 points) Suppose the customers who order pepperoni pizzas has to spend 5 more minutes in the shop compared with the ones who order cheese pizzas due to the adding pepperoni step. What are the flow time of customers who order pepperoni pizzas?
(i) (2 points) In average, how many customers who order pepperoni pizzas are in the shop?
2. Priority Queue (40 points)
Read the priority queue case slides and watch the video.
• System congestion (server utilization): 𝜌 = 𝜆(𝑝’/𝜇’ + 𝑝+/𝜇+).
• Waiting cost in the system:
In the document ‘predicted_scores’ you will find a table with the predicted probabilities of arriving patients to be Admitted given their collected information at the triage stage. Recall that these probabilities represent the actual output of the Machine Learning (ML) algorithm. Each row represents an arriving patient, where the first column (i.e., ‘prob.admit’) presents the predicted probability to be Admitted, and the second column (i.e., disposition’) presents the actual disposition of that patient (i.e., whether the patient was Admitted or Discharged).
(a) Recall that based on this prediction output, you define a classification threshold 𝑡. ∈ [0,1] such that patients with predicted probability equal or above 𝑡. are predicted as ‘Admitted’, while patients with predicted probability below such threshold are predicted as ‘Discharged’. Write a simple script to compute and draw the sensitivity and specificity of the prioritization as functions of threshold 𝑡.. Note that you have the real patient disposition and the predicted probability of patients in the ‘predicted-scores’ data set. When you define a particular threshold, you will have a predicted disposition for each patient. What happens to the sensitivity and specificity as you increase (decrease) the classification threshold 𝑡.? (10 points)
(b)Consider the case in which the medical provider is able to serve 𝜇 = 10 patients/hour on average.
The proportion of ‘Admitted’ patients is 𝑝’ = 0.3, while the proportion of ‘Discharged’ patients is 𝑝+
= 0.7. Moreover, the medical acuity of patients is captured by 𝑐+ = 1 and 𝑐’ = 10 which represent the
waiting cost per hour that each low acuity (i.e., Discharged) and each high acuity (i.e., Admitted)
patient experiences respectively. Create a simple script to compute the waiting cost C in the system
as a function of the threshold 𝑡.. Recall from previous question that a particular value for 𝑡. is
associated with a value for sensitivity and a value for specificity. Identify the threshold 𝑡∗ that .
minimizes the waiting cost for the following arrival rates 𝜆 scenarios (assume that the interarrival time is exponentially distributed) and fill in the following table: (10 points)
(c) Consider now the case in which the medical provider is able to serve 10 high acuity patients/hour,
and 50 low acuity patients/hour (assume that the service times for both patients’ types are
exponentially distributed). Identify the threshold 𝑡∗ that minimizes the waiting cost for the following .
arrival rates 𝜆 scenarios (assume that the interarrival time is exponentially distributed) and fill in the following table: (10 points)
(d) Finally, consider the case in which the medical provider is able to serve 10 high acuity patients/hour,
and 120 low acuity patients/hour (assume that the service times for both patients’ types are
exponentially distributed). Identify the threshold 𝑡∗ that minimizes the waiting cost for the following .
arrival rates 𝜆 scenarios (assume that the interarrival time is exponentially distributed) and fill in the following table: (10 points)
3. Linear Programming (35 points)
A firm has 6 plants (squares) to produce 6 products (circles). Each product has a demand independently uniformly distributed between [0,200]. The capacity of each plant is 100. Suppose the firm are considering two architectures: full flexibility or long chain. Facing different product demands, the firm need to decide the amount each plant to produce. This is easy for full flexibility setting: if the total demand is larger than 600 units, then each plant is fully utilized and the total production is 600 units; otherwise, the total production is the demand. For the long chain, the allocation is tricky: each product can be produced by two plants and you need to figure out the best amount of production from each plant, in order to maximize the total production. Please formulate a LP for this production allocation problem. (20 points)
Simulate 1000 times and find the average total production under each architecture. How close are they? (15 points)