Question

The Sentry Lock Corporation manufacturers a popular commercial security lock at plants in Macon, Louisville, Detroit and Pheonix. The per unit cost of production at each plant is $35.50, $37.50, $39.00, and $36.25, respectively, and the annual production capacity at each plant is 18,000, 15,000, 25,000, and 20,000, respectively. Sentry’s locks are sold to retailers through wholesale distributors in seven cities across the United States. The unit cost of shipping from each plant to each distributor is summarized in the following table along with the forecasted demand from each distributor for the coming year.

	Unit Shipping Cost to Distributor in
	Tacoma	San Diego	Dallas	Denver	St. Louis	Tampa	Baltimore
Macon	$2.50	$2.75	$1.75	$2.00	$2.10	$1.80	$1.65
Louisville	$1.85	$1.90	$1.50	$1.60	$1.00	$1.90	$1.85
Detroit	$2.30	$2.25	$1.85	$1.25	$1.50	$2.25	$2.00
Phoenix	$1.90	$0.90	$1.60	$1.75	$2.00	$2.50	$2.65
Demand	8,500	14,500	13,500	12,600	18,000	15,000	9,000

Sentry wants to determine the least expensive way of manufacturing and shipping locks from their plants to the distributor. Because the total demand from distributors exceeds the total production capacity of all plants, Sentry realized it will not be able to satisfy all the demand for its product, but wants to make sure each distributor will have the opportunity to fill at least 80% of the orders they receive. Let us suppose that the company operates optimally, what is the total cost incurred?

Answer 1

For cost min: (Note production need not be full)

The boxes in green, are decision variables
The boxes in blue are constrains
The boxes in orange is the objective function

The solution.

	Unit Shipping Cost to Distributor in
	Tacoma	San Diego	Dallas	Denver	St. Louis	Tampa	Baltimore	Prod cost
Macon	2.50	2.75	1.75	2.00	2.10	1.80	1.65	35.50
Louisville	1.85	1.90	1.50	1.60	1.00	1.90	1.85	37.50
Detroit	2.30	2.25	1.85	1.25	1.50	2.25	2.00	39.00
Phoenix	1.90	0.90	1.60	1.75	2.00	2.50	2.65	36.25
	Decision variable (Units to ship)
	Tacoma	San Diego	Dallas	Denver	St. Louis	Tampa	Baltimore	Supply constraint
Macon	0.00	0.00	0.00	0.00	0.00	12000.00	6000.00	18000.00	<=	18000
Louisville	0.00	0.00	600.00	0.00	14400.00	0.00	0.00	15000.00	<=	15000
Detroit	0.00	0.00	8600.00	10080.00	0.00	0.00	1200.00	19880.00	<=	25000
Phoenix	6800.00	11600.00	1600.00	0.00	0.00	0.00	0.00	20000.00	<=	20000
80% of demand constraint	6800.00	11600.00	10800.00	10080.00	14400.00	12000.00	7200.00
	>=	>=	>=	>=	>=	>=	>=
	6800	11600	10800	10080	14400	12000	7200
Demand	8500.00	14500.00	13500.00	12600.00	18000.00	15000.00	9000.00
Total cost	2805450
Obj func	Minimize

But if Production needs to be full:

Supply constraint
18000.00	=	18000
15000.00	=	15000
25000.00	=	25000
20000.00	=	20000

Is changed:

so the new solution is :

	Unit Shipping Cost to Distributor in
	Tacoma	San Diego	Dallas	Denver	St. Louis	Tampa	Baltimore	Prod cost
Macon	2.50	2.75	1.75	2.00	2.10	1.80	1.65	35.50
Louisville	1.85	1.90	1.50	1.60	1.00	1.90	1.85	37.50
Detroit	2.30	2.25	1.85	1.25	1.50	2.25	2.00	39.00
Phoenix	1.90	0.90	1.60	1.75	2.00	2.50	2.65	36.25
	Decision variable (Units to ship)
	Tacoma	San Diego	Dallas	Denver	St. Louis	Tampa	Baltimore	Supply constraint
Macon	0.00	0.00	0.00	0.00	0.00	12000.00	6000.00	18000.00	=	18000
Louisville	600.00	0.00	0.00	0.00	14400.00	0.00	0.00	15000.00	=	15000
Detroit	0.00	0.00	10801.00	12999.00	0.00	0.00	1200.00	25000.00	=	25000
Phoenix	6200.00	13800.00	0.00	0.00	0.00	0.00	0.00	20000.00	=	20000
80% of demand constraint	6800.00	13800.00	10801.00	12999.00	14400.00	12000.00	7200.00
	>=	>=	>=	>=	>=	>=	>=
	6800	11600	10800	10080	14400	12000	7200
Demand	8500.00	14500.00	13500.00	12600.00	18000.00	15000.00	9000.00
Total cost	3011341
Obj func	Minimize