Question

Zark is a new Spanish clothing company that is trying to implement Zara's supply chain management. Currently, Zark has two production plants. Coimbatore Plant (in South India) can produce up to 1,000 clothing items per day, whereas Surakarta plant (in Java, Indonesia) can produce as many as 2,000 clothing items per day. All items must be distributed to one of the warehouses the company owns in Spain (Cartagena and Gerona) for quality control and packaging, before being shipped to their retail stores in East US. The daily demand of Boston and New York stores are 1,800 and 1,200 clothing items, respectively. Transportation costs per item between the different points (factories, warehouses and stores) are shown in the following table.

	Cartagena Warehouse	Gerona Warehouse	Boston Store	New York Store
Coimbatore Plant	1	4	-	-
Surakarta Plant	3	2	-	-
Cartagena Warehouse	-	1	6	-
Gerona Warehouse	3	-	5	8
Boston Store	-	-	-	1

a) Represent graphically the previous problem.

b) Establish a linear problem, stating all the elements and labeling all constraints, that can solve the problem.

c) Use excel to solve the problem in part b and interpret the solution.

Note: Assume shipping cannot occur between points where a transportation cost does not appear in the table.

Answer 1

1. Represent graphically the previous problem.

Supply 2500 Supply 1000 Coimbatore Surakarta Demand 1800 1200 Demand New York Boston

2. Establish a linear problem, stating all the elements and labeling all constraints that can solve the problem.

Objective: MINIMIZE

1X1 + 4X2 +0X3 +0X4 +3X5 +2X6 +0X7 +0X8 + 0X9 +1X10 + 6X11 +0X12 +            3X13 +0X14 +5X15 +8X16 +0X17 +0X18 +0X19 +1X20

Constraints:

• 1X1 + 4X2 = 1000
• 3X5 + 2X6 = 2000
• 1X10 + 6X11 = 0
• 3X13 + 5X15 + 8X16 = 0
• 1X20 = 1800

3. Use excel to solve the problem in part b and interpret the solution.

AXIMIZE: Z =-1X1-4X2+0X3 19-1X20+0X21 +0X22 +0 X23 subject to 1000 ject to = 2000 |Xi+4X2+1X25 = 1000 Xi +1Xu) + 1X21 = 1800 = 1800

I don’t know solve using Excel so I used PHP solver:

Solution:

Tableau 1 0.0 | 0.0 0.0 0.01 0.0 | 0.0 0.0 0.01 0.01 0.0 | 0.0 | 0.01 0.0 0.0 | 0.0 | 0.0 | 0.01 0.0 | 0.01 0.0 |-10/-1.0|-1.0|-1.0|-1.0 Base CbPo СЬ P1 P2P3 P4 P5P6P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 1.0 1000.0 1.0 4.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 2000.0 0.0 0.0 0.0 0.03.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.06.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0 0.05.0-8.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 1800.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.01.0 0.0 0.0 0.0 0.0 -4800.0 | ー1.01-4.0 | 0.01 0.01-3.0 |-2.0 | 0.0 0.01 0.0 | 1.0 | 6.0 | 0.0 | 3.0 | 0.0 | 5.0 | 8.0 | 0.0 | 0.0 | 0.0 |-1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 Tableau 2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00.00.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 -1.0 -1.0-1.0 -1.0 P1 P2 P3 P4 PP6P7 P P P10 P11 P12 P13 P14 P15 P16P17 P18 P19 P20 P21 P22 P23 P24 P25 сь 0.0 250.0 0.3 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 1.01 2000.0 | 0.0 | 0.0 | 0.01 0.0 | 3.0 | 2.0 | 0.01 0.01 0.01 0.0 | 0.0 | 0.01 0.0 | 0.0| 0.0 | 0.0 0.0 0.01 0.01 0.0 | DOI 0 0.0 L 0.0 1.0 0.0 0.00.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 6.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0-3.0 0.05.08.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 1800.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0 3800.0 0.0 0.0 0.0 0.0-3.0-2.0 0.0 0.0 0.0 1.0 6.0 0.0 3.0 0.05.0 8.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 1.0

Tableau 3 0.0 | 0.0 0.0 0.01 0.0 | 0.0 0.0 0.01 0.0 0.0 0.0 0.0 0.0 0.0 0.0 | 0.0 | 0.01 0.0 0.0 0.0 1-1.01-1.01-1.01-1.01-1.0 сь Pa | P2 | P3 | P4 | P5 | P6 | P7 | P8 | P9 | P10 | P11 | P12 | P13 | P14 | P15 | P16 | P17 | P18 | P19 | P20 İ P21 | P22 | P23 | P24 | P25 0.0 250.0 0.3 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00.0 0.0 0.3 0.0 666.7 0.0 0.0 0.0 0.0 1.0 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0-6.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.03.0 0.0 -5.08.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 1800.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0 1800.0 0.0 0.0 0.00.0 0.0 0.0 0.0 0.0 .01.0 6.0 0.0 3.0 0.0 5.0 8.0 0.0 0.0 0.01.0 0.0 0.0 0.01.0 1.0 Tableau 4 0.0 | 0.0 0.0 0.0 0.0 | 0.0 | 0.01 0.01 0.01 0.0 | 0.0 10.01 0.0 10.01 0.0 | 0.0 10.0 0.000 0.0 |-1.01-1-이-1.01-1.0-1.0 СЬ P1 P2 P3 P4P5 P6P7 P P P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 0.0 250.0 0.3 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.0 666.7 0.0 0.0 0.0 0.01.0 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.00.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0-6.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0-3.0 0.0-5.08.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1800.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 6.0 0.03.0 0.0 5.0 8.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 1.0

Tableau 1			-1.0	-4.0	0.0	0.0	-3.0	-2.0	0.0	0.0	0.0	-1.0	-6.0	0.0	-3.0	0.0	-5.0	-8.0	0.0	0.0	0.0	-1.0
Base	Cb	P0	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12	P13	P14	P15	P16	P17	P18	P19	P20
P2	-4.0	250.0	0.3	1.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
P5	-3.0	666.7	0.0	0.0	0.0	0.0	1.0	0.7	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
P23	-1.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	-1.0	-6.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
P22	-1.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	-3.0	0.0	-5.0	-8.0	0.0	0.0	0.0	0.0
P20	-1.0	1800.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	1.0
Z		-4800.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	2.0	12.0	0.0	6.0	0.0	10.0	16.0	0.0	0.0	0.0	0.0

There are infinitely many values of X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20 for the optimal value Z = 4800, which are contained in the region of the space 0 X1 + 4 X2 + 0 X3 + 0 X4 + 3 X5 + 2 X6 + 0 X7 + 0 X8 + 0 X9 + 1 X10 + 6 X11 + 0 X12 + 3 X13 + 0 X14 + 5 X15 + 8 X16 + 0 X17 + 0 X18 + 0 X19 + 1 X20 = 4800 that satisfy the constraints of this problem. One of them is:
X1 = 0
X2 = 250
X3 = 0
X4 = 0
X5 = 666.66666666667
X6 = 0
X7 = 0
X8 = 0
X9 = 0
X10 = 0
X11 = 0
X12 = 0
X13 = 0
X14 = 0
X15 = 0
X16 = 0
X17 = 0
X18 = 0
X19 = 0
X20 = 1800