J.J. Bakery in Irvine sells a special kind of bread. The daily number of customers arriving at the store approximately follows a normal distribution with a mean of 100 and a standard deviation of 10. There is a 0.30 chance that a customer buys one load of sourdough bread, a 0.10 chance that a customer buys two loaves, and a 0.60 chance that a customer does not purchase bread.

The bread is baked once a day at 6 A.M. (before observing demand) and sold between 9 A.M and 9 P.M (when demand comes in). Each loaf of bread costs $1.00 to produce and sells for $3.00. Unsold loaves of bread at the end of the day are donated and the store gets a tax credit of $0.50 per loaf. Given the information, the store needs to determine the production quantity for each day’s bakery and estimate the daily profit given the chosen production quantity. You do not need to implement the problem in Excel using Analytic Solver software. Please answer the following two sub questions.

(a) Describe all controllable (or decision) variables, if any, fixed model parameters and random inputs (or variables). For random variables, please explicitly define their distribution types and the associated parameters for the distributions identified.

(b) Write the algebraic expression (function) that defines how the daily profit is calculated as a function of the controllable inputs, the fixed model parameters, and the random variables. If you would like to use any notation to write out the algebraic expression of the forecast/result function, please clearly define the notation used.

Question

J.J. Bakery in Irvine sells a special kind of bread. The daily number of customers arriving at the store approximately follows a normal distribution with a mean of 100 and a standard deviation of 10. There is a 0.30 chance that a customer buys one load of sourdough bread, a 0.10 chance that a customer buys two loaves, and a 0.60 chance that a customer does not purchase bread.

The bread is baked once a day at 6 A.M. (before observing demand) and sold between 9 A.M and 9 P.M (when demand comes in). Each loaf of bread costs $1.00 to produce and sells for $3.00. Unsold loaves of bread at the end of the day are donated and the store gets a tax credit of $0.50 per loaf. Given the information, the store needs to determine the production quantity for each day’s bakery and estimate the daily profit given the chosen production quantity. You do not need to implement the problem in Excel using Analytic Solver software. Please answer the following two sub questions.

(a) Describe all controllable (or decision) variables, if any, fixed model parameters and random inputs (or variables). For random variables, please explicitly define their distribution types and the associated parameters for the distributions identified.

(b) Write the algebraic expression (function) that defines how the daily profit is calculated as a function of the controllable inputs, the fixed model parameters, and the random variables. If you would like to use any notation to write out the algebraic expression of the forecast/result function, please clearly define the notation used.

BlackTom AI · Accepted Answer

The task presents a bakery planning problem with distinct types of inputs. Below I restate the scenario and then break down (a) the decision/fixed/random components and (b) the algebraic expression for daily profit.

First, restating the scenario in my own words:
- The store decides how many loaves to bake each day (the production quantity Q). This is the primary controllable decision variable.
- Daily demand is uncertain. The number of customers arriving per day follows a normal distribution with mean 100 and standard deviation 10. Each customer independently purchases 0, 1, or 2 loaves with probabilities 0.60, 0.30, and 0.10, respectively. The actual total demand D is the total number of loaves purchased by all customers that day.
- Costs and revenues:
  - It costs $1.00 to produce each loaf (cost = Q). 
  - Each loaf that is sold brings in $3.00 (revenue from sold loaves). 
  - Any unsold loaf at the end of the day is donated, and the store receives a tax credit of $0.50 for each donated loaf (i.e., for each unsold loaf, a credit of 0.50).
- Time frame: bread is baked at 6 AM and sold during the day; demand realization occurs during the selling window.

(a) Controllable (decision) variables, fixed model parameters, and random inputs
- Controllable (decision) variable:
  - Q: the number of loaves to produce each day. This is the principal decision variable that the store can set before demand is realized.

- Fixed model parameters (constants that do not depend on the random realization):
  - Prices and costs: selling price per loaf p = $3.00 and production cost per loaf c = $1.00.
  - Tax credit per donated loaf t = $0.50.
  - Demand model structure parameters: the per-customer probabilities P(X = 0) = 0.60, P(X = 1) = 0.30, P(X = 2) = 0.10, which define the distribution of how many loaves a single customer buys.
  - Customer arrival distribution parameters: N (the number of customers) is Normal with mean μ_N = 100 and standard deviation σ_N = 10. Note that N is a continuous-distribution model for the count of customers; in practice, you would work with a discrete approximation or a carefully defined interpretation, since the true number of customers is an integer.

- Random inputs (random variables, with their distributions):
  - N: the number of customers in the day, distributed as Normal(μ_N = 100, σ_N = 10). This represents the daily flow of potential buyers.
  - For each customer i = 1,…,N, let X_i denote the number of loaves purchased by that customer, with the distribution: P(X_i = 0) = 0.60, P(X_i = 1) = 0.30, P(X_i = 2) = 0.10. These X_i are independent and identically distributed given N (i.e., the purchase behavior is the same for each customer and does not depend on the total number of customers).
  - The total daily demand D is the sum of loaves purchased by all customers: D = X_1 + X_2 + ... + X_N. D is a random variable that depends on the realized N and the realized X_i values. In general this is a compound distribution (a random number of i.i.d. purchases per customer).

(b) Algebraic expression (function) for daily profit as a function of inputs
- Let Q be the production quantity (decision variable).
- Let D be the realized daily demand (random variable as described above).
- The number of loaves sold is min(D, Q) (you can only sell up to the amount you produced, and you cannot sell more than the demand).
- The number of loaves donated (unsold) is max(Q − D, 0).
- Profit for the day combines revenue from sold loaves, production costs, and the tax credit for donated loaves. A compact expression is:
  Profit(Q, D) = 3 · min(D, Q) − 1 · Q + 0.50 · max(Q − D, 0)
- This expression can be expanded in the two regions for intuition:
  - If D ≥ Q (enough demand to use all produced loaves):
    Sold = Q, Unsold = 0, Revenue = 3Q, Cost = Q, Credit = 0, so Profit = 3Q − Q = 2Q.
  - If D < Q (not enough demand to use all produced):
    Sold = D, Unsold = Q − D, Revenue = 3D, Cost = Q, Credit = 0.50 · (Q − D), so Profit = 3D − Q + 0.50(Q − D) = 2.5D − 0.5Q.
- In a compact, notation-clean form suitable for optimization or simulation, you can keep the single-figure expression:
  Profit(Q, D) = 3 · min(D, Q) − Q + 0.5 · max(Q − D, 0).
- If you want to express the daily profit as a function of the controllable input Q and the random inputs (via D, which itself depends on N and the X_i), you could write:
  Profit(Q; N, {X_i}) = 3 · min(D, Q) − Q + 0.5 · max(Q − D, 0), where D = ∑_{i=1}^N X_i and N ~ Normal(100, 10^2), X_i ~ with P(0)=0.60, P(1)=0.30, P(2)=0.10 for i = 1,…,N and the X_i are iid independent of N.
- Notational clarity you might use in a formal write-up:
  - Q ∈ {0,1,2, …} is the production quantity.
  - N ∼ Normal(μ_N, σ_N^2) with μ_N = 100, σ_N = 10.
  - For i = 1,…,N, X_i ∈ {0,1,2} with P(X_i = 0) = 0.60, P(X_i = 1) = 0.30, P(X_i = 2) = 0.10, independent of N and of other X_j.
  - D = ∑_{i=1}^N X_i is the total number of loaves demanded (purchased) that day.
  - The daily profit is Profit(Q, D) = 3 · min(D, Q) − Q + 0.5 · max(Q − D, 0).

Notes and possible extensions for a deeper understanding:
- The combination of N being Normal and the per-customer purchase distribution implies D has a compound distribution. In practice, one might approximate D with a normal or use simulation (Monte Carlo) to estimate expected profit for a given Q.
- Since the tax credit only applies to donated loaves, the credit only appears in the region where production exceeds demand (D < Q).
- If you were to compute expected daily profit E[Profit(Q, D)] for planning, you would integrate over the joint distribution of N and {X_i}, or simulate draws of N and the X_i to estimate E[Profit] for each candidate Q.
- For brevity, the key algebraic relation to memorize for the daily profit given realized D and planned Q is Profit(Q, D) = 3 · min(D, Q) − Q + 0.5 · max(Q − D, 0).

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