A&C Distributors produces several outdoor products distributed to discount store

Question

A&C Distributors produces several outdoor products distributed to discount stores, garden centers, and hardware stores. Currently, A&C is developing a three month shipment plan for two of their products (the Water Wave lawn sprinkler and the Spring Shower lawn sprinkler) to a warehouse belonging to one of the major customers. The table below provides the relevant information for shipping over the three month period. In the table, 8,000 represents the maximum number of lawn sprinklers that can be shipped in March (water wave and spring shower combined). The units shipped during a given month, but not sold during that month may be put in inventory to be sold during the next month. The inventory costs are given in the last column. Formulate the LP model to meet the minimum requirements and minimize shipping and inventory costs.

Month

Shipping

Capacity

Product

Minimum

Requirement

Unit Cost

to Ship

Per Unit

Inventory Cost

March

8000

Water Wave

3000

.30

.06

Spring Shower

1800

.25

.05

April

7000

Water Wave

4000

.40

.09

Spring Shower

4000

.30

.06

May

6000

Water Wave

5000

.50

Spring Shower

2000

.35

Objective function: _________________________________________________

ST

2a. How many total lawn sprinklers will be produced in May? ___________

2b. How many Water wave sprinklers will be put in inventory during the three months combined?

               ___________

2c. If only 7,000 sprinklers could be produced in March, what would be the total shipping and inventory costs?

                          ____________

2d. At what value would more Spring Shower sprinklers would be shipped during the month of April?

          ___________

Month

Shipping

Capacity

Product

Minimum

Requirement

Unit Cost

to Ship

Per Unit

Inventory Cost

March

8000

Water Wave

3000

.30

.06

Spring Shower

1800

.25

.05

April

7000

Water Wave

4000

.40

.09

Spring Shower

4000

.30

.06

May

6000

Water Wave

5000

.50

Spring Shower

2000

.35

Explanation / Answer

let xij be the number of sprinkelr i shipped in month j where where i=1,2 for ww and ss respe and j=1,2,3 for march, april ,may.

also assume yij be excess sprinkler in inventories.

cosst function for shipping= 0.3x11+0.25x12+ 0.4x12+0.3x22+ o.5x13+ 0.35x23............(1)

cost function for inventories= 0.06y11+0.05y21+0.09y12+0.06y22...............................(2)

assuming since data is not given for inventories in may, cost=o in may.

objective function would be (1)+(2)

constraints-

x11+x21<=8000

x12+x22<=7000

x13+x23<=6000

from shipped and inventory balancing

x11-y11=3000

x21-y21=1800

x12+y11-y12=4000

x22+y21-y22=4000

x13+y12=5000

x23+y22=2000

putting value of yij, x13 and x23 in above objective function and constriants to simplify we get,

objective function- 0.95x11+0.99 x12+0.71 x21+0.71x22-3578

subjected to- x11+x21<=8000

x12+ x22<=7000

0.59x11+0.59x12+0.41x21+ 0.41x22<=8738

solve these equation, furhter answer will be obtained easily.

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