1. Frandec Company manufactures, assembles, and rebuilds material handling equipment used in warehouses and distribution centers. One product, called a Liftmaster, is assembled from four components: a frame, a motor, two supports, and a metal strap. Frandecs production schedule calls for 5000 Liftmasters to be made next month. Frandec purchases the motors from an outside supplier, but the frames, supports, and straps may be either manufactured by the company or purchased from an outside supplier. Manufacturing and purchase costs per unit are shown.

Component	Manufacturing Cost	Purchase Cost
Frame	$31.00	$44.00
Support	$8.50	$12.00
Strap	$5.50	$6.50

Three departments are involved in the production of these components. The time (in minutes per unit) required to process each component in each department and the available capacity (in hours) for the three departments are as follows:

Department
Component	Cutting	Milling	Shaping
Frame	2.8	1.8	2.4
Support	1.1	1.4	1.9
Strap	0.7		1.4
Capacity (hours)	330	390	640

1. Formulate and solve a linear programming model for this make-or-buy application. How many of each component should be manufactured and how many should be purchased?

   Let

   FM = number of frames manufactured

   FP = number of frames purchased

   SM = number of supports manufactured

   SP = number of supports purchased

   TM = number of straps manufactured

   TP = number of straps purchased

2. 1. Min	FM	+	FP	+	SM	+	SP	+	TM	+	TP
      s.t.
      2.8FM	+	SM	+	TM
      FM	+	1.4SM
      FM	+	SM	+	1.4TM
      FM	+	FP
      SM	+	SP
      TM	+	TP
      FM, FP, SM, SP, TM, TP 0

      If required, round your answers to the nearest whole number.

      Frames

      Supports

      Straps

   2. What is the total cost of the manufacturing and purchasing plan? When required, round your answer to the nearest dollar.$ _______
   3. How many hours of production time are used in each department? If required, round your answer to two decimal places.

      Cutting	__?__hours
      Milling	___?__hours
      Shaping	___?___hours

   4. How much should Frandec be willing to pay for an additional hour of time in the shaping department? If required, round your answer to two decimal places.$_______ because there is ____(surplus/slack) of hours.
   6. AnothermanufacturerhasofferedtosellframestoFrandecfor$38each.CouldFrandecimproveitspositionbypursuingthisopportunity?YesorNo?Whyorwhynot?Ifrequired,roundyouranswerstothenearestcent.Becausethecurrentpurchasepriceis$______.Thereducedcostis$______whichmeansthatthesolutionmaybeimprovedifthecostis$_____orbelow.

2.

Western Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the steakhouse operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4-hour shifts during peak meal times. On Saturdays the steakhouse is open from 11:00 A.M. to 10:00 P.M. Management wants to develop a schedule for part-time employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $7.60 per hour. The total number of full-time and part-time employees needed varies with the time of day as shown.

One full-time employee comes on duty at 11:00 A.M., works 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00 P.M. and works the same 4-hours-on, 1-hour-off, 4-hours-on pattern.

1. Develop a minimum-cost schedule for part-time employees.Optimal schedule for part-time employees:

   11:00 A.M.
   12:00 P.M.
   1:00 P.M.
   2:00 P.M.
   3:00 P.M.
   4:00 P.M.
   5:00 P.M.
   6:00 P.M.

2. What is the total payroll for the part-time employees? If required, round your answer to the nearest dollar.Total daily salary cost = $ ___How many part-time shifts are needed?Use the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts.The input in the box below will not be graded, but may be reviewed and considered by your instructor.
3. Assume that part-time employees can be assigned either a 3-hour or a 4-hour shift. Develop a minimum-cost schedule for the part-time employees.Optimal schedule for part-time employees:

   11:00 A.M.
   12:00 P.M.
   1:00 P.M.
   2:00 P.M.
   3:00 P.M.
   4:00 P.M.
   5:00 P.M.
   6:00 P.M.
   7:00 P.M.

   How many part-time shifts are needed?What is the cost savings compared to the previous schedule? If required, round your answer to the nearest cent.Total cost reduced to $____ .

3.

Romans Food Market, located in Saratoga, New York, carries a variety of specialty foods from around the world. Two of the stores leading products use the Romans Food Market name: Romans Regular Coffee and Romans DeCaf Coffee. These coffees are blends of Brazilian Natural and Colombian Mild coffee beans, which are purchased from a distributor located in New York City. Because Romans purchases large quantities, the coffee beans may be purchased on an as-needed basis for a price 11% higher than the market price the distributor pays for the beans. The current market price is $0.48 per pound for Brazilian Natural and $0.69 per pound for Colombian Mild. The compositions of each coffee blend are as follows:

Brazilian Natural	70%	40%
Colombian Mild	30%	60%

Romans sells the Regular blend for $3.5 per pound and the DeCaf blend for $4.4 per pound. Romans would like to place an order for the Brazilian and Colombian coffee beans that will enable the production of 900 pounds of Romans Regular coffee and 450 pounds of Romans DeCaf coffee. The production cost is $0.77 per pound for the Regular blend. Because of the extra steps required to produce DeCaf, the production cost for the DeCaf blend is $1.09 per pound. Packaging costs for both products are $0.25 per pound. Formulate a linear programming model that can be used to determine the pounds of Brazilian Natural and Colombian Mild that will maximize the total contribution to profit.

Let	BR = pounds of Brazilian beans purchased to produce Regular
BD = pounds of Brazilian beans purchased to produce DeCaf
CR = pounds of Colombian beans purchased to produce Regular
CD = pounds of Colombian beans purchased to produce DeCaf

If required, round your answers to four decimal places. For subtractive or negative numbers use a minus sign even if there is a plus sign before the blank. (Example: -300)

The complete linear program is

Max	+	+	+
s.t.
+	=
+	=
=
+	=
BR, BD, CR, CD 0

What is the optimal solution and what is the contribution to profit? If required, round your answer to the nearest whole number.

Optimal solution:

BR =

BD =

CR =

CD =

If required, round your answer to the nearest cent.

Value of the optimal solution = $ _______

4.

Greenville Cabinets received a contract to produce speaker cabinets for a major speaker manufacturer. The contract calls for the production of 3,700 bookshelf speakers and 4,500 floor speakers over the next two months, with the following delivery schedule:

Model	Month 1	Month 2
Bookshelf	2,300	1,400
Floor	1,700	2,800

Greenville estimates that the production time for each bookshelf model is 0.7 hour and the production time for each floor model is 1 hour. The raw material costs are $12 for each bookshelf model and $14 for each floor model. Labor costs are $24 per hour using regular production time and $35 using overtime. Greenville has up to 2,600 hours of regular production time available each month and up to 1,200 additional hours of overtime available each month. If production for either cabinet exceeds demand in month 1, the cabinets can be stored at a cost of $7 per cabinet. For each product, determine the number of units that should be manufactured each month on regular time and on overtime to minimize total production and storage costs. If required, round your answers to the nearest whole number.

Regular Time Manufacture
Model	Month 1	Month 2
Bookshelf
Floor

Overtime Manufacture
Model	Month 1	Month 2
Bookshelf
Floor

5.

1. As part of the settlement for a class action lawsuit, Hoxworth Corporation must provide sufficient cash to make the following annual payments (in thousands of dollars):

   1	2	3	4	5	6
   170	210	250	310	340	500

   The annual payments must be made at the beginning of each year. The judge will approve an amount that, along with earnings on its investment, will cover the annual payments. Investment of the funds will be limited to savings (at 3.25% annually) and government securities, at prices and rates currently quoted in The Wall Street Journal.Hoxworth wants to develop a plan for making the annual payments by investing in the following securities (par value = $1000). Funds not invested in these securities will be placed in savings.

   1	$1075	6.55	3
   2	$1000	5.125	4

   Assume that interest is paid annually. The plan will be submitted to the judge and, if approved, Hoxworth will be required to pay a trustee the amount that will be required to fund the plan.
   1. Use linear programming to find the minimum cash settlement necessary to fund the annual payments.

      Let

      F= total funds required to meet the six years of payments

      G₁= units of government security 1

      G₂= units of government security 2

      S_i= investment in savings at the beginning of year i

      Note: All decision variables are expressed in thousands of dollars. If required, round your answers to four decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)

      Min	F
      s.t.
      F	+	G1	+	G2	+	S1	=
      G1	+	G2	+	S1	+	S2	=
      G1	+	G2	+	S2	+	S3	=
      G1	+	G2	+	S3	+	S4	=
      G2	+	S4	+	S5	=
      S5	+	S6	=

      Round your answer to the nearest dollar.

      Current investment required	$
      Investment in government security 1	$
      Investment in government security 2	$
      Investment in savings for year 1	$
      Investment in savings for year 2	$
      Investment in savings for year 3	$
      Investment in savings for year 4	$
      Investment in savings for year 5	$
      Investment in savings for year 6	$

   2. Use the shadow price to determine how much more Hoxworth should be willing to pay now to reduce the payment at the beginning of year 6 to $400,000. Round your answer to the nearest dollar. $
   3. Use the shadow price to determine how much more Hoxworth should be willing to pay to reduce the year 1 payment to $150,000. Round your answer to the nearest dollar. Hoxworth should be willing to pay anything less than $.
   4. Supposethattheannualpaymentsaretobemadeattheendofeachyear.Reformulatethemodeltoaccommodatethischange. Note:Alldecisionvariablesareexpressedinthousandsofdollars. Ifrequired,roundyouranswerstofourdecimalplaces.Forsubtractiveornegativenumbersuseaminussignevenifthereisa+signbeforetheblank.(Example:-300)

      Min	F
      s.t.
      1)	F	+	G1	+	G2	+	S1	=
      2)	G1	+	G2	+	S1	+	S2	=
      3)	G1	+	G2	+	S2	+	S3	=
      4)	G1	+	G2	+	S3	+	S4	=
      5)	G2	+	S4	+	S5	=
      6)	S5	+	S6	=
      7)	S6	+	S7	=

      HowmuchwouldHoxworthsaveifthischangecouldbenegotiated?Roundyouranswertothenearestdollar. $________