OSCM-471571_Homework1

OSCM 471/571 Optimization and Decision Support Modeling for Business Homework 1, Spring 2024 Notice for Homework 1 Instructor: Seokjun Youn ( syoun@email.arizona.edu )  Due date: Thursday 2/1, 11:59 pm o Please submit your files to D2L > Assignments > Homework 1 1. A Word file (or PDF) with your answers combined into a single document. 2. An Excel spreadsheet template with your answers (for some sub-questions).  This homework is made up of THREE questions (20 pts): o Q1: 10 sub-questions (8 pts) o Q2: 4 sub-questions (6 pts) o Q3: 2 sub-questions (6 pts)  Students may choose either handwriting or word processing (or both). o Handwriting: please properly scan or take photos and organize them into one file before uploading in D2L.  Please write down your solutions step-by-step for partial credit.  You may use: o Your textbook and notes from the class. o Notes or sources from a related class or internet source. o Discussion with the instructor. o Voluntary, mutual, and cooperative discussion with other students currently taking the class.  You may not use: o Solution manuals (printed or electronic). o Copying from other students in this class, including expecting them to reveal their solutions in “discussion.”  It is fine if your answer is not 100% correct. However, if you do not put enough effort to the assignment, your score for this homework will be lower than your expectation. So, please try to convince your logic to instructor. 1/11

OSCM 471/571 Optimization and Decision Support Modeling for Business Homework 1, Spring 2024 Naman Jain: 1. The Whitt Window Company is a company with only three employees that makes two different kinds of handcrafted windows: a wood-framed and an aluminum framed window. They earn $60 profit for each wood-framed window and $30 profit for each aluminum- framed window. Doug makes the wood frames and can make 6 per day. Linda makes the aluminum frames and can make 4 per day. Bob forms and cuts the glass and can make 48 square feet of glass per day. Each wood-framed window uses 6 square feet of glass and each aluminum-framed window uses 8 square feet of glass. The company wishes to determine how many windows of each type to produce per day to maximize total profit. a. Construct and fill in a table for this problem like the table of the Wyndor Glass Co. problem during the class, identifying both the activities and the resources. Answer: The Whitt Window Company's situation parallels that of the Wyndor Glass Co. challenge, as both require optimizing limited resources for maximum profitability. In the Wyndor case, the constraints were the production capacities of Plants 1, 2, and 3. Conversely, for Whitt Window Company, the limiting factors are the quantities of glass, wood, and aluminum available for use. Below is a table that aligns with the framework of Table 2.1 from the Wyndor Glass Co. case study, clearly outlining the activities (window production) and the constrained resources: Resources Wood-framed Window Aluminum-framed Window Resource Availability Glass (sq ft) 6 8 48 Wood (units/day) 1 0 6 Aluminum (units/day) 0 1 4 Unit profit $60 $30 In this structured approach, we translate the production of wood-framed and aluminum- framed windows into activities, while treating glass, wood, and aluminum as the resources that are subject to availability limits. This setup is instrumental in determining the optimal production mix to maximize the company’s daily profit, given the material and labor constraints. b. Identify verbally the decisions to be made, the constraints on these decisions, and the overall measure of performance for the decisions. 2/11

OSCM 471/571 Optimization and Decision Support Modeling for Business Homework 1, Spring 2024  Maximize Profit:  Profit = $60W + $30A  Subject to:  Glass production : 6W + 8A ≤ 48  Wood-framed production: W ≤ 6  Aluminum-framed production: A ≤ 4  Non-negativity constraints:  W ≥ 0  A ≥ 0 Identify the decision variables, objective function, nonnegativity constraints, functional constraints, and parameters in both the algebraic version and spreadsheet version of the model.  Decision Variables:  W = the number of wood-framed windows to produce  A = the number of aluminum-framed windows to produce  Constraints:  Available production capacity in each of the resources (functional constraints):  Glass: 6W + 8A ≤ 48  Wood: W ≤ 6  Aluminum: A ≤ 4  Non-negativity constraints:  W ≥ 0  A ≥ 0  Objective Function:  Profit = $60W + $30A  Total Profit (Excel Formula):  = SUMPRODUCT(B2:C2,B5:C5)  Parameters:  The profit per unit for wood-framed ($60) and aluminum-framed ($30) windows. f. Use the graphical method to solve this model. Answer: Corner Points:  (0, 0) : P = $ 0  (0, 4 ): P = $ 120  (6,0): P = $ 360  (6,4 ): P = $ 480− optimal solution 4/11

OSCM 471/571 Optimization and Decision Support Modeling for Business Homework 1, Spring 2024 g. A new competitor in town has started making wood-framed windows as well. This may force the company to lower the price it charges and so lower the profit made for each wood-framed window. How would the optimal solution change (if at all) if the profit per wood-framed window decreases from $60 to $40? From $60 to $20? Answer: If the profit per wood-framed window decreases from $60 to $40, we must re-evaluate the objective function while keeping the constraints the same. Original Objective Function: Profit=$60 W +$30 A New Objective Function with $40 Profit for Wood-Framed: Profit=$40 W +$30 A Further Decrease to $20 Profit for Wood-Framed: Profit=$20 W +$30 A Impact on Optimal Solution: 1. Decrease to $40:  The reduced profit margin for wood-framed windows may shift the optimal production balance towards producing more aluminum-framed windows since they become relatively more profitable.  We need to recalculate the profits at each corner point of the feasible region to find the new optimal solution. 2. Decrease to $20:  With a significant decrease to $20, the optimal solution is likely to favor producing aluminum-framed windows even more, as the profit from wood- framed windows is now much lower.  Again, the profits at the corner points of the feasible region must be recalculated to find the optimal solution. In both scenarios, the optimal solution will change if the profit from wood-framed windows becomes less than the profit from producing the maximum number of aluminum-framed windows within the glass constraint. We would calculate the profits at all feasible corner points with the new profit values to determine the optimal solution. h. Doug is considering lowering his working hours, which would decrease the number of wood frames he makes per day. How would the optimal solution change if he only makes 5 wood frames per day? Answer: If Doug decreases his working hours, resulting in a maximum production of 5 wood-framed windows per day instead of 6, the constraints in the linear programming model will change. Original Wood-framed Production Constraint: W ≤6 New Wood-framed Production Constraint with Doug's Reduced Hours: W ≤5 The optimal solution may change as follows: 5/11

OSCM 471/571 Optimization and Decision Support Modeling for Business Homework 1, Spring 2024 2. The Primo Insurance Company is introducing two new product lines: special risk insurance and mortgages. The expected profit is $5 per unit on special risk insurance and $2 per unit on mortgages. Management wishes to establish sales quotas for the new product lines to maximize total expected profit. The work requirements are shown below: a. Identify verbally the decisions to be made, the constraints on these decisions, and the overall measure of performance for the decisions. Answer: Decisions to be Made:  Determine the number of units to sell for special risk insurance.  Determine the number of units to sell for mortgages. Constraints on These Decisions:  The Underwriting department can provide a maximum of 2,400 work-hours, and they require 3 hours per unit of special risk insurance and 2 hours per unit for mortgages.  The Administration department has 800 work-hours available and requires 1 hour per unit for mortgages (and none for special risk insurance).  The Claims department has 1,200 work-hours available and requires 2 hours per unit of special risk insurance (and none for mortgages). Overall Measure of Performance:  The measure of performance is the total expected profit, with a profit of $5 per unit for special risk insurance and $2 per unit for mortgages. The objective is to maximize this total profit. b. Convert these verbal descriptions of the constraints and the measure of performance into quantitative expressions in terms of the data and decisions. Answer: Decision Variables:  Let S represent the number of units sold for special risk insurance.  Let M represent the number of units sold for mortgages. Objective Function:  Maximize total expected profit: Profit=5 S +2 M Constraints:  Underwriting Department Constraint: 3 S +2 M ≤2400 work-hours 7/11

OSCM 471/571 Optimization and Decision Support Modeling for Business Homework 1, Spring 2024  Administration Department Constraint: M ≤800 work-hours (since they only work on mortgages and require 1 hour per unit)  Claims Department Constraint: 2 S ≤1200 work-hours (since they only work on special risk insurance and require 2 hours per unit) Non-Negativity Constraints:  S ≥0 (Cannot sell a negative number of insurance units)  M ≥0 (Cannot sell a negative number of mortgage units) These expressions provide the linear programming model's mathematical structure to maximize profit while adhering to the constraints imposed by the available work- hours in each department. c. Formulate and solve a linear programming model for this problem on a spreadsheet.  Please include the screenshot of your final spreadsheet model here. Answer: d. Formulate this same model algebraically. Answer: Certainly, here's the linear programming problem reformulated for clarity: Let:  R represent the number of sales quotas for special risk insurance.  M represent the number of sales quotas for mortgages. Objective: Maximize the total profit Profit , which is calculated as Profit =5 R +2 M Constraints: 1. The total work-hours for underwriting should not exceed their availability, so: 3 R +2 M ≤2400 2. The total work-hours for administration are dedicated to mortgages only and should not exceed their availability, so: M ≤800 3. The total work-hours for claims are dedicated to special risk insurance only and should not exceed their availability, so: 2 R ≤1200 Non-negativity Constraints: 4. The sales quotas for special risk insurance must be non-negative, so: R ≥0 8/11

OSCM 471/571 Optimization and Decision Support Modeling for Business Homework 1, Spring 2024 3. The Learning Center runs a day camp for 6-10 year olds during the summer. Its manager, Elizabeth Reed, is trying to reduce the center’s operating costs to avoid having to raise the tuition fee. Elizabeth is currently planning what to feed the children for lunch. She would like to keep costs to a minimum, but also wants to make sure she is meeting the nutritional requirements of the children. She has already decided to go with peanut butter and jelly sandwiches, and some combination of apples, milk, and/or cranberry juice. The nutritional content of each food choice and its cost are given in the table that accompanies this problem. The nutritional requirements are as follows. Each child should receive between 300 and 500 calories, but no more than 30 percent of these calories should come from fat. Each child should receive at least 60 milligrams (mg) of vitamin C and at least 10 grams (g) of fiber. To ensure tasty sandwiches, Elizabeth wants each child to have a minimum of 2 slices of bread, 1 tablespoon (tbsp) of peanut butter, and 1 tbsp of jelly, along with at least 1 cup of liquid (milk and/or cranberry juice). Elizabeth would like to select the food choices that would minimize cost while meeting all these requirements. a. Formulate and solve a linear programming model for this problem on a spreadsheet.  Please include the screenshot of your final spreadsheet model here. Answer: b. Formulate this same model algebraically. 10/11

OSCM 471/571 Optimization and Decision Support Modeling for Business Homework 1, Spring 2024 Answer: Let:  B be the number of bread slices to use and buy.  PB be the amount of peanut butter to use per tablespoon.  J be the amount of jelly to use per tablespoon.  A be the number of apples to use and buy.  M be the amount of milk given per cup.  CJ be the amount of cranberry juice given per cup. Objective: Minimize the total cost ( Cost ), which is calculated as: Cost =0.06 B +0.05 PB +0.08 J +0.35 A +0.20 M +0.40 CJ  Total calories needed:  300 ≤ 80 B +100 PB +70 J +90 A +120 M +110 J ≤ 5 00  Calories from Fat needed: 15 B +80 PB +60 M ≤ 30% ×TotalCalories  Vitamin C (mg) needed: 4 J +6 A +2 M +80 CJ ≥ 60  Fiber (g) needed: 4 B +3 J +10 A + CJ ≥ 10  Slices of bread needed: B≥ 2  Amount of peanut butter (tbsp) needed: PB≥ 1  Amount of jelly (tbsp) needed: J ≥ 1  Liquid requirement: M + CJ≥ 1  B≥ 0  PB≥ 0  J ≥ 0  A ≥ 0  M≥ 0  CJ ≥ 0 11/11