Chapter 20 Exercises

Step-by-Step Exercises

Optimum Mix Under One Constraint

Rank the products from highest to lowest in order of throughput margin per unit of constrained resource.

Question 1

Parets, Inc. produces four products in a single department that has 4,000 hours of available time per week. The products have the following sales price per unit, direct materials cost per unit, and required time per unit:

	Product A	Product B	Product C	Product D
Sales price per unit	$14.00	$18.00	$12.00	$20.00
DM cost per unit	$12.00	$16.20	$8.50	$16.25
Required time	1 hour	1.2 hours	2 hours	1.5 hours

Rank the products from highest to lowest in order of throughput margin per unit of constrained resource.

Question 2

Millway Chemicals sells five different products, all of which require a chemical (XMZ) produced by the company. Millway can only produce a limited amount of the chemical per year. Information about the five products follows:

• Formula J sells for $25 per unit and costs $7 per unit for materials other than XMZ. Formula J requires 6 ounces of XMZ per unit.
• Formula Q sells for $28 per unit and costs $12 per unit for materials other than XMZ. Formula Q requires 4 ounces of XMZ per unit.
• Formula V sells for $18 per unit and costs $11 per unit for materials other than XMZ. Formula V requires 3.5 ounces of XMZ per unit.
• Formula W sells for $32 per unit and costs $10 per unit for materials other than XMZ. Formula W requires 2 ounces of XMZ per unit.
• Formula Y sells for $27 per unit and costs $15 per unit for materials other than XMZ. Formula Y requires 5 ounces of XMZ per unit.

Rank the products from highest to lowest in order of throughput margin per unit of constrained resource.

Question 3

Goldfinch, Inc. manufactures three products, each of which requires time in the Finishing department, which has 200 hours available per week. If Goldfinch were to completely fill demand for their products, they would require 350 hours of Finishing time per week. Soliloquy sells for $240 and requires $100 in direct materials. Phalanx sells for $320 and requires $80 in direct materials. Huzzah sells for $280 and requires $120 in direct materials. Finishing hours required are 5 hours per unit for Soliloquy, 10 hours per unit for Phalanx, and 8 hours per unit for Huzzah.

Rank the products from highest to lowest in order of throughput margin per unit of constrained resource.

Determine the units of each product in the optimum product mix.

Question 4

Jumbler Enterprises only has 600 machine hours per month to manufacture their three products:

• Basic has throughput margin per machine hour of $250 and requires 1 machine hour per unit. Monthly demand is 400 units.
• Mediocre has throughput margin per machine hour of $320 and requires 2 machine hours per unit. Monthly demand is 150 units.
• Superior has throughput margin per machine hour of $400 and requires 3 machine hours per unit. Monthly demand is 80 units.

Determine the units of each product in the optimum product mix.

Question 5

 Marston Corporation has 400 direct labor hours available per week. Each of their four products requires 4 hours to manufacture. Throughput margin per direct labor hour and demand are as follows:

• Model A: Throughput margin of $75 per machine hour and demand of 25 units per week
• Model B: Throughput margin of $105 per machine hour and demand of 30 units per week
• Model C: Throughput margin of $80 per machine hour and demand of 35 units per week
• Model D: Throughput margin of $90 per machine hour and demand of 40 units per week

Determine the units of each product in the optimum product mix.

Question 6

Bob & Fred’s Mowing has the resources to mow 400,000 square yards of lawn per month. They serve three kinds of customers: Residential, Commercial, and Golf. Information on the throughput margin per square yard and demand follows:

• Commercial: Throughput margin is $1.30 per square yard, and demand is for 250,000 square yards per month
• Golf: Throughput margin is $1.50 per square yard, and demand is for 100,000 square yards per month
• Residential: Throughput margin is $0.90 per square yard, and demand is for 300,000 square yards per month

Determine the units of each product in the optimum product mix.

Optimum Mix Under Multiple Constraints

State the objective function.

Question 7

Archer, Inc. manufactures two products: Basic and Deluxe. Throughput margin is $60 per unit for Basic and $90 per unit for Deluxe.

State the objective function.

Question 8

Tobico produces three products: Product A, Product B, and Product C. The throughput margin per unit is $9 for Product A, $7 for Product B, and $12 for Product C.

State the objective function.

Question 9

Ethamax sells two energy drinks: Zing and Wow. Zing’s throughput margin per unit is $0.50, and Wow’s throughput margin per unit is $1.20.

State the objective function.

Set up inequalities for each of the constraints.

Question 10

Porter Masonry makes two kinds of mortar, Basic and Strong, which are processed in two departments (Mixing and Packaging) and sold to contractors by the batch. Basic requires 30 minutes per batch in Mixing and 20 minutes in Packaging. Strong requires 1 hour per batch in Mixing and 15 minutes in Packaging. The Mixing department has 120 hours available per week, and the Packaging department has 80 hours available per week.

Set up inequalitites for each of the constraints.

Question 11

FeltFeast manufactures two kinds of felt: Regular and Adhesive. Each bolt of Regular felt takes 1 hour in the Processing Department and 1 hour in the Packaging Department. Each bolt of Adhesive takes 2 hours in the Processing Department and ½ hour in the Packaging department. The Processing Department has 800 hours available per week, and the Packaging Department has 600 hours available per week.

Set up inequalities for each of the constraints.

Question 12

Haughton Chess manufactures high-quality wooden chess sets in two processes: Turning and Staining. Haughton’s Grandmaster model requires 4 hours in Turning and 2 hours in Staining. The Master model requires 3 hours Turning and 3 hours in Staining. Both the Turning process and the Staining process run 40 hours per week.

Set up inequalities for each of the constraints.

Find the corner points of the area of feasible solutions.

Question 13

Each department’s constraints can be expressed as follows:

• Department A: 2X + 2Y < 2,000
• Department B: X + 4Y < 2,200

Find the corner points of the area of feasible solutions.

Question 14

Each department’s constraints can be expressed as follows:

• Department A: .5X + Y < 1,125
• Department B: X + .25Y < 500

Find the corner points of the area of feasible solutions.

Question 15

Each department’s constraints can be expressed as follows:

• Department A: 4X + 2Y < 1,800
• Department B: X + 2Y < 1,500

Find the corner points of the area of feasible solutions.

Plug the corner points into the objective function to determine which product mix earns the most throughput margin.

Question 16

16.The corner points of the area of feasible solutions are as follows:

• (100, 0)
• (0, 400)
• (50, 250)

The objective function is Max $40X + $50Y.

Plug the corner points into the objective function to determine which product mix earns the most throughput margin.

Question 17

The corner points of the area of feasible solutions are as follows:

• (2,000, 0)
• (0, 1,500)
• (1,000, 750)

The objective function is Max $200X + $130Y.

Plug the corner points into the objective function to determine which product mix earns the most throughput margin.

Question 18

The corner points of the area of feasible solutions are as follows:

• (18,000, 0)
• (0, 20,000)
• (15,000, 10,000)

The objective function is Max $4X + $3Y.

Plug the corner points into the objective function to determine which product mix earns the most throughput margin.

Complete Problems

Optimum Mix Under One Constraint

Question 19

Chao Manufacturing produces three products using one machine, which has 400 hours available for production each week. The following table presents the weekly demand, sales price, direct materials cost, and production time per unit for each of Chao’s three products:

	Product A	Product B	Product C
Weekly demand	50	100	75
Sales price	$7.50	$10.00	$4.00
Direct materials cost	$1.50	$5.00	$3.00
Production time	3 hours	2 hours	1 hour

Determine the optimum product mix.

Question 20

Magnificent Artworks manufactures all goods by hand, employing 200 skilled artisans who work 8 hours per day, 5 days per week. The following table presents the weekly demand, sales price, direct materials cost, and direct labor time per unit for each of Magnificent’s five products:

Determine the optimum product mix.

Question 21

Stuffies, Inc. manufactures stuffed animals, all of which must be stuffed on the company’s one stuffing machine. The machine operates 40 hours per week. The following table presents the weekly demand, sales price, direct materials cost, and stuffing time for each of the three sizes of stuffed animals manufactured at Stuffies.

	Small	Medium	Large
Weekly demand	500	200	120
Sales price	$15	$30	$100
Direct materials cost	5	15	20
Stuffing time	2 minutes	5 minutes	8 minutes

Determine the optimum product mix.

Optimum Mix Under Multiple Constraints

Question 22

Jenco manufactures two products (Whoosh and Swish) in two departments (Molding and Packaging). Molding has 1,000 hours available per week and Packaging has 1,500 hours available per week. Demand for both products is unlimited. Throughput margin is $80 per unit for Swish. Whoosh requires ½ hour per unit in Molding and ¼ hour per unit in Packaging. Swish requires ¼ hour per unit in Molding and ¾ hour per unit in Packaging.

Determine the optimum product mix if the throughput margin per unit for Whoosh is $20.

Determine the optimum product mix if the throughput margin per unit for Whoosh is $50.

Question 23

Nappo Company produces two products (Super and Ultra) in two departments. Department A has 2,000 hours available per week, and Department B has 1,600 hours available. Demand for both products is unlimited. The throughput margin, Department A time and Department B time, for each product are as follows:

	Super	Ultra
Throughput margin per unit	?	$70
Department A time	4 hours	2 hours
Department B time	2 hours	2 hours

a. Determine the optimum product mix if Super’s throughput margin per unit is $90.

b. Determine the optimum product mix if Super’s throughput margin per unit is $150.

Question 24

Swanni Company produces two products, Optimo and Ultimo (throughput margin = $5 per unit), in two departments, Assembly and Packaging. Assembly has 900 hours available each week, while Packaging has 800 hours available. Each unit of Optimo requires 1 hour in Assembly and 2 hours in Packaging, while each unit of Ultimo requires 3 hours in Assembly and 1 hour in Packaging. Demand is unlimited for both products.

a. Determine the optimum product mix if the throughput margin per unit for Optimo is $12 per unit.

b. Determine the optimum product mix if the throughput margin per unit for Optimo is $6 per unit.

Assignment Problem

Note: Check figures are not provided for assignment problems so your instructor may use them for homework.

Question 25

Goss, Inc. has the capability to manufacture five products, and has 1,500 labor hours available each week for production. Per-unit information about the five products is as follows:

	Sales Price	DM Cost	Production Time	Demand
Product A	$50	$10	1 hour	350
Product B	$35	$15	0.5 hours	350
Product C	$70	$20	2 hours	500
Product D	$100	$30	3.5 hours	250
Product E	$20	$5	0.25 hours	300

Determine the optimum product mix given current production constraints.

Challenge Problem

Question 26

Minerva Manufacturing produces three products, Platinum, Silver and Gold, each of which must be processed in Department A, Department B, and Department C. Customers demand 10 units of Platinum, 11 units of Silver, and 9 units of Gold per day.

Each of the three departments employs five workers, who each work eight hours per day. The only constraint in each department is labor time. Each unit of Platinum requires two hours of processing in Department A, two hours and 42 minutes of processing in Department B, and 24 minutes of processing in Department C. Each unit of Silver requires an hour of processing in both Department A and Department B, and three hours of processing in Department C. Each unit of Gold requires two hours of processing in both Department A and Department C, and one hour of processing in Department B.

Each unit of Platinum sells for $6,000 and requires $1,000 in direct materials. Each unit of Silver sells for $570 and requires $120 in direct materials. Each unit of Gold sells for $600 and requires $240 in direct materials.

Determine the optimum product mix given current production constraints.

Pre-Assessment Problems

Use these problems to check whether you are fully prepared for the assessment. Work the problems under assessment conditions – don’t use any notes or other materials!

Question 27

Delatronics, Inc. manufactures four products: Alpha, Beta, Gamma, and Epsilon. All four products are processed in the Manufacturing department, then Alpha and Beta are processed further in the Assembly department and the Finishing department. The demand, selling price, direct material cost, and time (in minutes) required in each department is listed for each product in the table below:

Product	Demand	Selling Price	DM Cost	Manufacturing Time	Assembly Time	Finishing Time
Alpha	500	$65	$20	20 min.	15 min.	30 min.
Beta	600	$75	$15	30 min.	20 min.	10 min.
Gamma	1,100	$40	$10	10 min.	0	0
Epsilon	1,500	$45	$20	10 min.	0	0

The Manufacturing Department has a capacity of 700 machine hours per week, but capacity is not constrained in the Assembly and Finishing departments.

Determine the optimum product mix.

Question 28

Ubertone, Inc. manufactures two products (Basic and Deluxe) in a two-step process. First, the goods are manufactured in the Machining department, which has 120 hours available per week. Next, they are polished and packaged in the Finishing department, which has 40 hours available per week. Each unit of Basic requires two machine hours in the Machining department and one direct labor hour in the Finishing department. Each unit of Deluxe requires three machine hours in the Machining department and ½ labor hour in the Finishing department. Each unit of Basic sells for $150 and uses direct materials that cost $60. Each unit of Deluxe sells for $200 and uses direct materials that cost $75. Ubertone can sell up to 50 units of each product per week.

Determine the optimum product mix and total throughput margin at the optimum point.