# Quantity Discount Contract

Questions 1-3 are related to the **Variable-Cost Contract**:

1. Develop a descriptive model that shows, for any given demand level and any quantity ordered by Predicto, the following measures: (i) Predicto’s profit; (ii) InTech’s profit; (iii) the total profit of Predicto and InTech. In order to do so:

Clearly list all inputs (i.e., decision variables), parameters, and outputs (objective function) and assign them a symbol if necessary. [2 pts]

Hint: You can refer to Art’s newsstand for an example.

Express the algebraic relationship between inputs/parameters and constraints/outputs/objective function. [6 pts]

Hint: Please list ALL relevant equations. You can refer to Art’s newsstand for an example.

2. Implement the model you developed in Q1. In the cell representing the demand, set its value to generate a random demand from a normal distribution with a mean of 10,000 and a standard deviation of 4,000. Do not worry about non-integer values, but use the function =max(mean+std*NORMINV(RAND(),0,1),0), to ensure that you do not get negative demand values. Run the simulation for an order quantity of 8,000 units, and provide a 99% confidence interval for Predicto’s expected profit, InTech’s expected profit, and total expected profit. Consider 5,000 runs of simulation. [12 pts]

Instruction: Please clearly indicate different component of your model (parameters, inputs, calculation, output). Seven points will be given for correctness and five points will be given for clarity of your implementation.

3. Create a table (or a graph) in which you show the __estimated expected profits__ (Predicto’s, InTech’s, and total) for all possible order quantities (5,000, 5,500, 6,000, …, 11,500, 12,000). Based on your estimates, what should be Predicto’s __optimal__ order quantity if it wanted to maximize its expected profit? [7 pts]

Instruction: Five points will be given for correctness and two points will be given for clarity of your implementation. You may include your answer to Q2 and Q3 in the same spreadsheet.

Questions 4-5 are related to the **Quantity Discount Contract**:

4. Modify your benchmark model in Q1 to represent the new setting. Specifically, list new parameters and adjust the algebraic relationship between inputs/parameters and constraints/outputs/objective function to incorporate all-unit quantity discount. We are still interested in (i) Predicto’s profit; (ii) InTech’s profit; (iii) the total profit of Predicto and InTech. [5 pts]

Questions 4-5 are related to the **Quantity Discount Contract**:

5. Implement your model in Q4 and repeat the analysis of Q3. What would be Predicto’s optimal order quantity? Would it be different at all from that in the benchmark case? What would be the expected profits (Predicto’s, InTech’s, and total) under the above quantity discount contract? Should InTech offer such contract? [12 pts]

Instruction: Please answer the above questions in your Excel file. In addition, report a 99% confidence interval for Predicto’s expected profit, InTech’s expected profit, and total expected profit for the optimal order quantity. Seven points will be given for correctness and five points will be given for clarity of your implementation.

Questions 6-7 are related to **Revenue Sharing Contract**

6. Modify your benchmark model in Q1 to represent the new setting. Specifically, update the list of parameters and adjust the algebraic relationship between inputs/parameters and constraints/outputs/objective function to incorporate the revenue sharing specifications. We are still interested in (i) Predicto’s profit; (ii) InTech’s profit; (iii) the total profit of Predicto and InTech. [5 pts]

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