The following question is based on the Buckeye Manufacturing Case that we covered in class. Buckeye produces three types of engine heads, P-Heads, H-heads, and R-heads. Each week, the production line is set up to produce one of the three products (the line can only produce the type of engine heads that it is set up to produce during a given week – production of the other two engine head types is not possible). Each weekend, a decision is made as to whether to keep the current setup for the production line or to change over the line. If the line is “changed-over” from one week to the next, then a $1000 cost is incurred. At present, the line is set up to make H-Heads. The requirements for the next four weeks, beginning (current) inventories, inventory holding costs ($/unit/week based on ending inventory), and maximum production per week are shown below:

	Week 1	Week 2	Week 3	Week 4	Week 5	Beginning Inventory	Holding Cost	Maximum Production
P-Heads	45	24	45	36	40	115 units	$15/week	90
H-Heads	33	30	48	58	30	115units	$19/week	75
R-heads	25	30	20	50	40	80 units	$14/week	80

An integer linear programming formulation is used to determine a production / changeover schedule for a five-week period. The objective is to minimize total costs: inventory + changeover, subject to constraints that meet demand and balance inventory, limit production, and pick-up changeovers. The decision variables are shown below:

x_j = the number of P-heads produced in week j; j = 1,2,3,4,5;

y_j = the number of H-heads produced in week j; j = 1,2,3,4,5;

z_j = the number of R-heads produced in week j; j = 1,2,3,4,5;

p_j = the ending inventory of P-heads in week j; j = 1,2,3,4,5;

h_j = the ending inventory of H-heads in week j; j = 1,2,3,4,5;

r_j = the ending inventory of R-heads in week j; j = 1,2,3,4,5;

s_j = 1 if the production line is setup to produce P-heads in week j, 0 otherwise j = 1,2,3,4,5;

t_j = 1 if the production line is setup to produce H-heads in week j, 0 otherwise j = 1,2,3,4,5;

u_j = 1 if the production line is setup to produce R-heads in week j, 0 otherwise j = 1,2,3,4,5;

c_j = 1 if the production line was “changed-over” from week j – 1 to week j, 0 if the setup for the line was the same for weeks j-1 and j; j = 1,2,3,4;

Objective function:

1. Write the constraints (for each week) for R-heads that will guarantee that demand for the R-heads is satisfied and also accurately balance the inventory for this item.

2. Write the constraints (for each week) that will guarantee that only one type of engine head is produced and also limit accordingly the production of each type of engine block.

3. Write constraints (for each week) that will collect changeovers if they occur