Consider the operations of the Universal Sporting Goods Company that makes skis in three plants throughout Europe. The plants supply four company-owned warehouses that distribute the skis directly to ski shops. Depending on which mode is cheaper, the product is air-freighted or trucked from the plants to warehouses. The monthly capacities of the plants in terms of the number of pairs of skis that can be made are

Plant	Capacity
1. Glasgow	250250
2. Prague	300300
3. Dresden	200200
Total	750750

and warehouse demands for the next month are

Warehouse	Demand
A. Berne	145145
B. Lienz	205205
C. Helsinki	230230
D. Goteborg	120120
Total	700700

The company wants to know how many pairs of skis per month each plant should manufacture to satisfy the demand at the minimum shipping cost. Also, the company needs to know how many pairs of skis should be shipped from each plant to each warehouse.

In the following table xijxij represents the number of pair of skis shipped from plant ii to warehouse jj, where

• i=i= 1 == Glasgow, i=i= 2 == Prague, i=i= 3 == Dresden
• j=j= A == Berne, j=j= B == Lienz, j=j= C == Helsinki, j=j= D == Goteborg

Also, the table provides the point-to-point costs of shipping a pair of skis.

	TO WAREHOUSE
FROM PLANT	A Berne	B Lienz	C Helsinki	D Goteborg
1. Glasgow	x1Ax1A $39$39	x1Bx1B $32$32	x1Cx1C $26$26	x1Dx1D $29$29
2. Prague	x2Ax2A $33$33	x2Bx2B $24$24	x2Cx2C $35$35	x2Dx2D $34$34
3. Dresden	x3Ax3A $28$28	x3Bx3B $18$18	x3Cx3C $26$26	x3Dx3D $32$32

Please help the company to set up a transportation LP model.

a) The objective function for this problem is:

• minimize Z=39x1A+33x1B+28x1C+32x1D+24x2A+18x2B+26x2C+35x2D+26x3A+29x3B+34x3C+32x3DZ=39x1A+33x1B+28x1C+32x1D+24x2A+18x2B+26x2C+35x2D+26x3A+29x3B+34x3C+32x3D
• minimize Z=39x1A+32x1B+26x1C+29x1D+33x2A+24x2B+35x2C+34x2D+28x3A+18x3B+26x3C+32x3DZ=39x1A+32x1B+26x1C+29x1D+33x2A+24x2B+35x2C+34x2D+28x3A+18x3B+26x3C+32x3D
• maximize Z=39x1A+33x1B+28x1C+32x1D+24x2A+18x2B+26x2C+35x2D+26x3A+29x3B+34x3C+32x3DZ=39x1A+33x1B+28x1C+32x1D+24x2A+18x2B+26x2C+35x2D+26x3A+29x3B+34x3C+32x3D
• maximize Z=39x1A+32x1B+26x1C+29x1D+33x2A+24x2B+35x2C+34x2D+28x3A+18x3B+26x3C+32x3DZ=39x1A+32x1B+26x1C+29x1D+33x2A+24x2B+35x2C+34x2D+28x3A+18x3B+26x3C+32x3D
• None of the above.

(b) Please choose the correct constraint:

• x1A+x2A+x3A750x1A+x2A+x3A750
• x1A+x1B+x1C+x1D750x1A+x1B+x1C+x1D750
• x1A+x1B+x1C+x1D250x1A+x1B+x1C+x1D250
• x1A+x1B+x1C+x1D250x1A+x1B+x1C+x1D250
• x1A+x1B+x1C+x1D750x1A+x1B+x1C+x1D750
• x1A+x2A+x3A250x1A+x2A+x3A250
• None of the above.

(c) Please choose the correct constraint:

• x2A+x2B+x2C+x2D300x2A+x2B+x2C+x2D300
• x2A+x2B+x2C+x2D750x2A+x2B+x2C+x2D750
• x2A+x2B+x2C+x2D300x2A+x2B+x2C+x2D300
• x1B+x2B+x3B300x1B+x2B+x3B300
• x2A+x2B+x2C+x2D750x2A+x2B+x2C+x2D750
• x1B+x2B+x3B750x1B+x2B+x3B750
• None of the above.

(d) Please choose the correct constraint:

• x1C+x2C+x3C750x1C+x2C+x3C750
• x3A+x3B+x3C+x3D750x3A+x3B+x3C+x3D750
• x3A+x3B+x3C+x3D200x3A+x3B+x3C+x3D200
• x1C+x2C+x3C200x1C+x2C+x3C200
• x3A+x3B+x3C+x3D200x3A+x3B+x3C+x3D200
• x3A+x3B+x3C+x3D750x3A+x3B+x3C+x3D750
• None of the above.

(e) Please choose the correct constraint:

• x1A+x2A+x3A>145x1A+x2A+x3A>145
• x1A+x2A+x3A>700x1A+x2A+x3A>700
• x1A+x2A+x3A<700x1A+x2A+x3A<700
• x1A+x2A+x3A=145x1A+x2A+x3A=145
• None of the above.

(f) Please choose the correct constraint in the standard form:

• x1B+x2B+x3B>205x1B+x2B+x3B>205
• x1B+x2B+x3B>700x1B+x2B+x3B>700
• x1B+x2B+x3B<700x1B+x2B+x3B<700
• x1B+x2B+x3B=205x1B+x2B+x3B=205
• None of the above.

(g) Please choose the correct constraint in the standard form:

• x1C+x2C+x3C=230x1C+x2C+x3C=230
• x1C+x2C+x3C>230x1C+x2C+x3C>230
• x1C+x2C+x3C>700x1C+x2C+x3C>700
• x1C+x2C+x3C<700x1C+x2C+x3C<700
• None of the above.

(h) Please choose the correct constraint in the standard form:

• x1D+x2D+x3D>120x1D+x2D+x3D>120
• x1D+x2D+x3D>700x1D+x2D+x3D>700
• x1D+x2D+x3D<700x1D+x2D+x3D<700
• x1D+x2D+x3D=120x1D+x2D+x3D=120
• None of the above.

(i) Non-negativity for this LP problem means that:

• all xij0xij0
• product of all xij0xij0
• x1D+x2D+x3D0x1D+x2D+x3D0
• x1B+x2B+x3B0x1B+x2B+x3B0
• x1A+x2A+x3A0x1A+x2A+x3A0
• x1C+x2C+x3C0x1C+x2C+x3C0
• xij0xij0
• at least one xij0xij0
• None of the above.