S-Problems/Soutions

S-Problems/Soutions

S-Problems/Soutions
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Problems

1 . Given this LP problem:

Maximize Image180

Image181
Image182
Image183

Solve the problem using the graphical method:

Plot and label the constraints.
Indicate the feasible solution space.
Plot the objective function and use it to identify the optimum point of the graph.
Use simultaneous equations to determine the optimal values of Image184 , and Image185
Compute the optimum value of the objective function.

2. Solve problem #1 by using an EXCEL spreadsheet.

How do you enter the objective function?
How do you enter the constraints?
What are the optimal values of Image186 and Image187 ?
What is the value of the objective function?
How much slack does each constraint, A and B, have?

3. Given this LP problem:

Minimize Image188

Image189
Image190
Image191
Image192

Solve this problem using the graphical method.

Plot and label the constraints.
Indicate the feasible solution space.
Plot the objective function and use it to identify the optimum vertex.
Uvse simultaneous equations to determine the optimal values of Image193 and Image194 .
Compute the optimal value of the objective function.

4. Solve Problem #3 by using an EXCEL spreadsheet.

How do you enter the objective function?
How do you enter the constraints?
What are the optimal values of Image186 and Image187 ?
What is the value of the objective function?
How much surplus does each constraint, A , B, and C have?

5. A dietitian in a hospital is required to devise a recipe for a food which will provide at least the following amounts of vitamins: 500 units of vitamin A. 500 units of vitamin B. and 700 units of vitamin C. The dietitian may use three ingredients, P, Q, and R in the recipe which are described below. At least one ounce of ingredient R must be used in the recipe.

Units per Ounce:

Ingredient	A	B	C	Cost per Ounce
P	20	30	60	$0.30
Q	60	30	0	$0.20
R	10	50	30	$0.15

Is this a problem in maximization or minimization?
What are the decision variables?
What is the objective function?
What are the constraints?

6. Solve problem #6 on an EXCEL spreadsheet.

How do you enter the objective function?
How do you enter the constraints?
How much of each ingredient should be used in the recipe?
How large is one serving of this food, in kilograms? In ounces? (I kg. = 2.2 lbs., or 35.2 oz.)
What is the cost of one serving?
How much extra of each vitamin does this recipe provide?

7. Refer to your EXCEL spreadsheet for Problem #6.

What is the shadow prices for each constraint?
If it is necessary to economize on the recipe, which should be reduced: the requirements for vitamin A. or for vitamin B. or for vitamin C?
Find the range of feasibility for each constraint.
What is the interpretation of the range of feasibility?
Find the range of optimality for each coefficient in the objective function.
What is the interpretation of the range of optimality?

8. The Fairley and Winn Company produces hockey sticks, hockey pucks, and baseball bats; it is necessary to plan the production schedule for next week. The hockey sticks, hockey pucks, and baseball bats are made of oak, of which the company has 600 board feet. A hockey stick requires 4 board feet, a puck requires 2 board feet, and a baseball bat requires 3 board feet. The company has a power saw for cutting the oak boards into the appropriate pieces; a hockey stick requires 3.0 minutes, a puck requires 1.5 minutes, and a baseball bat requires 1.5 minutes. The power saw is expected to be available for 3.6 hours next week. After cutting, the pieces of work in process are hand finished in the finishing department, which consists of 4 skilled and experienced craftsmen, each of whom can complete any of the products. A hockey stick requires 60 minutes of finishing, a puck requires 30 minutes, and a baseball bat requires 90 minutes. The finishing department is expected to operate for 40 hours next week. Hockey sticks sell for $29.95 and have a unit variable cost of $17.95; a puck sells for $11.95 and has a unit variable cost of $4.95; a baseball bat sells for $16.95 and has a unit variable cost of $8.95. The company has an order for two dozen hockey sticks which must be filled. It wishes to produce no more than 50 hockey pucks. Determine the linear programming form of this problem.

Is this a problem in maximization or minimization?
What are the decision variables? Suggest symbols for them.
What is the objective function?
What are the constraints?

9. Solve Problem #8 on an EXCEL spreadsheet.

How do you enter the objective function?
How do you enter the constraints?
How much of each product should be produced?
How much profit will be generated by selling all of the output?
How much oak will be unused?
How much idle time will there be in the finishing department?
How much idle time will there be in the finishing department?

10. Use your EXCEL spreadsheet in Problem #9 to solve this problem.

What is the shadow price of each constraint?
If it is necessary to produce more baseball bats, would more oak, more saw time, or more finishing time be required?
What is the range of feasibility of each constraint?
What are the interpretations of the ranges of feasibility?
What is the range of optimality of each objective function coefficient?
What are the interpretations of the ranges of optimality?

Solutions

1. a,b,c.

d. Image195 ; Image196 .
e. Z = 4(4) + 3(4.67) = $30.00.

2.

Image197

a. In cell E3: =B3*C7+C3*C8.
b In cell E4: =B4*C7+C4*C8.
In cell E5: =B5*C7+C5*C8.

(It is not necessary to enter the non-negativity constraints.)
Hint: Format your answer cells. Click on Format; Click on Cells; Select Number and 3 or 4 decimal places.

c. Click on Tools; Click on Solver. Set Target Cell $E$3. Maximize. By Changing Cells $C$7,$C$8.
Click on Add. Cell Reference $E$4, <=, 84 (for constraint A). Click on OK.
Click on Add. Cell Reference $E$5, <=, 84 (for constraint B). Click on OK.
Click on Solve. Image198 ; Image199 .

d. In the SOLVER box, click on the answer report. Did you get the same answer as in Problem #1?
e. Both constraints equal their RHS and have no slack.

3. a,b,c.

d. Image200 ; Image201 .
e. Z = 5(35) + 8(15) = $295.00.

4.

Image202

a. In cell E3: =B3*C8+C3*C9.
b. In cell E4: =B4*C8+C4*C9 for constraint A
In cell E5: =B5*C8+C5*C9 for constraint B
In cell E6: =B6*C8+C6*C9 for constraint C.

(It is not necessary to enter the non-negativity constraints.)
Hint: Format your answer cells. Click on Format; Click on Cells; Select Number and 3 or 4 decimal places.

c. Click on Tools; Click on Solver. Set Target Cell $E$3. Minimize. By Changing Cells $C$8,$C$9.
Click on Add. Cell Reference $E$4, >=, 100 (for constraint A). Click on OK.
Click on Add. Cell Reference $E$5, >=, 700 (for constraint B). Click on OK
Click on Add. Cell Reference $E$6, >=, 15 (for constraint C). Click on OK
Click on Solve. Image203 ; Image204 .

d. In the SOLVER box click on the answer report Did you get the same answer as in Problem #3?
e. Constraints A and C equal their RHS and have no surplus; constraint B has a surplus of 175.

Note: EXCEL calls this "slack," but since this is a minimization problem you must interpret it as a surplus.

5. a. Since the problem contains information about the cost of the ingredients, it will involve minimization.
b. The dietician can decide how much of each ingredient to use in the recipe. Use P ,Q and R as symbols.
c. Minimize Z = .30P + .20Q + .15R.
d. The food must provide certain quantities of vitamins.
A) 20P + 60Q + 10R > 500 units
B) 30P + 30Q + 50R > 500 units
C)60P + 0Q + 30R > 700 units
D) R > 1

6.

Image205

a. In cell F3: B3*C9+C3*C10+D3*C11.
b. In Cell F4: B4*C9+C4*C10+D4*C11
etc.
c. Click on Tools; click on Solver; click on options; select assume linear model; click on OK. Set Target Cell $F$3. Minimize. By Changing Cells $C$9,$C$10,$C$11.

Click on Add. Cell Reference $F$4, >=, 500 (for constraint A). Click on OK.
Click on Add. Cell Reference $F$5,>=, 500 (for constraint B). Click on OK
Click on Add. Cell Reference $F$6, >=, 700 (for constraint C). Click on OK
Click on Add. Cell Reference $F$7, >=, 1 (for constraint D). Click on OK.
Click on Solve. P = 11.17, Q = 4.44, R = 1.

d. In the SOLVER RESULTS box, select the ANSWER report. Use 11.17 oz. of P, 4.44 oz. of Q, and 1 oz. of R. The recipe will make 16.61 oz.
e. One batch costs $4.39.
f. The recipe provides no extra vitamin A or vitamin C, but 18.33 extra units of vitamin B.

Note: EXCEL calls this "slack," but since this is a minimization problem you must interpret it as a surplus.

a. In the SOLVER RESULTS box, select the SENSITIVITY report.
b. Vitamin C has the largest shadow price ($.0039). If the RHS of the C constraint could reduced by one unit, to 699, the value of the objective function (and the cost of the recipe) would be reduced by almost 4 cents.
c On the SENSITIVITY report: the allowable increase and decrease for the RHS of each constraint is given. The RHS of vitamin A may increase by 1E+30; read this as infinity; it may increase without limit. It may decrease by 36.67 to 463.33. Obtain the other ranges in the same manner.
d. Within the range of feasibility, the shadow price indicates the effect on the value of the objective function of a one unit change in the RHS of the constraint.
(The shadow price of vitamin B is zero. An increase in the RHS up to 18.33 or any decrease will have no effect on the value of the objective function. You may use the SOLVER facility to test this result by changing the RHS of the B constraint.)
e. For the coefficient of ingredient P, the allowable increase is 0 and the allowable decrease is .23 to .07. Handle the rest of the coefficients in the same manner.
f. Within the range of optimality, the values of the decision variables in the solution will not change.

8. a. Since selling prices are given, it is a problem in maximization.
b. The management can decide how many hockey sticks, hockey pucks, and baseball bats to produce next week. We suggest using H = Hockey sticks, P = Hockey pucks, B = baseball bats. You may have thought of others.
c. Maximize Z = 12H + 7P + 8B.
d. Oak: 4H + 2P + 3B < 600 board feet

Saw: .05H + .025P + .025B < 3.6 hours
Finishing: H + .5P + 1.5B < 160 hours
Sticks: H > 24
Pucks: P < 50.

9.

Image206

a. In cell F3: B3*C10+C3*C11+D3*C12.
b. In cell F4: B4*C10+C4*C11+D4*C12
etc.
c. Click on Tools; click on Solver; click on options; select assume linear model; click on OK. Set Target Cell $F$3. Maximize. By Changing Cells $C$10,$C$11,$C$12.
Click on Add. Cell Reference $F$4, >=, 600 (for oak). Click on OK.
Click on Add. Cell Reference $F$5, >=, 3.6 (for the saw). Click on OK
Click on Add. Cell Reference $F$6, >=, 160 (forfinishing). Click on OK
Click on Add. Cell Reference $F$7, >=, 24 (for hockey sticks). Click on OK.
Click on Add. Cell Reference $F$8, <, 50 (for hockey pucks). Click on OK
Click on Solve. Produce 24 hockey sticks, 50 hockey pucks, and 56 baseball bats.
d. In the SOLVER RESULTS box, select the ANSWER report. The profit will be $1,086.00.
e. There will be 236 board feet of oak left over.
f. There will be no free time on the saw.
g. There will be 27 unused hours of finishing time.

10 a. In the SOLVER RESULTS box, select the SENSITIVITY report.
b. The saw has the largest shadow price ($320). If the RHS of the saw constraint could be increased, the value of the objective function (and the profits) would be increased by $320.per hour.
c. On the SENSITIVITY report: the allowable increase and decrease for the RHS of each constraint is given. The RHS of the saw may increase up to .45 of and hour (27 minutes) and may decrease infinitely. Obtain the other ranges in the same manner.
d. Within the range of feasibility, the shadow price indicates the effect on the value of the objective function of a one unit change in the RHS of the constraint.
(The shadow price of oak is zero. Any increase in the RHS or a decrease of 236 board feetwill have no effect on the value of the objective function. You may use the SOLVER facility to test this result by changing the RHS of the B constraint.)
e. For the coefficient of hockey pucks, the allowable increase is $0.75 to $8.75 and the allowable decrease is $200 to $6.00. Obtain the other ranges in the same manner.
f. Within the range of optimality, the values of the decision variables in the solution will not change.

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Home > Chapter 6 > S-Problems/Soutions

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	Course-wide Content
		Data Sets Excel Templates PowerWeb Web Links Operations Management ... Interactive OM Updates and Errata

	Instructor Resources
		Instructors Manual PowerPoint Presentations

	News, Articles & Links
		News Feeds PowerWeb Articles Weekly Update Archive
	Quizzes
		Multiple Choice Quiz True or False Self Quiz from CD
	Interactive Exercises
		Interactive OM
	More Resources
		Problems/Solutions Glossary Supplement to Chapter S-Multiple Choice Quiz S-True or False S-Problems/Soutions S-Glossary S-Self Quiz from CD