Problem $10-03$

Tri$-$County Utilities, Inc., supplies natural gas to customers in a three$-$county area. The company purchases natural gas from two companies: Southern Gas and Northwest Gas. Demand forecasts for the coming winter season are as follows: Hamilton County, $400$ units; Butler County, $200$ units; and Clermont County, $300$ units. Contracts to provide the following quantities have been written: Southern Gas, $500$ units; and Northwest Gas, $400$ units. Distribution costs for the counties vary, depending upon the location of the suppliers. The distribution costs per unit $($in thousands of dollars$)$ are as follows:

Southern Gas $-10($hamilton$),20($butler$),15($clermont$)$

northwest gas $-12($hamilton$,15($butler$),18($clermont$)$

Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs.

Let xij $=$ amount shipped from supply node i to demand node j$.$

Min fill in the blank $2$

x$11+$ fill in the blank $3$

x$12+$ fill in the blank $4$

x$13+$ fill in the blank $5$

x$21+$ fill in the blank $6$

x$22+$ fill in the blank $7$

x$23$

s$.$t$.$

fill in the blank $8$

x$11+$ fill in the blank $9$

x$12+$ fill in the blank $10$

x$13<=$ fill in the blank $11$

fill in the blank $12$

x$21+$ fill in the blank $13$

x$22+$ fill in the blank $14$

x$23<=$ fill in the blank $15$

fill in the blank $16$

x$11+$ fill in the blank $17$

x$21=$ fill in the blank $18$

fill in the blank $19$

x$12+$ fill in the blank $20$

x$22=$ fill in the blank $21$

fill in the blank $22$

x$13+$ fill in the blank $23$

x$23=$ fill in the blank $24$

xij $>=0$ for all i$,$ j

Describe the distribution plan and show the total distribution cost.

Amount Cost

Southern $-$ Hamilton fill in the blank $25$

$fill in the blank $26$

Southern $-$ Butler fill in the blank $27$

fill in the blank $28$

Southern $-$ Clermont fill in the blank $29$

fill in the blank $30$

Northwest $-$ Hamilton fill in the blank $31$

fill in the blank $32$

Northwest $-$ Butler fill in the blank $33$

fill in the blank $34$

Northwest $-$ Clermont fill in the blank $35$

fill in the blank $36$

Total Cost $fill in the blank $37$

Recent residential and industrial growth in Butler County has the potential for increasing demand by as much as $100$ units. Which supplier should Tri$-$County contract with to supply the additional capacity?

Amount Cost

Southern $-$ Hamilton fill in the blank $38$

$fill in the blank $39$

Southern $-$ Butler fill in the blank $40$

fill in the blank $41$

Southern $-$ Clermont fill in the blank $42$

fill in the blank $43$

Northwest $-$ Hamilton fill in the blank $44$

fill in the blank $45$

Northwest $-$ Butler fill in the blank $46$

fill in the blank $47$

Northwest $-$ Clermont fill in the blank $48$

fill in the blank $49$

Total Cost $fill in the blank $50$

From the new solution we see that Tri$-$County should contract with

Gas for the additional $100$ units.