Web Mercantile sells many household products through an online catalog. The company needs substantial warehouse space for storing its goods. Plans now are being made for leasing warehouse storage space over the next 5 months. Just how much space will be required in each of these months is known. However since these space requirements are quite different, it may be most economical to lease only the amount needed to each month on a month by month basis. On the other hand the additional cost for leasing space for additional months is much less than for the first month, so it may be less expensive to lease the maximum amount needed for the entire 5 months. Another option is the intermediate approach of changing the total amount of space leased(by adding a new lease and/or having an old lease expire) at least once but not every month. The space requirement and the leasing costs for the various leasing periods are as follows:

Month Required space (Sq. Ft)

1 30,000

2 20,000

3 40,000

4 10,000

5 50,000

Leasing period(Months) Cost per Sq. Ft leased

1 $65

2 $100

3 $135

4 $160

5 $190

The objective is to minimize the total leasing cost for meeting the space requirements.

A) Formulate a linear programming model for this problem

B) Solve this model by the simplex method

PLEASE SEE AND REFERENCE THE ABOVE PROBLEM WITH SOLUTION IN ORDER TO SOLVE THE FOLLOWING

-Redefine the decision variable Xij = Area rented in the beginning of the month i for duration of j months

-set up the constraints as equalities

a) compare and comment on the result of the model with (>=) type of constraints

b) validate one shadow price and one reduced cost of your choice

PLEASE SHOW ALL STEPS

Let xi; = the amount of space leased in month i for a period of j months, for i=1,2,3,4,5 and j=1,...,6-i. The objective is to minimize total cost of leasing while satisfying all space weekly requirements. The constraints are the minimum space requirement per day. Constraint 1: the required space in month 1 is 30,000. The space can be satisfied by renting a month in each month. Hence: X1 + X21 + x31 + X41 + X512 30,000. Constraint 2: the required space in month 2 is 20,000. The space can be satisfied by renting in month 1 for 2 months, 3 months, 4 months, and 5 months. Also this space can be satisfied by renting in month 2 for 1 month, 2 month, 3 months, and 4 months. Hence the constraint is: X12 + x3 + x2 + x3 + x21 + xy2 + x3 + x24 > 20,000. The same logic applies to the remaining monthly space demands. MIN C = 65.(X11 + x21 + X31 + x41 + X5i) +100 (X12 + x 2 + x32 +X42) +135. (X13 + X23 + X33) +160 (X14 + X24) +190 (X15) S.T. X11 + x12 + x13 + X14 + X15 > 30,000 X12 + X13+ X14 + X15 + X21 + X22 + X23 + X24 20,000 X13 + X14 + X1 + X22 + X23+ X24 + X31 + X32 + x33 40,000 X14 + X1 + X23 + X24 + X32 + x33 + X41 + x42 2 10,000 X15 + X24 + X33 + X42 + Xs > 50,000 xij 2 0; i = 1,...,5; j = 1,...6-i b) Using LINGO to solve the model, we get C*=$765,000 The optimal solution: X15 = 30,000: X31 = 10,000: X51 = 20,000. This means in month 2 there will b extra space of 10,000 sq.ft., and in month 4 there will be extra space of 20,000sq.ft. b) OPTIMAL SOLUTION Objective Function Value = 7650000.000 Variable Value Reduced Costs x11 x12 x13 x14 x15 x21 x22 x23 0.000 0.000 0.000 0.000 30000.000 0.000 0.000 0.000 5.000 40.000 10.000 35.000 0.000 65.000 35.000 70.000 x24 X31 x32 x33 X41 x42 x51 0.000 10000.000 0.000 0.000 0.000 0.000 20000.000 30.000 0.000 35.000 5.000 65.000 35.000 0.000 Let xi; = the amount of space leased in month i for a period of j months, for i=1,2,3,4,5 and j=1,...,6-i. The objective is to minimize total cost of leasing while satisfying all space weekly requirements. The constraints are the minimum space requirement per day. Constraint 1: the required space in month 1 is 30,000. The space can be satisfied by renting a month in each month. Hence: X1 + X21 + x31 + X41 + X512 30,000. Constraint 2: the required space in month 2 is 20,000. The space can be satisfied by renting in month 1 for 2 months, 3 months, 4 months, and 5 months. Also this space can be satisfied by renting in month 2 for 1 month, 2 month, 3 months, and 4 months. Hence the constraint is: X12 + x3 + x2 + x3 + x21 + xy2 + x3 + x24 > 20,000. The same logic applies to the remaining monthly space demands. MIN C = 65.(X11 + x21 + X31 + x41 + X5i) +100 (X12 + x 2 + x32 +X42) +135. (X13 + X23 + X33) +160 (X14 + X24) +190 (X15) S.T. X11 + x12 + x13 + X14 + X15 > 30,000 X12 + X13+ X14 + X15 + X21 + X22 + X23 + X24 20,000 X13 + X14 + X1 + X22 + X23+ X24 + X31 + X32 + x33 40,000 X14 + X1 + X23 + X24 + X32 + x33 + X41 + x42 2 10,000 X15 + X24 + X33 + X42 + Xs > 50,000 xij 2 0; i = 1,...,5; j = 1,...6-i b) Using LINGO to solve the model, we get C*=$765,000 The optimal solution: X15 = 30,000: X31 = 10,000: X51 = 20,000. This means in month 2 there will b extra space of 10,000 sq.ft., and in month 4 there will be extra space of 20,000sq.ft. b) OPTIMAL SOLUTION Objective Function Value = 7650000.000 Variable Value Reduced Costs x11 x12 x13 x14 x15 x21 x22 x23 0.000 0.000 0.000 0.000 30000.000 0.000 0.000 0.000 5.000 40.000 10.000 35.000 0.000 65.000 35.000 70.000 x24 X31 x32 x33 X41 x42 x51 0.000 10000.000 0.000 0.000 0.000 0.000 20000.000 30.000 0.000 35.000 5.000 65.000 35.000 0.000