Round Tree Manor is a hotel that provides two types of rooms with three rental classes

1. Problem 8-21 (Algorithmic)

Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:

Rental Class
Room		Super Saver	Deluxe	Business
	Type I	$32	$43	—
	Type II	$17	$35	$39

Type I rooms do not have wireless Internet access and are not available for the Business rental class.

Round Tree's management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 120 rentals in the Super Saver class, 70 rentals in the Deluxe class, and 55 rentals in the Business class. Round Tree has 105 Type I rooms and 120 Type II rooms.

1. 1. Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types.

Variable	# of reservations
SuperSaver rentals allocated to room type I
SuperSaver rentals allocated to room type II
Deluxe rentals allocated to room type I
Deluxe rentals allocated to room type II
Business rentals allocated to room type II

1. 1. 
      Is the demand by any rental class not satisfied?
      
      No 
      
      Explain.
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       
   2. How many reservations can be accommodated in each rental class?

Rental Class	# of reservations
SuperSaver
Deluxe
Business

1. 1.  
   2. Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5, should this incentive be offered?
      
       
   3. With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a Type I or a Type II room?
      
       
      
      Why?
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       
   4. Could the linear programming model be modified to plan for the allocation of rental demand for the next night?
      
       
      
      What information would be needed and how would the model change?
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       
2. Problem 8-29 (Algorithmic)

La Jolla Beverage Products is considering producing a wine cooler that would be a blend of a white wine, a rose wine, and fruit juice. To meet taste specifications, the wine cooler must consist of at least 50% white wine, at least 20% and no more than 30% rose, and exactly 20% fruit juice. La Jolla purchases the wine from local wineries and the fruit juice from a processing plant in San Francisco. For the current production period, 11000 gallons of white wine and 6500 gallons of rose wine can be purchased; an unlimited amount of fruit juice can be ordered. The costs for the wine are $1 per gallon for the white and $1.5 per gallon for the rose; the fruit juice can be purchased for $0.5 per gallon. La Jolla Beverage Products can sell all of the wine cooler it can produce for $3 per gallon.

1. 1. Is the cost of the wine and fruit juice a sunk cost or a relevant cost in this situation?
      
       
      
      Explain.
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       
   2. Formulate a linear program to determine the blend of the three ingredients that will maximize the total profit contribution. Solve the linear program to determine the number of gallons of each ingredient La Jolla should purchase and the total profit contribution it will realize from this blend. If required, round your answers to one decimal place. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank.

	Let W = gallons of white wine
	Let R = gallons of rose wine
	Let F = gallons of fruit wine

1. 1.  

Max	W	+	R	+	F
s.t.
	W	+	R	+	F	≥		% white
	W	+	R	+	F	≥		% rose minimum
	W	+	R	+	F	≤		% rose maximum
	W	+	R	+	F	=		% fruit juice
	W					≤		Available white
			R			≤		Available rose
W, R, F ≥ 0

1. 1.  

Optimal Solution:
W
R
F

1. 1. 
      Profit Contribution = 
   2. If La Jolla could obtain additional amounts of the white wine, should it do so?
      
       
      
      If so, how much should it be willing to pay for each additional gallon, and how many additional gallons would it want to purchase?
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       
   3. If La Jolla Beverage Products could obtain additional amounts of the rosé wine, should it do so?
      
       
      
      If so, how much should it be willing to pay for each additional gallon, and how many additional gallons would it want to purchase?
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       
   4. Interpret the shadow price for the constraint corresponding to the requirement that the wine cooler must contain at least 50% white wine. What is your advice to management given this shadow price?
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       
   5. Interpret the shadow price for the constraint corresponding to the requirement that the wine cooler must contain exactly 20% fruit juice. What is your advice to management given this shadow price?
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       

 

 

Problem 7-15

Par, Inc., is a small manufacturer of golf equipment and supplies. Par’s distributor believes a market exists for both a medium-priced golf bag, referred to as a standard model, and a high-priced golf bag, referred to as a deluxe model. The distributor is so confident of the market that, if Par can make the bags at a competitive price, the distributor will purchase all the bags that Par can manufacture over the next three months. A careful analysis of the manufacturing requirements resulted in the following table, which shows the production time requirements for the four required manufacturing operations and the accounting department’s estimate of the profit contribution per bag:

 

The director of manufacturing estimates that 630 hours of cutting and dyeing time, 600 hours of sewing time, 708 hours of finishing time, and 135 hours of inspection and packaging time will be available for the production of golf bags during the next three months.

Select the correct graph that shows the optimal solution if Par's management encounters the following independent situations:

 

1. The accounting department revises its estimate of the profit contribution for the deluxe bag to $18 per bag. Is it i  ii  iii iv

 

A new low-cost material is available for the standard bag, and the profit contribution per standard bag can be increased to $20 per bag. (Assume that the profit contribution of the deluxe bag is the original $9 value.) Is it i  ii  iii iv

 

 

New sewing equipment is available that would increase the sewing operation capacity to 750 hours. (Assume that 10A + 9B is the appropriate objective function.) Is it i  ii  iii iv

 

1. Problem 8-25 (Algorithmic)

Georgia Cabinets manufactures kitchen cabinets that are sold to local dealers throughout the Southeast. Because of a large backlog of orders for oak and cherry cabinets, the company decided to contract with three smaller cabinetmakers to do the final finishing operation. For the three cabinetmakers, the number of hours required to complete all the oak cabinets, the number of hours required to complete all the cherry cabinets, the number of hours available for the final finishing operation, and the cost per hour to perform the work are shown here:

	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
Hours required to complete all the oak cabinets	47	40	27
Hours required to complete all the cherry cabinets	64	52	36
Hours available	40	30	35
Cost per hour	$34	$41	$52

For example, Cabinetmaker 1 estimates that it will take 47 hours to complete all the oak cabinets and 64 hours to complete all the cherry cabinets. However, Cabinetmaker 1 only has 40 hours available for the final finishing operation. Thus, Cabinetmaker 1 can only complete 40/47 = 0.85, or 85%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 40/64 = 0.63, or 63%, of the cherry cabinets if it worked only on cherry cabinets.

1. 1. Formulate a linear programming model that can be used to determine the proportion of the oak cabinets and the proportion of the cherry cabinets that should be given to each of the three cabinetmakers in order to minimize the total cost of completing both projects.

Let	O1 = proportion of Oak cabinets assigned to cabinetmaker 1
	O2 = proportion of Oak cabinets assigned to cabinetmaker 2
	O3 = proportion of Oak cabinets assigned to cabinetmaker 3
	C1 = proportion of Cherry cabinets assigned to cabinetmaker 1
	C2 = proportion of Cherry cabinets assigned to cabinetmaker 2
	C3 = proportion of Cherry cabinets assigned to cabinetmaker 3

1. 1.  

Min

O1

+

O2

+

O3

+

C1

+

C2

+

C3

s.t.

O1

C1

≤

Hours avail. 1

O2

+

C2

≤

Hours avail. 2

O3

+

C3

≤

Hours avail. 3

O1

+

O2

+

O3

=

Oak

C1

+

C2

+

C3

=

Cherry

O1, O2, O3, C1, C2, C3 ≥ 0

1. 1.  
   2. Solve the model formulated in part (a). What proportion of the oak cabinets and what proportion of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost of completing both projects? If required, round your answers for the proportions to three decimal places, and for the total cost to two decimal places.

	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
Oak	O1 =	O2 =	O3 =
Cherry	C1 =	C2 =	C3 =

1. 1. 
      Total Cost = $  
   2. If Cabinetmaker 1 has additional hours available, would the optimal solution change?
      
       
      
      Explain.
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       
   3. If Cabinetmaker 2 has additional hours available, would the optimal solution change?
      
       
      
      Explain.
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       
   4. Suppose Cabinetmaker 2 reduced its cost to $38 per hour. What effect would this change have on the optimal solution? If required, round your answers for the proportions to three decimal places, and for the total cost to two decimal places.

	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
Oak	O1 =	O2 =	O3 =
Cherry	C1 =	C2 =	C3 =

1. 1. 
      Total Cost = $  
      
      Explain.
      
      The input in the box below will not be graded, but may be reviewed and considered by your instructor.
       
2. Problem 8-19

Better Products, Inc., manufactures three products on two machines. In a typical week, 40 hours are available on each machine. The profit contribution and production time in hours per unit are as follows:

 

Two operators are required for machine 1; thus, 2 hours of labor must be scheduled for each hour of machine 1 time. Only one operator is required for machine 2. A maximum of 100 labor-hours is available for assignment to the machines during the coming week. Other production requirements are that product 1 cannot account for more than 50% of the units produced and that product 3 must account for at least 20% of the units produced.

1. 1. How many units of each product should be produced to maximize the total profit contribution?

Product	# of units
1
2
3

1. 1. 
      What is the projected weekly profit associated with your solution?
      
      $  
   2. How many hours of production time will be scheduled on each machine? If required, round your answers to two decimal places.
      
      Machine Hours Schedule: Machine 1:  Hours; Machine 2:  Hours
   3. What is the value of an additional hour of labor? If required, round your answers to two decimal places.
      
      $  
   4. Assume that labor capacity can be increased to 120 hours. Develop the optimal product mix, assuming that the extra hours are made available.

Product	# of units
1
2
3

1. 1. 
      Profit: $  
      
      Would you be interested in using the additional 20 hours available for this resource?
      
       
2.  

Problem 7-25 (Algorithmic)

George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the inheritance George finally decides to commit to the trust fund, he wants to invest at least 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%.

1. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives. If required, round your answers to three decimal places.

Let B	=	percentage of funds invested in the bond fund
S	=	percentage of funds invested in the stock fund

1.  

Max	B	+	S
s.t.
	B			≥		Bond fund minimum
	B	+	S			Minimum return
	B	+	S			Percentage requirement

1.  
2. Solve the problem using the graphical solution procedure. If required, round the answers to one decimal place.
   
   Optimal solution: B = , S = 
   
   Value of optimal solution is %