Question

Sweet Taste Fruit Company contracts with growers in Ohio, Pennsylvania, and New York to purchase grapes. The grapes are processed into juice at the farms and stored in refrigerated vats. Then the juice is shipped to two plants, where it is processed into bottled grape juice and frozen concentrate. The juice and concentrate are then transported to three distribution centers. The transportation costs per ton from the farms to the plants and from the plants to the distribution centers, and the supply at the farms and demand at the distribution centers are summarized in the following tables:

                                                          Plant

      Farm                              4. Indiana        5. Georgia            5. Georgia                   16                      105 Supply (1,000 tons)

ters are summarized in the following tabSupply (1,000 tons)

1. Ohio                               $16                 $21                      72
2. Pensylvania                      18                  16                    105
3. New York                         22                  25                      83                

                                                              Distribution Center

       Plant                           6. Virginia          7. Kentucky           8. Louisiana

4. Indiana                      $23                    $15                           $29
5. Georgia                       20                      17                             24

      Demand (1000 tons)         90                      80                           120         

Formulate this problem as a linear programming model.

Answer 1

Let us assume

Farms			DC
Ohio	Pennsylvania	New York	Total	Plant	Virginia	Kentucky	Louisiana	Total
Xoi	Xpi	Xni	Xoi + Xpi + Xni	Indiana	Xiv	Xik	Xil	Xiv + Xik + Xil
Xog	Xpg	Xng	Xog + Xpg + Xng	Georgia	Xgv	Xgk	Xgl	Xgv + Xgk +Xgl
Xoi+Xog	Xpi+Xpg	Xni+Xng	Xoi + Xpi + Xni + Xog + Xpg + Xng		Xiv+Xgv	Xik+Xgk	Xil+Xgl	Xiv + Xik + Xil + Xiv + Xik + Xil
<=	<=	<=			<=	<=	<=
				Demand	90,000	80,000	120,000
72,000	105,000	83,000		Supply

Here Xoi = quantity moved from Farm ohio to Plant Indiana, Xiv = quantitiy moved from Plant Indiana to DC Virginia and like wise

Therefore, Constraints Will be

1. Quantity moved from a Farm  < Total Supply from the Farm

Xoi+Xog < = 72,000 (Ohio)

Xpi+Xpg < = 105,000 (Pennsylvania)

Xni+Xng < = 83,000 (New York)

2. Total Quantity received by a DC > = Total Demand for that DC ( Here we can use only equal to also)

Xiv+Xgv > = 90,000 (Virginia)

Xik+Xgk > = 80,000 (Kentucky)

Xil+Xgl > = 120,000 (Louisiana)

3. Total Quantity Received by a Plant from a Farm = Total quantity sent by the Plan to a DC

Xoi + Xpi + Xni = Xiv + Xik + Xil (Indiana)

Xog + Xpg + Xng = Xgv + Xgk +Xgl (Georgia)

4. Positivity Constraint

Xoi,Xpi,Xni,Xog,Xpg,Xng,Xiv,Xik,Xil,Xgv,Xgk,Xgl >=0

For objective Function, the transportation costs will be given by, Unit Cost * Corresponding quantity moved

e.g. Transportation cost from Farm Ohio to Plant Indiana will be given by Unit Cost of transportation from Ohio to Indiana (which is16 as given in the problem) * Quantity moved (which is Xoi)

=16*Xoi and like wise

Thus Objective Function will be

Minimization of Total Cost 16*Xoi +18* Xpi + 22*Xni + 21*Xog + 16*Xpg + 25*Xng + 23*Xiv + 15*Xik + 29*Xil + 20*Xgv + 17*Xgk +24*Xgl

	Cost
	Farms			DC
	Ohio	Pennsylvania	New York	Total	Plant	Virginia	Kentucky	Louisiana	Total
	16*Xoi	18*Xpi	22*Xni	16*Xoi + 18*Xpi + 22*Xni	Indiana	23*Xiv	15*Xik	29*Xil	23*Xiv + 15*Xik + 29*Xil
	21*Xog	16*Xpg	25*Xng	21*Xog + 16*Xpg + 25*Xng	Georgia	20*Xgv	17*Xgk	24*Xgl	20*Xgv + 17*Xgk +24*Xgl
				16*Xoi +18* Xpi + 22*Xni + 21*Xog + 16*Xpg + 25*Xng	Total				23*Xiv + 15*Xik + 29*Xil + 20*Xgv + 17*Xgk +24*Xgl
Overall Transportation Cost	16*Xoi +18* Xpi + 22*Xni + 21*Xog + 16*Xpg + 25*Xng + 23*Xiv + 15*Xik + 29*Xil + 20*Xgv + 17*Xgk +24*Xgl

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