Assignment # 2

QUESTION ONE

Decision Variables

Let,

* X1 = number of full-time tellers

* Y1 = number of part-time tellers starting at 9 a.m. (leaving at 1 p.m.) * Y2 = number of part-time tellers starting at 10 a.m. (leaving at 2 p.m.) * Y3 = numbers of part-time tellers starting at 11 a.m. (leaving at 3 p.m.) * Y4 = number of part-time tellers starting at noon ( leaving at 4 p.m.) * Y5 = number of part-time tellers starting at 1 p.m. (leaving at 5 p.m.)

Objective Function

* MIN 90X1 + 28(Y1 + Y2 + Y3 + Y4 + Y5)

Contraints

* X1 + Y1 ≥ 10 (9am-10am)

* 0.5X1 + Y1 + Y2 ≥ 12 (10am – 11am)

* 0.5X1 + Y1 + Y2 + Y3 ≥14 (11am – noon) * X1 + Y1 + Y2 +Y3 + Y4 ≥ 16 (noon – 1pm)

* X1 + Y1 + Y2 + Y3 + Y4 + Y5 ≥ 18 (1pm - 2pm)

* X1 + Y3 + Y4 + Y5 ≥ 17 (2pm – 3pm)

* X1 + Y4 + Y5 ≥ 15 (3pm – 4pm)

* X1 + Y5 ≥ 10 (4pm – 5pm)

* X <12

* X1, Y1, Y2, Y3, Y4, Y5 ≥ 0

Part-time workers cannot work more than 50% of the total hours required each day. Therefore, 4(Y1+Y2+Y3+Y4+Y5) ≤ 0.50(10+12+14+16+18+17+15+10)

Optimal Solution

* X1 = 10

* Y1 = 0

* Y2 = 7

* Y3 = 2

* Y4 = 5

* Y5 = 0

Optimal Value

$ 1,292 is the optimal value to minimize the total cost of employees working.

QUESTION TWO

A)

Decision Variables

Let,

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

Objective Function

* Min 1800O1 + 1764O2 + 1650O3 + 2160C1 + 2016C2 + 1925C3 Contraints

* 50O1 + 60C1 ≤40

* 42O2 + 48C2≤30

* 30)3 + 35C3≤35

* O1 + O2 + O3=1

* C1 + C2 + C3=1

* O1, O2, O3, C1, C2, C3≥0

B)

| Cabinet Maker 1| Cabinet Maker 2| Cabinet Maker 3| Oak| 0.271| 0| 0.729|

Cherry| 0| 0.625| 0.375|

Therefore, to complete both projects is assigning 27.1% of the oak cabinets to Cabinet Maker 1 and 72.9% to Cabinet Maker 2. For the Cherry, 62.5% should be assigned to Cabinet Maker 2 and 37.5% to Cabinet Maker 3. Total cost of completing both projects = $ 3672.500

C) Cabinetmaker 1 has a slack of 26.458 hours and a dual price of 0. This increases the right hand side of constraint 1 will not change the value of the optimal solution.

D) The dual price for constraint 2 is 1.75. The current Right-Hand-Side is 30 and allowable increase is 11.143. The upper limit on Right-Hand-Side range is 41.143. This means for each extra hour of time for Cabinet Maker 2, the total cost will decrease by $1.75/hour, to a maximum of 41.143 hours.

E) If cabinetmaker 2 reduces his cost to $38 per hour, the new objective function coefficients for O2 and C2 would be: 42(38) = 1596 and 48(38) = 1824. The optimal solution does not change but the total cost decreases to $3552.50.

QUESTION THREE

Decision Variables

Let,

* S =t the amount invested in stocks

* B = the amount invested in bonds

* M = the amount invested in mutual funds

* C = the amount invested in Cash

* R = the amount of Risk

* AR= annual return

Objective Functions

* R = 0.8S + 0.2B +0.3M + 0.0C

* AR = 0.1S + 0.03B + 0.04M + 0.01C

Constraints

* 10 ≤ C ≤ 30

* M ≥ B

* S ≤ 75

* S, B, M, C ≥ 0

A) R = 0.8S + 0.2B + 0.3M + 0C

R = 0.8(0.409) + 0.2(0.145) + 0.3(0.145) + 0(0.3)

R = 0.3997

Therefore the optimal allocation for this scenario is as follows; S = 40.9%, B = 14.5%, M = 14.5%, C = 30% which will create the optimal amount of risk, 0.3997,...