GCU Interpretations of The Slack Variables Risk Management Worksheet

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, 9000 gallons of white wine and 7500 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. Is the cost of the wine and fruit juice a sunk cost or a relevant cost in this situation?

   Sunk costRelevant cost

   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 wineLet R = gallons of rose wineLet F = gallons of fruit wineMaxfill in the blank 3W+fill in the blank 4R+fill in the blank 5Fs.t.fill in the blank 6W+fill in the blank 7R+fill in the blank 8F?fill in the blank 9% whitefill in the blank 10W+fill in the blank 11R+fill in the blank 12F?fill in the blank 13% rose minimumfill in the blank 14W+fill in the blank 15R+fill in the blank 16F?fill in the blank 17% rose maximumfill in the blank 18W+fill in the blank 19R+fill in the blank 20F=fill in the blank 21% fruit juicefill in the blank 22W?fill in the blank 23Available whitefill in the blank 24R?fill in the blank 25Available roseW, R, F ? 0Optimal Solution:Wfill in the blank 26Rfill in the blank 27Ffill in the blank 28
   Profit Contribution = fill in the blank 29
3. If La Jolla could obtain additional amounts of the white wine, should it do so?

   YesNo

   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. If La Jolla Beverage Products could obtain additional amounts of the rosé wine, should it do so?

   YesNo

   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.

5. 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.

6. 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.

(S) Bshaji:

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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, 9000 gallons of white wine and 7500 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. Is the cost of the wine and fruit juice a sunk cost or a relevant cost in this situation?

   Sunk costRelevant cost

   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 wineLet R = gallons of rose wineLet F = gallons of fruit wineMaxfill in the blank 3W+fill in the blank 4R+fill in the blank 5Fs.t.fill in the blank 6W+fill in the blank 7R+fill in the blank 8F?fill in the blank 9% whitefill in the blank 10W+fill in the blank 11R+fill in the blank 12F?fill in the blank 13% rose minimumfill in the blank 14W+fill in the blank 15R+fill in the blank 16F?fill in the blank 17% rose maximumfill in the blank 18W+fill in the blank 19R+fill in the blank 20F=fill in the blank 21% fruit juicefill in the blank 22W?fill in the blank 23Available whitefill in the blank 24R?fill in the blank 25Available roseW, R, F ? 0Optimal Solution:Wfill in the blank 26Rfill in the blank 27Ffill in the blank 28
   Profit Contribution = fill in the blank 29
3. If La Jolla could obtain additional amounts of the white wine, should it do so?

   YesNo

   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. If La Jolla Beverage Products could obtain additional amounts of the rosé wine, should it do so?

   YesNo

   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.

5. 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.

6. 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 7-41

Southern Oil Company produces two grades of gasoline: regular and premium. The profit contributions are $0.30 per gallon for regular gasoline and $0.50 per gallon for premium gasoline. Each gallon of regular gasoline contains 0.3 gallons of grade A crude oil and each gallon of premium gasoline contains 0.6 gallons of grade A crude oil. For the next production period, Southern has 18,000 gallons of grade A crude oil available. The refinery used to produce the gasolines has a production capacity of 50,000 gallons for the next production period. Southern Oil’s distributors have indicated that demand for the premium gasoline for the next production period will be at most 20,000 gallons.

1. Formulate a linear programming model that can be used to determine the number of gallons of regular gasoline and the number of gallons of premium gasoline that should be produced in order to maximize total profit contribution. If required, round your answers to two decimal places.
   Let R=number of gallons of regular gasoline producedP=number of gallons of premium gasoline producedMaxMinfill in the blank 2 R+fill in the blank 3 Ps.t.fill in the blank 4 R+fill in the blank 5 P??=fill in the blank 7Grade A crude oil availablefill in the blank 8 R+fill in the blank 9 P??=fill in the blank 11Production capacityfill in the blank 12 P??=fill in the blank 14Demand for premiumR, P??=fill in the blank 16
2. What is the optimal solution?
   Gallons of regular gasolinefill in the blank 17Gallons of premium gasolinefill in the blank 18Total profit contribution$fill in the blank 19
3. What are the values and interpretations of the slack variables?

   ConstraintValue of Slack Variable
   Interpretation1fill in the blank 20All available grade A crude oil is usedAll available grade A crude oil is not used2fill in the blank 22Total production capacity is usedTotal production capacity is not used3fill in the blank 24Premium gasoline production is 10,000 gallons less than the maximum demandPremium gasoline production is 10,000 gallons greater than the maximum demand

4. What are the binding constraints?
   Grade A crude oil availableBindingNon-bindingProduction capacityBindingNon-bindingDemand for premiumBindingNon-binding

   Problem 7-49 (Algorithmic)

PharmaPlus operates a chain of 30 pharmacies. The pharmacies are staffed by licensed pharmacists and pharmacy technicians. The company currently employs 100 full-time-equivalent pharmacists (combination of full time and part time) and 175 full-time-equivalent technicians. Each spring management reviews current staffing levels and makes hiring plans for the year. A recent forecast of the prescription load for the next year shows that at least 280 full-time-equivalent employees (pharmacists and technicians) will be required to staff the pharmacies. The personnel department expects 10 pharmacists and 30 technicians to leave over the next year. To accommodate the expected attrition and prepare for future growth, management states that at least 15 new pharmacists must be hired. In addition, PharmaPlus’s new service quality guidelines specify no more than two technicians per licensed pharmacist. The average salary for licensed pharmacists is $35 per hour and the average salary for technicians is $15 per hour.

1. Determine a minimum-cost staffing plan for PharmaPlus. How many pharmacists and technicians are needed?
   Let P=number of full-time equivalent pharmacistsT=number of full-time equivalent techniciansMaxMinfill in the blank 2 P+fill in the blank 3 Ts.t.fill in the blank 4 P+fill in the blank 5 T??=fill in the blank 7Full-time-equivalent employeesfill in the blank 8 P-fill in the blank 9 T??=fill in the blank 11Quality guidelinefill in the blank 12 P??=fill in the blank 14Number of pharmacists
   The optimal solution requires fill in the blank 15 full-time equivalent pharmacists and fill in the blank 16 full-time equivalent technicians. The total cost is $fill in the blank 17 per hour.
2. Given current staffing levels and expected attrition, how many new hires (if any) must be made to reach the level recommended in part (a)?
   New Hires RequiredPharmacistsfill in the blank 18Techniciansfill in the blank 19
   What will be the impact on the payroll?

   The payroll cost will go upgo down by $ fill in the blank 21 per hour.

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